Find the value of the coefficient of
1/x
in the expansion of (2x −
1/x )^5
. ...?
Final answer:
To find the coefficient of 1/x in the expansion of (2x - 1/x)^5, we use the binomial theorem, specifically the term where the exponent of x is -1. We find that the coefficient of 1/x is -80.
Explanation:
To find the coefficient of 1/x in the expansion of (2x - 1/x)^5, we can use the binomial theorem. The binomial theorem states that:
(a + b)^n = Σ (n choose k) a^(n-k) b^k
where 'n choose k' is the binomial coefficient, computed as:
n! / [(n-k)!k!]
In our case, a = 2x and b = -1/x, and we want the term where the power of x is -1, which occurs when k = 4:
(5 choose 4) (2x)^(5-4) (-1/x)^4 = 5 * 2 * 1 * 1/x^4 = 10/x^4
However, to get the term with 1/x, we need a positive power of 1 for x. We achieve this when k = 1:
(5 choose 1) (2x)^(5-1) (-1/x)^1 = 5 * 16x^4 * (-1/x) = -80
Therefore, the coefficient of 1/x in the expansion is -80.
What is closer to 1/3 0.3 0.33 or 0.333
What is 598% divided by 26%
In a game of blackjack, a 2-card hand consisting of an ace and a face card or a 10 is called a blackjack. (Round your answers to four decimal places.)
(a) If a player is dealt 2 cards from a standard deck of 52 well-shuffled cards, what is the probability that the player will receive a blackjack?
(b) If a player is dealt 2 cards from 2 well shuffled standard decks, what is the probability that the player will receive a blackjack?
To calculate the probability of getting a blackjack from a single well-shuffled deck, you must consider the probability of drawing an Ace and then a 10-value card or vice versa, and add those probabilities together. The same process applies to two decks with adjusted numbers. You will then round the final probabilities to four decimal places.
Explanation:To answer the question of the probability of getting a blackjack in a single deck, consider that there are 4 aces in a deck and 16 cards that are either a face card or a 10 (each suit has 4 such cards - Jacks, Queens, Kings, and 10s). When one card is drawn, the remaining cards are one fewer, so we need to adjust our probability for the second card.
Calculate the probability of drawing an Ace first and then a card with a value of ten: P(Ace first, then 10) = (4/52) × (16/51).Calculate the probability of drawing a 10-value card first and then an Ace: P(10 first, then Ace) = (16/52) × (4/51).Add the two probabilities together for the final answer.For the probability of a blackjack from two decks, follow the same steps, but adjust the numbers for a 104-card deck.
P(Ace first, then 10) = (8/104) × (32/103).P(10 first, then Ace) = (32/104) × (8/103).Add the two probabilities together for the final answer.After doing the calculations, we round the probabilities to four decimal places as instructed.
If the integral of f(x)dx = 2.3 from 1 to 3 and F'(x) = f(x), what is the value of F(3) - F(0). The graph will be inserted as a picture. ...?
Three-fourths of a number increased by one-fifth is no less than seven-eighths.
Which equation represents this problem?
Answer:
yea, most likely it´s 3/4 y + 1/5 ≥ 7/8
Step-by-step explanation:
A lake was stocked with 80 trout, 20 bass, 70 walleye and 30 catfish. What is the probability that a fisherman catches a trout first then a second trout.
The probability of a fisherman catching a trout first and then a second trout from this lake is 0.158.
Explanation:The subject of this question pertains to the field of probability in mathematics and requires an understanding of fundamental concepts related to this topic.
The total number of fish in this lake is 200 (80 trout + 20 bass + 70 walleye + 30 catfish). The probability of catching a trout first can be calculated by dividing the number of trout by the total number of fish, which is 80/200 = 0.4. After catching the first trout, there are now 79 trout left, and the total number of fish is 199. Therefore, the probability of catching a second trout is 79/199. To find the overall probability of both events happening, we multiply the probabilities of each individual event:0.4 x (79/199) = 0.158.
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The profit P (in thousands of dollars) for a company spending an amount s (in thousands of dollars on advertising is
P= -(1/10)s^3 + 6s^2 + 400
Find the amount of money the company should spend on advertising in order to yield a maximum profit
To find the maximum profit, take the derivative of the profit function and set it equal to zero, then solve for s. This gives the advertising spend that yields the maximum profit.
Explanation:To find the amount of money the company should spend on advertising in order to yield a maximum profit, we first need to find the derivative of the profit function P(s) = -(1/10)s^3 + 6s^2 + 400. The derivative, P'(s), gives the rate of change of profit with respect to advertising spending.
The derivative of the function is P'(s) = -(3/10)s^2 + 12s. We find the maximum of the function by setting the derivative equal to zero and solving for s.
Setting P'(s)=0 we get: -(3/10)s^2 + 12s = 0
Solving this equation will provide the advertising spend for maximum profit.
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In order to apply the law of sines to find the length of the side of a triangle, it is enough to know which of the following?
A. Two angles and a side
B. All three angles
C. Two sides and one angle
D. The area of the triangle ...?
Answer:
We must have two angles and a side.
A is the correct option.
Step-by-step explanation:
For any triangle ABC, the law of sine is given by
[tex]\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}[/tex]
From this formula it is clear that in order to find the length of the side of the triangle, we must have two angles and a side.
Let us understand this by assuming that we need to find a (length of the side). From the formula, we have
[tex]\frac{a}{\sin A}=\frac{b}{\sin B}\\\\a=\frac{b\sin A}{\sin B}[/tex]
Thus, to find the length a, we must have b, sin A and sin B.
Hence, o find the length of the side of the triangle, we must have two angles and a side.
the sum of two numbers is s.if one of the numbers is n,the second number can be expressed as
1. s+n
2. s÷n
3. s-n
4. n-s
Given that two numbers sum to 's', and one of the numbers is 'n', the other number can be expressed as 's - n'. In essence, if you subtract 'n' from the sum 's', you obtain the second number.
Explanation:The student's question relates to expressing one number in terms of another when their sum is given. Given that the sum of the two numbers is s and one of the numbers is n, the second number can be expressed as s - n. This is because when two numbers add to give 's', subtracting one of the numbers (n, in this case) from 's' would give the other number.
For instance, if s = 10 and n = 3, then the second number would be 10 - 3 = 7. Hence, the correct answer from the options provided would be 3. s - n.
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Round to the nearest tenth 848.607
The ratio of two numbers is 1 to 4, and the sum of the numbers is 45. What is the smaller number?
10
9
8
7
9 i think
The average number of vehicles waiting in line to enter a sports arena parking lot is modeled by the function w(x)=x^2/2(1-x), where x is a number between 0 and 1 known as the traffic intensity. Find the average number of vehicles waiting if the traffic intensity is 0.92. ...?
To find the average number of vehicles waiting at a traffic intensity of 0.92, we substitute 0.92 into the function w(x) = x² / (2(1-x)) to arrive at the answer 5.29.
Explanation:The average number of vehicles waiting in line is modeled by the function w(x) = x^2 / (2(1-x)) where x represents traffic intensity. The question is asking us to calculate the average number of vehicles waiting when the traffic intensity is 0.92. By plugging this value into the function, we get w(0.92) = 0.92^2 / (2(1-0.92)), which simplifies to w(0.92) = 0.8464 / 0.16. The result is 5.29, so the average number of vehicles waiting would be 5.29 when the traffic intensity is 0.92. To find the average number of vehicles waiting when the traffic intensity is 0.92, we substitute the value of x into the given function w(x)=x^2/2(1-x). In this case, x=0.92. So, the average number of vehicles waiting can be found by evaluating w(0.92).
w(0.92) = (0.92)^2 / 2(1 - 0.92)
w(0.92) = 0.8464 / (2 * 0.08)
w(0.92) = 10.58
Therefore, when the traffic intensity is 0.92, the average number of vehicles waiting is approximately 10.58.
I need help on the back side
The average income, I, in dollars, of a lawyer with an age of x years is modeled with the following function:
What is the youngest age for which the average income of a lawyer is $450,000?
Round to the nearest year.
To find the youngest age for a lawyer to have an average income of $450,000, you solve the income function equation and round to the nearest year.
Explanation:The youngest age for which the average income of a lawyer is $450,000 can be found by solving the given function:
I(x) = 10000 + 2000x
Given I(x) = $450,000, the equation becomes:
450,000 = 10000 + 2000x2000x = 440,000x = 220Rounded to the nearest year, the youngest age would be 220 years old.
If B=A, where on the number line will B appear?
A.to the right of A
B.in exactly the same location as A
C.to the left of A
find the x-intercepts of the parabola with vertex(6,27) and y-intercept (0,-81). write answer in form (x1,y1)(x2,y2). if necessary ,round to the nearest hundredth .
Answer:
(8.33) , (236.67)
Step-by-step explanation:
This is correct
how to factor x3-x2-4x-4
When the following expression is written in simplest form, what is the coefficient of the variable term?
-2.5y + 8y - 10.5y
the cost of a pencil is 25 cents more than the cost of an eraser. if the cost of 8 pencils and 10 erasers is 12.80 dollars, find the cost of each
State the postulate or theorem that proves that x||y.
A.converse of the consecutive interior angles theorem
B.converse of the alternate interior angles theorem
C.converse of the corresponding angles postulate
D.converse of the alternate exterior angles theorem
The theorem that proves x is parallel to y is the Converse of the Alternate Interior Angles Theorem, which states that if alternate interior angles are congruent, the lines are parallel. The correct answer to the question of which postulate or theorem proves that x is parallel to y is B. Converse of the Alternate Interior Angles Theorem.
Explanation:According to this theorem, if two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel. The other options, such as the converse of the corresponding angles postulate or the converse of the consecutive interior angles theorem, are related but specifically address different sets of angles formed when two lines are cut by a transversal.
In geometry, it is crucial to apply the correct postulates or theorems to prove specific properties, like parallel lines. The reliability of these mathematical proofs, like those in trigonometry or physics, comes from the logical sequence of applying these fundamental principles accurately.
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Angie and kenny play online video games. angie buys 1 software package and 1 month of game play. kenny buys 1 software packages and 5 months of game play. each software package costs $20. if their total cost is $154 what is the cost of one month of game play? the cost of one month of game play is ______
The hypotenuse of a 45°-45°-90° triangle measures 128 cm.What is the length of one leg of the triangle?
Answer:
Since, a 45°-45°-90° triangle is a special type of isosceles right triangle, where the two legs are congruent to one another and the non-right angles are both equal to 45 degrees.
In 45°−45°−90° triangle
Sides are in the proportion [tex]1:1:\sqrt{2}[/tex]
then,
the measures of the sides are [tex]x , x ,\sqrt{2}x[/tex]
The length of hypotenuse= [tex]\sqrt{2}[/tex] times the length of the leg. ....[1]
Given: length of hypotenuse = 128 cm.
then,
from [1] we have;
128 = [tex]\sqrt{2}\cdot x[/tex]
or
[tex]x=\frac{128}{\sqrt{2}}[/tex] = [tex]\frac{128}{\sqrt{2} } \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{128\sqrt{2}}{\sqrt{2}\cdot \sqrt{2}}[/tex]
Simplify:
[tex]x =\frac{128 \cdot \sqrt{2}}{2} =64\sqrt{2}[/tex] cm.
Therefore, the length of one leg of the triangle is; [tex]64\sqrt{2}[/tex] cm.
4 is equivalent to how hundreths
the exterior angles of a triangle have measures of (x+10) degrees, (2x+20) and 3x degrees. what is the measure of the smallest interior angle of the triangle ...?
Answer: X= 55
Step-by-step explanation:
The sum of exterior angles of any polygon will always equal to 360 degrees. which means, we can write:
360 = (x+10) + (2x+20) + 3x
x = 55
The exterior angles have values of 65, 130 and 165. The exterior that will have the largest value will correspond to the smallest interior angle and therefore have a value of 15 degree!. ;)
Final answer:
By calculating the value of x and using the relationship that each exterior angle and its corresponding interior angle are supplementary, we find that the measure of the smallest interior angle of the triangle is 115 degrees.
Explanation:
To find the measure of the smallest interior angle of the triangle, we first need to use the fact that the sum of the exterior angles of a triangle is always 360 degrees. So, we set up the equation (x+10) + (2x+20) + 3x = 360 and solve for x. This gives us:
x + 10
2x + 20
3x
Combining like terms, we get 6x + 30 = 360. Subtracting 30 from both sides, we get 6x = 330. Dividing both sides by 6, we find that x = 55 degrees. Therefore, the exterior angles are 65 degrees, 130 degrees, and 165 degrees. To find the smallest interior angle, we take the smallest exterior angle, 65 degrees and subtract it from 180 degrees (since the exterior angle and interior angle at the same vertex are supplementary). Thus, the smallest interior angle is 180 degrees - 65 degrees = 115 degrees.
A fisherman rows up stream at 4mph and downstream at 6mph. He started rowing upstream until he got tired and then rowed downstream to his starting point. How far did the fisherman row in the whole trip in 7 hours?
By setting the upstream and downstream distances equal and solving for the time rowed upstream, we find that the fisherman rowed 4.2 hours upstream and 2.8 hours downstream. Since he rowed back to his starting point, the total distance rowed upstream and downstream is the same, amounting to 33.6 miles for the entire trip.
To solve this problem, we need to determine how much time the fisherman spent rowing upstream and downstream. Let's call the upstream speed U, which is 4 mph, and the downstream speed D, which is 6 mph. He rows for a total of 7 hours. Let's call the time spent rowing upstream T and the time spent rowing downstream (7 - T) since the entire trip takes 7 hours.
Since distance equals rate times time, the upstream distance will be U × T and the downstream distance will be D × (7 - T). Because the fisherman returns to his starting point, the upstream and downstream distances are equal.
Setting them equal gives us the equation:
4T = 6 (7 - T)
Solving for T gives us:
T = 42 / 10 = 4.2 hours
Hence, the time spent rowing downstream is 7 - 4.2 = 2.8 hours.
To find the total distance rowed, we use the downstream distance (since it's equal to the upstream distance) which is calculated as:
Distance = D × (7 - T) = 6 × 2.8 = 16.8 miles
Therefore, the fisherman rowed a total of 16.8 miles upstream and 16.8 miles downstream, summing up to 33.6 miles for the entire trip.
the perimeter of a rectangle is twice the sum of its length and width. The perimeter is 34 meters and its length is 2 meters more than twice its width. What is its length? ...?
Write the following in terms of sin(θ) and cos(θ); then simplify if possible.
cos(θ) + sin(θ)cot(θ) = ...?
Why are their bad people in the world?
The population of foxville is about 12 times 10 to the power of 3. which is another way to write this number
Answer:
The population of foxville is 12000.
Step-by-step explanation:
It is given that the population of foxville is about 12 times 10 to the power of 3.
Population of foxville = [tex]12\times 10^3[/tex]
[tex]12\times 10^3[/tex] is a scientific notation of a number.
We need to write this number in another way.
[tex]12\times 10^3=12\times 1000[/tex]
[tex]12\times 10^3=12000[/tex]
Therefore, the population of foxville is 12000.