Final answer:
The values of k that ensure the points (-5, 5) and (k, 0) are [tex]\(\sqrt{29}\)[/tex]units apart are -3 and -7, derived using the Pythagorean theorem and algebraic manipulation.
Explanation:
To find all values of k so that the points (-5, 5) and (k, 0) are \(\sqrt{29}\) units apart, we use the distance formula derived from Pythagoras’ theorem, which is[tex]d = \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)[/tex]. Here, [tex]\(x_1 = -5\), \(y_1 = 5\), \(x_2 = k\), and \(y_2 = 0\). Substituting the given values, we get:[/tex]
[tex]\(\sqrt{29} = \sqrt{(k + 5)^2 + (0 - 5)^2}\)[/tex]
Squaring both sides to eliminate the square root gives:
[tex]29 = (k + 5)^2 + 25[/tex]
Subtracting 25 from both sides, we get:
[tex]4 = (k + 5)^2[/tex]
Therefore, \(k + 5 = \pm2\), which simplifies to:
[tex]\(k + 5 = 2\) giving \(k = -3\)[/tex]
[tex]\(k + 5 = -2\) giving \(k = -7\)[/tex]
The values of k are -3 and -7, so the correct option is c. -3, -7.
The correct answer is option (c) -3, -7.
How is it so?
To find the values of k such that the given points (-5, 5) and (k, 0) are [tex]\(\sqrt{29}\)[/tex] units apart, use the distance formula between two points in a plane:
[tex]\[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Given that the distance between the points is [tex]\(\sqrt{29}\)[/tex], we have:
[tex]\[ \sqrt{(k - (-5))^2 + (0 - 5)^2} = \sqrt{29} \][/tex]
[tex]\[ \sqrt{(k + 5)^2 + 5^2} = \sqrt{29}[/tex]]
[tex]\[ \sqrt{k^2 + 10k + 25 + 25} = \sqrt{29} \][/tex]
[tex]\[ \sqrt{k^2 + 10k + 50} = \sqrt{29} \][/tex]
Squaring both sides to eliminate the square root:
[tex]\[ k^2 + 10k + 50 = 29 \][/tex]
[tex]\[ k^2 + 10k + 21 = 0 \][/tex]
Solve this quadratic equation
[tex]\[ k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Where a = 1, b = 10, and c = 21.
[tex]\[ k = \frac{-10 \pm \sqrt{10^2 - 4 \cdot 1 \cdot 21}}{2 \cdot 1} \][/tex]
[tex]\[ k = \frac{-10 \pm \sqrt{100 - 84}}{2} \][/tex]
[tex]\[ k = \frac{-10 \pm \sqrt{16}}{2} \][/tex]
[tex]\[ k = \frac{-10 \pm 4}{2} \][/tex]
[tex]\[ k = \frac{-10 + 4}{2} \] or \( k = \frac{-10 - 4}{2} \)[/tex]
[tex]\[ k = \frac{-6}{2} \] or \( k = \frac{-14}{2} \)[/tex]
[tex]\[ k = -3 \] \:or \( k = -7 \)[/tex]
So, the correct answer is option (c) -3, -7.
what are three equivalent fractions for 80/100 ...?
Three equivalent fractions for [tex]\( \frac{80}{100} \) are \( \frac{4}{5} \), \( \frac{160}{200} \), and \( \frac{400}{500} \)[/tex].
To find three equivalent fractions for [tex]\( \frac{80}{100} \)[/tex], we can simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 20 in this case.
1. Divide both the numerator and denominator by 20:
[tex]\[ \frac{80 \div 20}{100 \div 20} = \frac{4}{5} \][/tex]
So, [tex]\( \frac{80}{100} \)[/tex] is equivalent to [tex]\( \frac{4}{5} \)[/tex].
2. Multiply both the numerator and denominator by the same non-zero integer:
[tex]\[ \frac{80 \times 2}{100 \times 2} = \frac{160}{200} \][/tex]
So, [tex]\( \frac{80}{100} \) is also equivalent to \( \frac{160}{200} \)[/tex].
3. Similarly, we can multiply both the numerator and denominator by another non-zero integer:
[tex]\[ \frac{80 \times 5}{100 \times 5} = \frac{400}{500} \][/tex]
So, [tex]\( \frac{80}{100} \) is also equivalent to \( \frac{400}{500} \)[/tex].
An equation for a line in the plane allows you to find the x- and y-coordinates of any point on that line.
A. True
B. False
An equation for a line in the plane allows you to find the x- and y-coordinates of any point on that line.
Explanation:The statement is True.
An equation for a line in the plane, such as y = mx + b, allows you to find the x- and y-coordinates of any point on that line. The x-coordinate can be found by substituting a given y-value into the equation and solving for x. Similarly, the y-coordinate can be found by substituting a given x-value into the equation and solving for y.
Answer: TRUE!!!!!
Step-by-step explanation:
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Elena has 32 hair ties. Melanie has 8 hair ties the ratio of elena's hair ties to Melanie's hair ties is
The new ratio of Elena’s hair ties to Melanie’s hair ties is 8 : 3.
Given that Elena has 32 hair ties and Melanie has 8 hair ties, let’s analyze the situation:
Initial Ratio:
The ratio of Elena’s hair ties to Melanie’s hair ties is: [ \text{Elena : Melanie} = 32 : 8 = 4 : 1 ]
Melanie Gets 4 More Hair Ties:
After Melanie receives 4 additional hair ties, her total becomes: [ \text{Melanie’s new total} = 8 + 4 = 12 ]
New Ratio:
Now, let’s find the new ratio of Elena’s hair ties to Melanie’s hair ties: [ \text{Elena : Melanie} = 32 : 12 = 8 : 3 ]
Therefore, the new ratio of Elena’s hair ties to Melanie’s hair ties is 8 : 3.
Complete question:
Elena has 32 hair ties. Melanie has 8 hair ties. The ratio of Elena’s hair ties to Melanie’s hair ties is : 1. If, Melanie gets 4 more hair ties, the new ratio of Elena’s hair ties to Melanie’s hair ties is 8 :
Tarriq begins to solve the equation 50 + 2x = –190.
50 + 2x = –190
50 – 50 + 2x = –190 – 50
2x = –240
To finish solving the equation using the multiplication property of equality, which factor must Tarriq use?
A)-2
B)-1/2
C)1/2
D)2
Tarriq must use "2" as the Multiplication Property of equality.
What is the Multiplication Property of Equality?The Multiplication Property of Equality for any numbers a, b, and c, If we multiply both sides of an equation by the identical number, we always have equivalency.
Rearrange unfamiliar terms to the left side of the equation then
2x = -190 - 50
Calculate the sum or difference
2x = -240
Divide both sides of the equation by the coefficient of variable 2, and we get
x = -240 [tex]$ \div[/tex] 2
Calculate the product or quotient then we get
x = -120
Therefore, the correct answer is option D) 2.
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Tarriq must use factor C) 1/2, as this represents dividing both sides of the equation by 2 to isolate x, resulting in x = –120.
To finish solving the equation 2x = –240 using the multiplication property of equality, Tarriq must divide both sides of the equation by 2, which is the coefficient of x.
This step can be shown as:
2x / 2 = –240 / 2
The division property of equality allows us to simplify by canceling out the 2s on the left, resulting in:
x = –240 / 2
Therefore, x equals:
x = –120
This shows that the factor Tarriq must use to finish solving the equation is C) 1/2 since that represents the inverse operation of multiplying by 2.
Convert the function into intercept form. Show your work. y=-x^2+5x+36
find the slope of the line through each pair of points.
(1, -19), (-2, -7)
The slope of the line that passes through the points (1, -19) and (-2, -7) is calculated to be -4.
The slope of the line through the pair of points (1, -19) and (-2, -7), you can use the slope formula: slope (m) = (y₂ - y₁) / (x₂ - x₁), where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points.
Using the points given:
x₁ = 1, y₁ = -19
x₂ = -2, y₂ = -7
Substitute these values into the formula:
m = (-7 - (-19)) / (-2 - 1)
m = (12) / (-3)
m = -4
Therefore, the slope of the line through the points (1, -19) and (-2, -7) is -4.
Conver the equation y=2 to polar form. Then solve the resulting equation for r.
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Two cars, car X and car Y , start moving from the same point P on a cross intersection. Car X is travelling east and car Y is travelling north. Some time later car X is 60 km east of point P and travelling in an easterly direction at 80 km/h and car Y is 80 km north of point P and travelling in a northerly direction at 100 km/h. How fast is the distance between car X and car Y changing?
Samuel and Jason spend 3/4 of their combined earnings from Wednesday to buy a gift. How much do they spend? Is there enough left over from Wednesday's earnings to buy a card that costs $3.25? Explain.
Earnings:
Samuel - (12.5×0.40)
Jason - (7.1×0.40)
Answer:
they spend $ 5.88 to buy a gift but there is not enough money left to buy a card of $3.25
Step-by-step explanation:
The earnings form Wednesday of Samuel and Jason are:
Samuel earnings = 12.5x0.40 = $5
Jason earnings = 7.1x0.40 = $ 2.84
The combined earnings form Wednesday are:
Combined earnings = Samuel earnings + Jason earnings
Combined earnings= $5 + $ 2.84 = $ 7.84
The spended earnings are:
Spended earnings = combined earnings x ¾
Spended earnings = 7.84 x ¾ = $ 5.88
The left over is:
Left over = combined earnings – spended earnings
Left over = 7.84 – 5.88 = $ 1.96
So, they spend $ 5.88 to buy a gift but there is not enough money left to buy a card of $3.25
Which shows the correct substitution of the values a, b, and c from the equation 0 = 4x2 + 2x – 1 into the quadratic formula below?
Quadratic formula: x =
Answer:
[tex]x=\frac{-2\pm \sqrt{(2)^{2}-4(4)(-1)}}{2(4)}[/tex] is the answer.
Step-by-step explanation:
Given quadratic equation is 4x²+ 2x - 1 = 0
To find the solution of the equation we use the quadratic formula
[tex]x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}[/tex]
Now we put the values of a = 4, b = 2 and c = (-1)
Therefore, quadratic formula for the equation given will be
[tex]x=\frac{-2\pm \sqrt{(2)^{2}-4(4)(-1)}}{2(4)}[/tex]
Lee Wong receives an annual salary of $65,000 from CVS Pharmacy. Today his boss informs his that he will be getting a $3,000 raise. The percent increase rounded to the nearest tenth percent is:
Lee Wong's annual salary raise of $3,000 represents a 4.6% increase from his original salary of $65,000 when rounded to the nearest tenth of a percent.
To calculate the percentage increase of Lee Wong's salary, we use the formula for percentage change: Percentage Change = (Change in Quantity / Original Quantity) × 100%. In this case, the change in quantity is the raised amount of $3,000, and the original quantity is the original salary of $65,000.
Percentage Increase = ($3,000 / $65,000) × 100% = 0.04615 × 100% = 4.615%. Rounded to the nearest tenth of a percent, Lee Wong's salary increase is 4.6%.
-x^2=23 in standard form
Luis bought stock at $83.60. The next day, the price increased $15.35. This new price changed by -4 and 3/4 (mixed number) the following day. What was the final stock price? Is your answer reasonable? Explain.
After buying stock at $83.60, the price increased by $15.35 the next day, followed by a 4.75% decrease the day after. The final stock price was therefore $94.25, and this is a reasonable fluctuation in the stock market.
Luis purchased stock at an initial price of $83.60. On the next day, the stock price increased by $15.35, resulting in a new price of $83.60 + $15.35 = $98.95. The following day, the stock price experienced a decrease by 4 and 3/4 percent. To find the change in price, we multiply $98.95 by 4.75% (or 0.0475 in decimal form), which equals approximately $4.70. So, the final stock price after the decrease would be $98.95 - $4.70 = $94.25.
Is this answer reasonable? Yes, a price fluctuation in stock is common, and a change of $15.35 followed by a percentage decrease the following day is a typical scenario in the stock market.
5t + 5 = 30 solve for t
a rectunglar pool is 7ft wide. it is 3 time as long as it is wide
If f(x) = 5x + 40, what is f(x) when x = –5?
A. -9
B. -8
C. 7
D. 15
Answer:
D) 15
Step-by-step explanation:
The value of the function f(x) = 5x + 40 at x = -5 will be 15. Then the correct option is D.
What is a function?A statement, principle, or policy that creates the link between two variables is known as a function. Functions are found all across mathematics and are required for the creation of complex relationships.
The function is given below.
f(x) = 5x + 40
The value of the function f(x) = 5x + 40 at x = -5, then the value of the function will be
f(-5) = 5 (-5) + 40
f(-5) = - 25 + 40
f(-5) = 15
Then the value of the function f(x) = 5x + 40 at x = -5 will be 15.
Then the correct option is D.
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What is the sum of the first five terms of a geometric series with a1 = 20 and r = 1/4?
Answer: The required sum of first terms of the series is [tex]\dfrac{1705}{64}.[/tex]
Step-by-step explanation: We are given to find the sum of the first five terms of a geometric series with first term and common ratio as follows :
[tex]a_1=20~~~~~\textup{and}~~~~~r=\dfrac{1}{4}.[/tex]
We know that
the sum of first n terms of a geometric series with first term [tex]a_1[/tex] and common ratio r is given by
[tex]S_n=\dfrac{a(1-r^n)}{1-r}.[/tex]
Therefore, the sum of first 5 terms of the given geometric series is given by
[tex]S_5\\\\\\=\dfrac{a(1-r^5)}{1-r}\\\\\\=\dfrac{20(1-(\frac{1}{4})^5)}{1-\frac{1}{4}}\\\\\\=\dfrac{20\left(1-\frac{1}{1024}\right)}{\frac{3}{4}}\\\\\\=20\times\dfrac{4}{3}\times\dfrac{1023}{1024}\\\\\\=20\times\dfrac{341}{256}\\\\\\=\dfrac{5\times 341}{64}\\\\\\=\dfrac{1705}{64}.[/tex]
Thus, the required sum of first terms of the given geometric series is [tex]\dfrac{1705}{64}.[/tex]
An audience of 450 people is seated in an auditorium. Each row contains the same number of seats and each seat in the auditorium is occupied. With three fewer seats per row, and five extra rows, the same audience could still be seated, occupying all seats. How many rows does the auditorium have? ...?
Answer with Step-by-step explanation:
Let there be r rows and s seats in every row.
An audience of 450 people is seated in an auditorium and each seat is occupied.
i.e. rs=450
or s=450/r
With three fewer seats per row, and five extra rows, the same audience could still be seated, occupying all seats.
i.e. (s-3)(r+5)=450
s(r+5)-3(r+5)=450
rs+5s-3r-15=450
450+5s-3r-15=450
Subtracting both sides by 450,we get
5s-3r-15=0
i.e. 5s-3r=15
5(450/r)-3r=15
Dividing both sides by 3 and multiplying by r, we get
750-r²=5r
r²+5r-750=0
r² + 30r - 25r - 750 = 0
r(r + 30) - 25(r + 30) = 0
(r + 30)(r - 25) = 0
either r+30=0 or r-25=0
either r= -30 or r=25
r can't be negative
Hence, number of rows in auditorium are:
25
Sam and jesse can wash five cars each hour they work for seven hours each day over two days how many cards did sam and jesse wash?
Sam and Jesse wash a total of 70 cars over two days, by washing five cars each hour for seven hours each day.
Explanation:Sam and Jesse work together to wash cars. They can wash five cars each hour, and they work for seven hours each day. Since they do this over the course of two days, we need to calculate the total number of cars they wash in this period.
First, we find out how many cars they can wash in one day by multiplying the number of cars they can wash in an hour by the number of hours they work:
5 cars/hour * 7 hours/day = 35 cars/dayNext, since they work for two days, we multiply the number of cars washed in one day by two:
35 cars/day * 2 days = 70 carsSam and Jesse wash a total of 70 cars over the two days.
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The inverse of the function f(x) = 1/2x + 10 is shown.
h(x) = 2x – ?
What is the missing value?
Answer:
d. 20
Step-by-step explanation:
Replace the x and y values in the equation
(y = 0.5x + 10) to (x = 0.5y + 10)
Now, solve for the y value in the new equation. You'll get y = 2x + 20.
So, the intercept of the inverse function is 20.
A children's book has dimensions 20 cm by 24 cm.
What scale factor should be used to make an enlarged version that has dimensions 25 cm by 30 cm?
A. 5
B. 1.5
C. 1.25
D. 0.8
Answer:
We should use C. 1.25
Step-by-step explanation:
We know that the children's book dimensions are 20 cm x 24 cm
We need to find a scale factor (which is a number) that will turn the dimensions 20 cm x 24 cm ⇒ 25 cm x 30 cm
We can write :
(20 cm) . a = 25 cm
Where ''a'' is the scale factor.
Solving for a :
[tex](20cm).a=25cm\\a=\frac{25cm}{20cm}=1.25 \\a=1.25[/tex]
This result is reasonable because the scale factor won't have units
A scale factor of 1.25 turns 20 cm ⇒ 25 cm
We can use the another dimension to verify :
(24 cm) . a = 30 cm
(24 cm) . (1.25) = 30 cm
30 cm = 30 cm
The scale factor is option C. 1.25
Andrea borrowed 2,240 at 15% apr for 18 months. how much interest will she pay?is the answer $336
Answer:
No, the answer is not $ 336
The interest that she will pay is $ 504.
Step-by-step explanation:
Given : Andrea borrowed 2,240 at 15% apr for 18 months.
We have to calculate the interest will she pay.
Using formula for Simple interest.
[tex]SI =P\times r\times t[/tex]
Where P is principal amount
r is rate of interest
t is time period.
Given : P = $ 2240
t = 18 months
In years , [tex]\frac{18}{12}[/tex]
r = 15% = 0.15
Substitute, we have,
[tex]SI=2240\cdot0.15\cdot\frac{18}{12}[/tex]
Simplify, we have,
SI = 504
Thus, The interest that she will pay is $ 504.
No, the answer is not $ 336
Andrea will pay $504 in interest on a loan of $2,240 at 15% APR over 18 months, not $336. The calculation involves converting the loan duration into years and applying the formula for simple interest.
To determine how much interest Andrea will pay on a loan of $2,240 at 15% APR for 18 months, we first need to understand that APR (Annual Percentage Rate) is the interest rate for a whole year (annual), rather than just a monthly fee or rate. Since APR is annual but our loan term is in months, we convert the duration into years to match the APR's annual nature. 18 months is equivalent to 1.5 years. Therefore, the interest calculation would be as follows:
Principal (the amount borrowed) = $2,240
Rate (APR) = 15%
Time = 1.5 years
Interest = Principal × Rate × Time
Interest = $2,240 × 15% × 1.5 = $2,240 × 0.15 × 1.5
Interest = $504
Therefore, Andrea will pay $504 in interest, not $336 as presumed. It's essential to accurately convert the time to years when dealing with APR to ensure the calculation is correct.
What is the average rate of change of the function f(x)=120(0.1)^x from x = 0 to x = 2?
The average rate of change of the function f(x)=120(0.1)^x from x=0 to x=2 is calculated as (f(2) - f(0)) / (2 - 0), which gives -59.4.
Average Rate of Change
The average rate of change of a function over an interval is calculated by the difference in function values at the endpoints divided by the difference in input values. For the function f(x) = 120(0.1)^x, we want to find this rate from x = 0 to x = 2. First, we find the function values:
f(0) = 120(0.1)⁰ = 120
f(2) = 120(0.1)² = 120(0.01) = 1.2
Then we use the formula for average rate of change:
Average rate of change = (f(2) - f(0)) / (2 - 0) = (1.2 - 120) / 2 = -118.8 / 2 = -59.4
Therefore, the average rate of change of the function on the interval from x = 0 to x = 2 is -59.4.
If line segment AB = 12 feet, what is the length of line segment AC?
Select one of the options below as your answer:
A.
6 feet
B.
10 feet
C.
12 feet
D.
24 feet
9% percent are in the band. the band has 180 members.how many people are in school
The correct answer is that there are 2000 people in the school.
To solve this problem, we can set up a proportion based on the information given. We know that 9% of the school's population is in the band, and the band has 180 members. We want to find the total population of the school, which we will denote as [tex]\( P \)[/tex].
The proportion can be written as:
[tex]\[ \frac{9}{100} = \frac{180}{P} \][/tex]
Now, we can solve for [tex]\( P \)[/tex] by cross-multiplying:
[tex]\[ 9P = 180 \times 100 \][/tex]
Divide both sides by 9 to isolate [tex]\( P \)[/tex]:
[tex]\[ P = \frac{180 \times 100}{9} \][/tex]
Simplify the right side of the equation:
[tex]\[ P = \frac{18000}{9} \][/tex]
[tex]\[ P = 2000 \][/tex]
Therefore, the total population of the school is 2000 people. This matches the correct answer given in the options.
A caterer has 5 rolls. He is ordering more rolls. He can order up to 9 packages of rolls and each package contains 12 rolls. The caterer cannot order partial packages. The function that models the number of rolls the caterer has is f(p)=12p+5f(p)=12p+5, where p is the number of packages the he orders.
What is the practical domain of the function?
A: all real numbers from 1 to 9, inclusive
B: {17, 29, 41, 53, 65, 77, 89, 101, 113}{17, 29, 41, 53, 65, 77, 89, 101, 113}
C: all integers from 1 to 9, inclusive
D: all real numbers
The practical domain of the function f(p) = 12p + 5 is all integers from 1 to 9, inclusive.
The practical domain of the function f(p) = 12p + 5 refers to the set of all possible values that p, the number of packages of rolls, can take given the constraints of the situation described. Since the caterer cannot order partial packages and can order up to 9 packages, p must be an integer from 1 to 9 inclusive.
Moreover, these are the only values that make sense in the context of this problem since ordering 0 packages will not change the quantity of rolls they already have and ordering more than 9 packages is beyond the caterer's limit.
Parabola: The _____ value looks like the bottom of a valley. ...?
use the geometric mean to find the 7th term in a geomtric sequence if the 6th term is 75 and the 8th term is 48.
Which of these inequalities has no solutions?
Which is the safest way to invest money??
Invest in more than one type of investment.