Quadrilateral ABCD is a parallelogram if both pairs of opposite sides are congruent. Show that quadrilateral ABCD is a parallelogram by finding the lengths of the opposite side pairs. What is the length of BC?
In parallelogram ABCD, opposite sides AB and DC are both 8 units long, and AD and BC are both 5 units long.
To show that quadrilateral ABCD is a parallelogram, we need to demonstrate that both pairs of opposite sides are congruent. We can do this by finding the lengths of the opposite side pairs and showing that they are equal.
Let's denote the points as follows:
- A, B, C, D are the vertices of the quadrilateral ABCD.
- AB and DC are opposite sides, and AD and BC are opposite sides.
Given that AB = 8 units and AD = 5 units, we need to find the lengths of DC and BC.
Since ABCD is a parallelogram, opposite sides are congruent. Therefore, DC = AB = 8 units.
To find BC, we know that BC = AD = 5 units.
Thus, we have:
- AB = DC = 8 units
- AD = BC = 5 units
Since both pairs of opposite sides are congruent, quadrilateral ABCD is a parallelogram.
The length of BC is 5 units.
In the given case, The length of BC is 8 units.
To show that quadrilateral ABCD is a parallelogram, we need to demonstrate that both pairs of opposite sides are congruent.
Let's denote the lengths of the sides as follows: AB, BC, CD, and DA.
Given that ABCD is a parallelogram, we know that opposite sides are equal in length.
Therefore, we can equate the lengths of AB to CD and BC to DA.
Let's assume that the length of side AB (and thus CD, since they are opposite and equal) is given as 8 units.
We are asked to find the length of side BC (which will be equal to the length of side DA).
Since we do not have any additional information such as angles or diagonals, we cannot calculate the length of BC directly.
However, if we are given that ABCD is a parallelogram, then by definition, the lengths of opposite sides are equal.
Therefore, without loss of generality, we can state that the length of BC is also 8 units, which is equal to the length of AB.
Thus, we have shown that both pairs of opposite sides are congruent:
AB = CD = 8 units
BC = DA
Since AB = 8 units and AB = CD, it follows that CD = 8 units.
Similarly, since AB = CD and AB = 8 units, by the properties of a parallelogram, BC must also be equal to 8 units.
Therefore, the length of BC is 8 units, confirming that quadrilateral ABCD is indeed a parallelogram.
In the triangle below, what ratio is sin θ?
Answer:
[tex]\displaystyle \frac{12}{13} = sin\:θ\:[0,3743340836π ≈ θ][/tex]
Step-by-step explanation:
Remember this?!
Extended Information on Trigonometric Ratios
[tex]\displaystyle \frac{OPPOSITE}{HYPOTENUSE} = sin\:θ \\ \frac{ADJACENT}{HYPOTENUSE} = cos\:θ \\ \frac{OPPOSITE}{ADJACENT} = tan\:θ \\ \frac{HYPOTENUSE}{ADJACENT} = sec\:θ \\ \frac{HYPOTENUSE}{OPPOSITE} = csc\:θ \\ \frac{ADJACENT}{OPPOSITE} = cot\:θ[/tex]
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The derivative of the function B is given by B′(t)=8e0.2cost, and B(2.2)=4.5. If the linear approximation to B(t) at t=2.2 is used to estimate B(t), at what value of t does the linear approximation estimate that B(t)=9 ?
Answer:
t=2.83
Step-by-step explanation:
Linear Approximation Of Functions
The equation of a line is given by
[tex]y=y_o+m(x-x_o)[/tex]
Where m is the slope of the line and [tex](x_o,y_o)[/tex] are the coordinates of a point through which the line goes.
Given a function B(t), we can build an approximate line to model the function near one point. The value of m is the derivative of B in a specific point [tex](t_o, B_o)[/tex]. The equation becomes
[tex]B(t)=B_o+B'(t_o)(t-t_o)[/tex]
Let's collect our data.
[tex]B'(t)=8e^{0.2cost},\ B_o=B(2.2)=4.5[/tex]
Let's find the required values to build the approximate function near [tex]t_0=2.2[/tex]. We evaluate the derivative in 2.2
[tex]B'(2.2)=8e^{0.2cos2.2}=7.11[/tex]
The function can be approximated by
[tex]B(t)=4.5+7.11(t-2.2)[/tex]
Once we have B(t), we are required to find the value of t, such that
[tex]B(t)=9[/tex]
Or equivalently:
[tex]4.5+7.11(t-2.2)=9[/tex]
Rearranging
[tex]\displaystyle t-2.2=\frac{9-4.5}{7.11}[/tex]
Solving for t
[tex]\displaystyle t=\frac{9-4.5}{7.11}+2.2[/tex]
[tex]\boxed{t=2.83}[/tex]
We find the equation of the linear approximation at t=2.2 using B(2.2) and B'(2.2). We then solve this equation for t when the linear approximation equals 9. This gives us the time at which the linear approximation estimates B(t) = 9.
Explanation:To solve for the time when the linear approximation gives an output of 9, we need to find the equation of the tangent line (i.e., the linear approximation) at t = 2.2. The linear approximation to a function at a particular point is given by the formula L(t) = f(a) + f'(a)*(t-a), where f is the function, a is the point, and f' denotes the derivative of the function.
Given that B(2.2) = 4.5 and B'(t) = 8e0.2cos(t), we can substitute these into the linear approximation formula to get L(t) = 4.5 + 8e0.2cos(2.2)*(t-2.2).
Next, we solve this equation for t when L(t) = 9: 9 = 4.5 + 8e0.2cos(2.2)*(t-2.2). Solving the equation for t gives the time at which the linear approximation estimates B(t) = 9.
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Two chess players, A and B, are going to play 7 games. Each game has three possible
outcomes: a win for A (which is a loss for B), a draw (tie), and a loss for A (which is
a win for B). A win is worth 1 point, a draw is worth 0.5 points, and a loss is worth 0
points.
(A) How many possible outcomes for the individual games are there, such that overall
player A ends up with 3 wins, 2 draws, and 2 losses?
(B) How many possible outcomes for the individual games are there, such that A ends
up with 4 points and B ends up with 3 points?
(C) Now assume that they are playing a best-of-7 match, where the match will end as
soon as either player has 4 points. For example, if after 6 games the score is 4 to 2 in
favor of A, then A wins the match and they don’t play a 7th game. How many possible
outcomes for the individual games are there, such that the match lasts for 7 games and
A wins by a score of 4 to 3?
Answer:
A) 210
B) 357
C) 267
Step-by-step explanation:
A) Among 7 games, we can first choose 3 wins, and then among remaining 4 games, we can choose 2.
To calculate the possibility, we will use Combination.
[tex]C(7,3)*C(4,2) = 35 *6 = 210[/tex]
B) Player A can get 4 points with the following cases:
4 wins and 3 loses
3 wins, 2 draws and 2 loses
2 wins, 4 draws and 1 lose
1 win and 6 draws
Indeed, these cases matches for Player B too to get 3 points.
So again, we will use Combination to calculate the possibility.
[tex]C(7,4) + C(7,3)*C(4,2) + C(7,2)*C(5,4) + C(7,1) = 357[/tex]
C) Here, we need to find two possibilities after 6 games and add them, while Player A has 3 points and wins the 7th game, and Player A has 3.5 points and draws the 7th game.
[tex][C(6,3)+C(6,2)C(4,2)+C(6,1)C(5,4)+C(6,6)] + [C(6,3)C(3,1)+C(6,2)C(4,3)+C(6,1)] =[20+90+30+1]+[60+60+6]=141+126=267[/tex]
DeAndre is 15 miles into a 51-mile backpacking trip in the wilderness. DeAndre can hike 9 miles per day. How many more days does DeAndre need to finish?
Answer:
Step-by-step explanation:
miles left to travel=51-15=36
more days required=36/9=4 days.
Answer: it will take 4 more days before DeAndre finishes
Step-by-step explanation:
DeAndre is 15 miles into a 51-mile backpacking trip in the wilderness. This means that he has already covered 15 miles and the total number of miles is 51. Number of miles left is 51 - 15 = 36 miles.
DeAndre can hike 9 miles per day. The number of days that it will take him to cover 36 miles would be 36/9 = 4 days
Maggie graphed the image of a 90° counterclockwise rotation about vertex A of △ABC. Coordinates B and C of △ABC are (2, 6) and (4, 3) and coordinates B’ and C’ of it’s image are (–2, 2) and (1, 4). What is the coordinate of vertex A?
Answer:
A(2, 2)
Step-by-step explanation:
I find it useful to graph the given points. The center of rotation is at the place where the perpendicular bisectors of BB' and CC' meet. That point is ...
A = (2, 2).
__
The graph shows you the slope of BB' is 1, so its perpendicular bisector will have a slope of -1. The midpoint of BB' is (2-2, 6+2)/2 = (0, 4). This is the y-intercept of the line, so the perpendicular bisector of BB' has equation ...
y = -x +4
The slope of CC' is -1/3, so its perpendicular will have a slope of -1/(-1/3) = 3. The midpoint of CC' is (4+1, 3+4)/2 = (5/2, 7/2). In point-slope form the equation of the perpendicular bisector of CC' is ...
y -7/2 = 3(x -5/2)
2y -7 = 3(2x -5) . . . . multiply by 2
2(-x+4) -7 = 3(2x -5) . . . substitute for y
-2x +1 = 6x -15 . . . . .eliminate parentheses
16 = 8x . . . . . . . . . . add 2x+15
x = 2 . . . . . . divide by 8
y = -2+4 = 2 . . . . from the equation for y
The intersection of the perpendicular bisectors of BB' and CC' is the center of rotation:
A = (2, 2)
Final answer:
The coordinate of vertex A is (2, 2), found by equating B' to (y, x) and using the given B' coordinates.
Explanation:
To find the coordinates of vertex A, we can use the properties of a rotation. A 90° counterclockwise rotation about a point involves switching the coordinates and negating the y-coordinate. Let's denote the coordinates of A as (x, y). After the rotation, A becomes B', so we have:
B' = (y, x)
From the given coordinates of B' (–2, 2), we can equate:
y = 2
-2 = -x
Solving these equations, we find that x = 2. Therefore, the coordinates of vertex A are (2, 2). So, Maggie's image graph depicts a 90° counterclockwise rotation about vertex A (2, 2) of triangle ABC.
A transformation named T maps triangle XYZ to triangle X'Y'Z'. The transformation is a(n) _____.
The transformation T could be translation, rotation, reflection, or dilation, altering the position, orientation, or size of the triangle.
The transformation T mapping triangle XYZ to triangle X'Y'Z' could be any of the basic rigid transformations in geometry: translation, rotation, reflection, or dilation.
- A translation involves shifting the entire triangle by a certain distance in a certain direction without changing its orientation or shape.
- A rotation rotates the triangle around a fixed point (e.g., the origin or a specific vertex), altering its orientation but maintaining its shape and size.
- A reflection flips the triangle across a line (e.g., the x-axis, y-axis, or an arbitrary line), producing a mirror image with reversed orientation.
- A dilation scales the triangle uniformly, either enlarging or shrinking it while maintaining its shape and proportions.
To determine which transformation T specifically represents, additional information is needed, such as the coordinates of the vertices X, Y, Z and X', Y', Z', or any constraints or properties of the transformation provided. Each type of transformation has distinct characteristics that can be identified through geometric properties and transformations of coordinates.
A(n) is a typical, highly representative example of a concept. It is often described as an "average" or an "ideal member."
Answer:
Prototype
Step-by-step explanation:
Before a product is made, an original model of the product is first formulated. This is called a prototype. It is not simply a part of the design, but a very important part that the process of creating the final product solely depends on it. It is largely described as a highly representative example of a concept or product.
A prototype is a typical, highly representative example of a concept. It is often described as an "average" or an "ideal member."
What is a prototype?In Engineering, a prototype can be defined as a rudimentary (early) model, release, or sample of a particular product, so as to avail the developers and manufacturers an opportunity and ability to test a concept, functionality, or process within it.
For instance, the main purpose of the prototype of a robot is to determine and show whether or not the robot and all of its features are in tandem with the design specifications and market requirements.
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Let f(x) be a polynomial such that f(cos θ) = cos(4) θ for all θ. Find f(x). (This is essentially the same as finding cos(4) θ in terms of cos θ we structure the problem this way so that you can answer as a polynomial. Be sure to write your polynomial with the terms in order of decreasing degree.)
Answer:
f(x) = 8x⁴-8x²+1
Step-by-step explanation:
I will assume that f(cos θ) = cos(4θ). Otherwise, f would not be a polynomial. lets divide cos(4θ) in an expression depending on cos(θ). We use this properties
cos(2a) = cos²(a) - sin²(b)sin(2a) = 2sin(a)cos(a)sin²(a) = 1-cos²(a)cos(4θ) = cos(2 * (2θ) ) = cos²(2θ) - sin²(2θ) = [ cos²(θ)-sin²(θ) ]² - [2cos(θ)sin(θ)]² = [cos²(θ) - ( 1 - cos²(θ) ) ]² - 4cos²(θ)sin²(θ) = [2cos²(θ)-1]² - 4cos²(θ) (1 - cos²(θ) ) = 4 cos⁴(θ) - 4 cos²(θ) + 1 - 4 cos²(θ) + 4 cos⁴(θ) = 8cos⁴(θ) - 8 cos²(θ) + 1
Thus f(cos(θ)) = 8 cos⁴(θ) - 8 cos²(θ) + 1, and, as a result
f(x) = 8x⁴-8x²+1.
Suppose you can work at most a total of 25 hours per week. Baby-sitting, x, pays $6 per hour and working at the grocery store, y, pays $9 per hour. You need to earn $171 a week to pay for your expenses.
Answer:
y=7 number of hours at grocery store
x=18 number of hours at baby- sitting
Step-by-step explanation:
According to the information provided.
x is number of hours at baby- sitting
y is number of hours at grocery store
total number of hours worked
1) x+y =25
total earn in a week
2) x*$6 + y* $9 = $171
from equation 1
x+y=25
x= 25-y
we place the above derived equation in equation 2
x*$6 + y* $9 = $171
(25-y)*$6 + y* $9 = $171
(25*6) -6y +9y =171
150+3y=171
3y=171-150
3y=21
y=7 number of hours at grocery store
x= 25-y
x= 25-7
x=18 number of hours at baby- sitting
Brayden has math and reading homework tonight. Brayden can solve each math problem in 4 minutes and he can read each page in 2.5 minutes. The number of pages Brayden read is twice the number of math problems he solved. And it took him 45 minutes to complete all of his homework. Determine the number of math problems Brayden solved and the number of pages he read.
Answer:
Brayden Solved 5 math problem and read 10 pages.
Step-by-step explanation:
Let the number of Math Problem solved be x.
Also Let the Number of Pages he read be y.
Given:
The number of pages Brayden read is twice the number of math problems he solved.
Hence equation can be framed as;
[tex]y =2x[/tex]
Also Given:
Ayden can solve each math problem in 3 minutes and he can read each page in 1 minute and it took him 50 minutes to complete all of his homework.
Hence the equation can be framed as;
[tex]4x+2.5y =45[/tex]
Now Substituting the value of y in above equation we get;
[tex]4x+2.5(2x) =45\\\\4x+5x=45\\\\9x=45\\\\x=\frac{45}{9}=5[/tex]
Now Substituting the value of x to find the value of y we get;
[tex]y =2x =2\times5 =10[/tex]
Hence Brayden Solved 5 math problem and read 10 pages.
A construction company needs to remove tons of dirt from a construction site. They can remove tons of dirt each hour. How long will take to remove the dirt.
Question:
A construction company needs to remove 24 tons of dirt from a construction site. They can remove 3/4 tons of dirt each hour. How long will it take to remove dirt
Answer:
It takes 32 hours to remove the dirt
Step-by-step explanation:
Given:
Total amount dirt to be removed = 24 tons
Dirt that can removed in one hour = 3/4 tons
To Find:
Time taken to remove all the dirt =?
Solution:
Let the time taken to remove the dirt from the company be x.
Then
x = [tex]\frac{ \text { total amount of dirt in the company}}{\text{ amount of dirt removed in one hour}}[/tex]
Substituting the given values , we get
x = [tex]\frac{24}{\frac{3}{4}}[/tex]
x = [tex]24\times \frac{4}{3}[/tex]
x = [tex] \frac{96}{3}[/tex]
x= 32
An ice cream shop sold 48 vanilla milkshake in a day which is 40% of the total number of milkshake sold that day.What was the total number of milkshake that the ice cream shop sold that day?
Answer:the total number of milkshake that the ice cream shop sold that day is 120
Step-by-step explanation:
Let x represent the total number of
vanilla milkshakes that the ice cream shop sold that day.
The ice cream shop sold 48 vanilla milkshakes in a day which is 40% of the total number of milkshake sold that day. It means that
40/100 × x = 48
0.4 × x = 48
0.4x = 48
x = 48/0.4 = 120
The reflector of a flashlight is in the shape of a parabolic surface. The casting has a diameter of 8 inches and a depth of 4 inches.
How far should The light bulb should be placed from the vertex?
Answer:
1 in
Step-by-step explanation:
We are given that
Diameter of casting=8 in
Radius of casting=[tex]\frac{Diameter}{2}=4 in[/tex]
Depth of casting=4 in
We have to find the distance of light bulb should be from the vertex
It means the reflector passing through the point (4,4).
Equation of parabola along y-axis is given by
[tex]x^2=4ay[/tex]
Using the equation substitute x=4 and y=4
[tex]16=16a[/tex]
[tex]a=\frac{16}{16}=1[/tex]
The focus of parabola is at (0,a).
Therefore, the focus of reflector=(0,1)
Hence, the light bulb should be placed 1 in far from the vertex.
The distance of light bulb can be calculated using equation of parabola. The parabola is a plane curve which is U-shaped.
The distance of light bulb is [tex]1\:\rm in[/tex].
Given:
The diameter is [tex]8\:\rm in[/tex].
The radius is [tex]=\frac{d}{2}=\frac{8}{2}=4 \:\rm in[/tex].
The depth is [tex]4\:\rm in[/tex].
Since the depth is [tex]4\:\rm in[/tex] and radius is [tex]4\:\rm in[/tex] so reflector passes through the point [tex](4,4)[/tex].
Write the equation of parabola for y-axis.
[tex]x^2=4ay[/tex]
Putting [tex]4[/tex] for [tex]x[/tex] and [tex]4[/tex] for [tex]y[/tex].
[tex]4^2=4\times a\times 4\\a=\frac{16}{16}\\a=1[/tex]
The focus of a parabola is [tex](0,1)[/tex].
The distance of light bulb is [tex]1\:\rm in[/tex].
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Two planes are flying around the world. The first plane is flying at 300 miles per hour (mph) and is 350 miles ahead of the second plane. The second plane is flying at 400 mph. How long will it take for the second plane to catch the first plane?
Answer:
It takes 3.5 hours for the second plane to catch the first plane.
Step-by-step explanation:
From the information given:
Rate of the first plane = 300 miles per hourRate of the second plane = 400 miles per hourThe first plane is 350 miles ahead of the second plane.To find when the second plane catches the first plane, the distances of both planes must be equal.
We can use Distance = Rate x Time.
Let t be the time.
Distance of the first plane = Rate x Time = [tex]300\cdot t + 350[/tex]
Distance of the second plane = Rate x Time = [tex]400\cdot t [/tex]
Distance of the second plane = Distance of the first plane
[tex]400\cdot t=300\cdot t + 350[/tex]
Solving for t.
[tex]100\cdot t = 350[/tex]
t = 3.5 hours
It takes 3.5 hours for the second plane to catch the first plane.
The distance travelled by the two planes is an illustration of a linear function.
It will take the second plane 3.5 hours to catch up with the first plane
Let t represent time and d represent distance
The distance traveled by the first plane is represented as:
[tex]\mathbf{d_1 = 350 + 300t}[/tex]
The distance traveled by the second plane is represented as:
[tex]\mathbf{d_2 = 400t}[/tex]
Both planes will be at the same distance, when d1 = d2.
So, we have:
[tex]\mathbf{400t = 350 + 300t}[/tex]
Subtract 300t from both sides
[tex]\mathbf{400t - 300t = 350}[/tex]
Subtract
[tex]\mathbf{100t = 350}[/tex]
Divide both sides by 100
[tex]\mathbf{t = 3.50}[/tex]
Hence, it will take the second plane 3.5 hours to catch up with the first plane
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Grant is trying to make money to help pay for college by taking a job with Brian's bike taxis if Baltimore. He has an agreement with Brian to rent the bike for $35.00 a night. He charges customers $3.75 for every mile he transports them. If grant needs to make at least $15.00 a day, how many miles would he need to ride?
Answer: he would need to ride at least 13.3 miles
Step-by-step explanation:
The total amount that Grant needs to make in a day is greater than or equal to $15.
He has an agreement with Brian to rent the bike for $35.00 a night.
He charges customers $3.75 for every mile he transports them. If he transports the customers over x miles, his total revenue would be
3.75 × x = 3.75x
Profit = revenue - cost. Therefore,
his profit would be
3.75x - 35
Therefore,
3.75x - 35 ≥ 15
3.75x ≥ 15 + 35 = 50
x ≥ 50/3.75
x ≥ 13.3
There are 34 students in anatomy class and 37 students in physics class. If there are 48 students in total, how many students are in both classes?
Answer: 23
Step-by-step explanation:
let x=no. of students in both classes
no. of students in anatomy = 34
no. of students in anatomy only= 34 -x
no. of students in physics= 37
no. of students in physics only=37-x
48 = 34-x + 37-x + x
48 = 34 - x + 37
48 = 71 - x
x = 71 - 48
x = 23
Therefore the number of students in both classes are 23
Final answer:
To find the number of students in both the anatomy and physics classes, we can use the concept of sets and intersection. The number of students in both classes is 23.
Explanation:
To find the number of students in both the anatomy and physics classes, we can use the concept of sets and intersection. Let's assume A represents the set of students in the anatomy class, B represents the set of students in the physics class, and U represents the universal set of all students.
According to the given information, |A| = 34, |B| = 37, and |U| = 48.
The number of students in both classes can be found by using the formula:
|A ∩ B| = |A| + |B| - |U|
Substituting the values, we get:
|A ∩ B| = 34 + 37 - 48
|A ∩ B| = 71 - 48
|A ∩ B| = 23
Therefore, there are 23 students in both the anatomy and physics classes.
The function s(w) = 4w + 40 gives the total number of skateboards a company built w weeks after its office was relocated.
What is the meaning of the y-intercept?
A.No skateboards were built 10 weeks before the office was relocated.
B.The company built a total of 10 skateboards the week its office was relocated.
C.The company built a total of 40 skateboards the week its office was relocated.
D.No skateboards were built 40 weeks before the office was relocated.
Answer:
Option C. The company built a total of 40 skateboards the week its office was relocated
Step-by-step explanation:
Let
s ----> the total number of skateboards
w ---> the number of weeks
we have
[tex]s(w)=4w+40[/tex]
This is the equation of the line in slope intercept form
where
The slope is [tex]m=4\ \frac{skateboard}{week}[/tex]
The y-intercept is [tex]b=40\ skateboard[/tex]
Remember that
The y-intercept is the value of the function s when the value of variable w is equal to zero
In the context of the problem
The company built a total of 40 skateboards the week its office was relocated
If m arc CD = 143°, and m arc AB = 39°, what is m∠ DEC ?
Answer:
[tex]m\angle DEC=91^o[/tex]
Step-by-step explanation:
see the attached figure to better understand the problem
we know that
The measure of the interior angle is the semi-sum of the arcs comprising it and its opposite
so
[tex]m\angle DEC=\frac{1}{2}[arc\ AB+arc\ DC][/tex]
substitute the given values
[tex]m\angle DEC=\frac{1}{2}[39^o+143^o][/tex]
[tex]m\angle DEC=\frac{1}{2}[182^o][/tex]
[tex]m\angle DEC=91^o[/tex]
Chris has 7/8 yards of purple ribbon and 1 1/6 yards of green ribbon. Chris has a total of 3 2/3 yards of ribbon. How many yards of Chris's ribbon are neither purple nor green.
Answer:The number of yards of Chris's ribbon are neither purple nor green is 1.625 yards
Step-by-step explanation:
Chris has a total of 3 2/3 yards of ribbon. Converting 3 2/3 yards to improper fraction, it becomes 11/3 yards. Chris has 7/8 yards of purple ribbon. He also has 1 1/6 yards of green ribbon. Converting to improper fraction, it becomes 7/6 yards of green ribbon.
The total length of purple and green ribbon that Chris has would be 7/8 + 7/6 = (21+28)/24 = 49/24
The number of yards of Chris's ribbon are neither purple nor green is 11/3 - 49/24 = (88 - 49)/24
= 39/24 = 1.625 yards
One pound of tomatoes costs $1.59.One pound of bananas costs $1.87. How much more do 5 pounds of tomatoes costs than 2 pounds of bananas? Show your work
when you multiply $1.59 by 5 you get $7.95
when you multiply $1.87 by 2 you get $4.74
so when you subtract them you get $3.21
A truck hauled 136 cubic feet of sand to a construction site. The sand is dumped into a cone-shaped pile 6 feet in height. What is the approximate diameter of the pile of sand, in feet
18.614 because a=πh(r^2)/3 pretty sure thats the formula but the answer is correct
The approximate diameter of the pile of sand in the cone that sand was dumped in is 9.30 feet.
What is the approximate diameter?A cone is a 3-dimensional object that consists of a ciruclar base and a vertex. The diameter is twice the length of the radius.
Radius = √[volume / (1/3 x π x height)]
√[136 / (1/3 x 22/7 x 6)]
√[136 / 6.29 = 4.65
Diameter = 4.65 x 2 = 9.30 feet
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If it is snowing, there is an 80% chance that class will be canceled. If it is not snowing, there is a 95% chance that class will go on. Generally, there is a 5% chance that it snows in NJ in the winter. If we are having class today, what is the chance that it is snowing?
Answer:
[tex]P(S|\bar{C} ) = 0.1739[/tex]
Step-by-step explanation:
We define the probabilistic events how:
S: Today is snowing
C: The class is canceled
If it is snowing, there is an 80% chance that class will be canceled, it means
P( C | S ) = 0.8 conditional probability
If it is not snowing, there is a 95% chance that class will go on
[tex]P( \bar{C} | \bar{S}) = 0.95[/tex]
and P(S) = 0.05
We need calculate
[tex]P( S |\bar{C} ) = \frac{P(\bar{C} | S) P(S)}{P(\bar{C})}[/tex]
[tex]P(\bar{C}) = P( \bar{C}|S)P(S) + P( \bar{C}|\bar{S})P(\bar{S})[/tex]
How
[tex]P(C | S) = 0.8[/tex] then [tex]P( \bar{C} | S) = 0.2[/tex]
[tex]P (\bar{C})[/tex] = (0.2)(0.5) + (0.95)(0.5)
=0.575
[tex]P(S |\bar{C} ) = \frac{(0.2)(0.5)}{(0.575)}[/tex]
[tex]P(S|\bar{C} ) = 0.1739[/tex]
The second hand on a clock is 8 \text{ cm}8 cm8, start text, space, c, m, end text long. What is the distance the tip of the second hand travels in 101010 minutes?
The second hand of a clock is 8 cm long what is the distance of the tip of the second hand travel in 10 minutes
Answer:
The distance travelled by tip of second hand in 10 minutes is 502.4 cm
Solution:
Length of second hand = 8 cm long
In one revolution it travels a circumference of a circle of which radius is 8 cm
The circumference of circle = [tex]2 \pi r[/tex]
[tex]c = 2 \times \pi \times 8 = 16\pi[/tex]
We are aksed to find the distance of the tip of the second hand travel in 10 minutes
Second hand complete 1 revolution in 1 minute.
Therefore, in 10 minutes it revolves 10 revolution,
In 1 revolution tip of second covers = 16π
Hence distance travelled is given as:
[tex]\rightarrow 16 \pi \times10 = 160 \pi[/tex]
We know that π is a constant equals to 3.14
[tex]\rightarrow 160 \pi = 160 \times 3.14 = 502.4[/tex]
Thus the distance travelled by tip of second hand in 10 minutes is 502.4 cm
Answer:
502.4
Step-by-step explanation:
Mister Rogers is fencing another new rectangular garden in his neighborhood. One side of the garden faces the road and needs to be pretty. The other three sides just need to be functional. The pretty fencing costs $35 per linear foot and the functional fencing costs $18 per linear foot. Mr. Rogers has $ 3000 to build his fence. What dimensions of the garden give him the maximum area?
Answer:
length of the pretty side and length of the side oppositte to the pretty side = 37.91 ft
length of the other two sides = 27.52 ft
Step-by-step explanation:
The mathematical problem is:
Max A = b1*h
subject to: 35*b1 + 18*(2*h + b2) <= 3000
Where
A: area of the garden
b1: length of the pretty side
b2: length of the side oppositte to the pretty side
h: length of the other two sides
Replacing with b1 = b2 and taking only the equality sign in the restriction (in the maximum all the money will be spent), we get:
35*b1 + 18*(2*h + b1) = 3000
35*b1 + 36*h + 18*b1 = 3000
53*b1 + 36*h = 3000
b1 = 3000/53 - (36/53)*h
Substituing in Area's formula
A = (3000/53 - (36/53)*h)*h
A = (3000/53)*h - (36/53)*h^2
In the maximum, the derivative of A is equal to zero
dA/dh = 3000/53 - 2*(36/53)*h =
3000/53 - 72/35*h = 0
h = (3000/53)*(35/72)
h = 27.52 ft
then,
b1 = 3000/53 - (36/53)*27.52
b1 = 37.91 ft =b2
The dimensions of the garden that give Mr. Rogers the maximum area are approximately: Length (L) = 28.3 feet and Width (W) = 41.7 feet.
To find the dimensions of Mr. Rogers's garden that will give him the maximum area while staying within his budget, we need to set up and solve a problem involving optimization with constraints.
First, let's define the variables:
Let the length of the garden (which faces the road and requires pretty fencing) be denoted as [tex]L[/tex].Let the width of the garden be denoted as [tex]W[/tex].The cost of fencing:
Pretty fencing costs $35 per linear foot, and it is only needed for one side of the garden (the length L).Functional fencing costs $18 per linear foot, and it is needed for the other three sides (two widths W and the remaining length L).The total cost of fencing can be expressed as follows:
[tex]35L + 18(2W + L) = 3000[/tex]
Simplifying this equation:
[tex]35L + 36W + 18L = 3000[/tex]
[tex]53L + 36W = 3000[/tex]
To find the dimensions that maximize the area, we need to express the area in terms of one variable and use calculus to find the maximum. Let's solve for one variable in terms of the other. We'll solve for [tex]W[/tex]:
[tex]53L + 36W = 3000[/tex]
[tex]36W = 3000 - 53L[/tex]
[tex]W = \frac{3000 - 53L}{36}[/tex]
Now, express the area [tex]A[/tex] as a function of [tex]L[/tex]:
[tex]A = L \cdot W[/tex]
[tex]A = L \left(\frac{3000 - 53L}{36}\right)[/tex]
[tex]A = \frac{3000L - 53L^2}{36}[/tex]
To find the maximum area, we take the derivative of [tex]A[/tex] with respect to [tex]L[/tex] and set it to zero:
[tex]\frac{dA}{dL} = \frac{3000 - 106L}{36}[/tex]
Set the derivative to zero and solve for [tex]L[/tex]:
[tex]\frac{3000 - 106L}{36} = 0[/tex]
[tex]3000 - 106L = 0[/tex]
[tex]106L = 3000[/tex]
[tex]L = \frac{3000}{106}[/tex]
[tex]L \approx 28.3[/tex]
Now we use this value of [tex]L[/tex] to find the corresponding value of [tex]W[/tex]:
[tex]W = \frac{3000 - 53 \times 28.3}{36}[/tex]
[tex]W \approx \frac{3000 - 1499.9}{36}[/tex]
[tex]W \approx \frac{1500.1}{36}[/tex]
[tex]W \approx 41.7[/tex]
A sandwich shop has three types of sandwiches: ham, turkey, and chicken. Each sandwich can be ordered with white bread or multi-grain bread. Customers can add any combination of the six available toppings
To calculate the number of different sandwich combinations at a sandwich shop, one multiplies the basic sandwich configurations (3 types of sandwiches on 2 types of bread, totaling 6) by the possible topping combinations (2^6 = 64, including the option of no toppings), resulting in 384 different sandwich combinations.
Explanation:The question revolves around combinatorial mathematics, focusing on calculating the number of different sandwich combinations available at a sandwich shop with a given set of ingredients. The shop offers three types of sandwiches (ham, turkey, and chicken), each of which can be ordered on either white bread or multi-grain bread. Additionally, customers can add any combination of the six available toppings to their sandwiches. To calculate the total number of possible sandwich combinations, one would need to consider the choices for the type of sandwich, the bread, and the combinations of toppings.
For the sandwich and bread choices, since there are three types of sandwiches and two types of bread, there are a total of 3 * 2 = 6 basic sandwich configurations. For the toppings, since customers can choose any combination of the six available toppings, including the option of having no toppings at all, the total number of topping combinations can be calculated using the formula for combinations of a set: 2n, where n is the number of items (toppings) to choose from. Therefore, there are 26 = 64 possible topping combinations.
The total number of different sandwich combinations available can be calculated by multiplying the basic sandwich configurations by the topping combinations, which gives 6 * 64 = 384 different sandwich combinations. This calculation showcases the versatility of the menu and the vast array of options available to customers at the sandwich shop.
Let's say:
Ham sandwich: H
Turkey sandwich: T
Chicken sandwich: C
White bread: W
Multigrain bread: M
The representation using set notations would be:
[ (H,W), (H,M), (T,W), (T,M), (C,W), (C,M) ]
Complete question is here:
Represent the sample space using set notation.A sandwich shop has three types of sandwiches: ham, turkey, and chicken. Each sandwich can be ordered with white bread or multi- grain bread.
Find a polynomial equation that has zeros at x = 0, x = -5 and x = 6
Answer:
The answer to your question is x³ + 11² + 30x
Step-by-step explanation:
Data
x = 0; x = - 5; x = 6
Process
1.- Equal the zeros to zero
x₁ = 0; x₂ + 5 = 0; x₃ + 6 = 0
2.- Multiply the results
x(x + 5)(x + 6) = x [ x² + 6x + 5x + 30]
3.- Simplify
= x [ x² + 11x + 30]
4.- Result
= x³ + 11² + 30x
The polynomial equation with zeros at x = 0, x = -5, and x = 6 is x^3 - x^2 - 30x.
To find a polynomial equation with zeros at x = 0, x = -5, and x = 6, you would use the relationship between zeros and factors of a polynomial. Each zero corresponds to a factor of the polynomial; for x = 0, the factor is x, for x = -5, the factor is (x + 5), and for x = 6, the factor is (x - 6). Therefore, the polynomial equation that has these zeros can be constructed by multiplying these factors together.
The result is the polynomial equation:
f(x) = x(x + 5)(x - 6)
Expanding this product gives:
f(x) = x³ - x² - 30x
PLEASE HELP ASAP!!!!
There are 30 students in Bobby's class. 40% of the students have blue eyes. How many students in Bobby's class have blue eyes?
A. 12
B. 18
C. 28
D. 42
E. 75
Answer:
12
Step-by-step explanation:
Number of students in Bobby's class = 30
40% of the students have blue eyes.
It means 40 out of 100 students will have blue eyes.
Then we have to find how many out of 30 students will have blue eyes.
Let us take the number of blue-eyed students out of 30 to be x.
Then:
[tex]\frac{40}{100}=\frac{x}{30}\\x=\frac{40}{100}\times30\\x=12[/tex].
Hence 12 out of 30 students will have blue eyes.
look at the figure below:
Which step should be used to prove that point P is equidistant from points R and Q?
If any one side and any one common angle are equal in triangles PQR and PRS, then their corresponding sides are also equal.
If two sides and one included angle are equal in triangles PQS and PRS, then their third sides are equal.
In triangles PQR and PQS, if one side and one angle are equal, then their corresponding sides and angles are also equal.
In triangles PRS and PQS, all three angles are equal.
Answer:
The correct option is
If two sides and one included angle are equal in triangles PQS and PRS, then their third sides are equal.
Step-by-step explanation:
Given:
RS ≅ SQ
∠PSR ≅ ∠PSQ = 90°
To Prove:
point P is equidistant from points R and Q
i.e PR ≅ PQ
Proof:
In ΔPSR and Δ PSQ
PS ≅ PS ……….{Reflexive Property}
∠PSR ≅ ∠PSQ = 90° …………..{Measure of each angle is 90° given}
RS ≅ QS ……….{Given}
ΔPSR ≅ ΔPSQ ….{By Side-Angle-Side Congruence test}
∴ PR ≅ PQ .....{Corresponding Parts of Congruent Triangles}
i.e point P is equidistant from points R and Q .......Proved
A T-shirt makes wants to open his first store. If he chooses the store on Main Street, he will pay $640 in rent and will charge $30 per T-shirt. If he chooses the store on Broad Street, he will pay $450 in rent and will charge $25 per T-shirt. How many T-shirts would he have to sell in 1 month to make the same profit at either location?
Answer:38 t-shirts
Step-by-step explanation:
P=30t-640 main street
p=25t-450 broad street
30t-640=25t-450
30t-25t-640=-450
5t-640+640=-450+640
5t/5=190/5
t=38
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