The histogram below gives the distribution of test scores for a sample of
students in a school in Alaska. Approximately how many students received a
score between 70.5 and 80?
Answers
Answer:
B. 200 students
Step-by-step explanation:
Related Questions
Determine the equation of straight line passing through the point (1,0) and (2, - 3)
Answers
first you need to find the gradient of the line and to do that you divide the height of a part of the line by its length. so to find the height we do -3-0=-3 and to find the length we do 2-1=1. then divide -3 by 1 and the gradient is -3. so now we know the equation is y=-3x+b. to find b we replace y with the y coordinate of one of the 2 points. lets take the first point. so it will be 0=-3+b. Now we can easily find b by doing 0-(-3) which is 3. so the equation of the line is y=-3x+3. Also please mark this as brainliest answer if you found it helpful thanks.
Answer:
y = - 3x + 3
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
calculate m using the slope formula
m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
with (x₁, y₁ ) = (1, 0 ) and (x₂, y₂ ) = (2, - 3 )
m = [tex]\frac{-3-0}{2-1}[/tex] = [tex]\frac{-3}{1}[/tex] = - 3 , then
y = - 3x + c ← is the partial equation
to find c substitute either of the 2 points into the partial equation
using (2, - 3 )
- 3 = - 3(2) + c = - 6 + c ( add 6 to both sides )
3 = c
y = - 3x + 3 ← equation of line
75% of flights arriving in Memphis are on time if the FAA Selects 60 random flights what is the probability that more than 80% of a simple flights are on time?
Answers
The mean of the distribution of sample proportion is equal to the population proportion, which is 0.75.
The standard deviation of the distribution of sample proportion is given by:
σ = sqrt[ (p * (1 - p)) / n ]
where p is the population proportion (0.75), and n is the sample size (60).
σ = sqrt[ (0.75 * (1 - 0.75)) / 60 ]
= 0.0625
To find the probability that more than 80% of the sample flights are on time, we will use a z-score calculation.
z = (x - μ) / σ
where x is the value we want to find the probability of, μ is the mean of the distribution (0.75), and σ is the standard deviation of the distribution (0.0625).
z = (0.8 - 0.75) / 0.0625
= 0.8
Using a standard normal table or calculator, we can find that the probability of a z-score of 0.8 or higher is approximately 0.2119.
Therefore, the probability that more than 80% of the sample flights are on time is approximately 0.2119.
Felicia is deciding on her schedule for next semester. She must take each of the following classes: English 102, Spanish 102, History 102, and College Algebra. If there are 15 sections of English 102, 9 sections of Spanish 102, 12 sections of History 102, and 13 sections of College Algebra, how many different possible schedules are there for Felicia to choose from? Assume there are no time conflicts between the different classes.
Answers
Okay, here are the steps to solve this problem:
* There are 15 sections of English 102 to choose from
* There are 9 sections of Spanish 102 to choose from
* There are 12 sections of History 102 to choose from
* There are 13 sections of College Algebra to choose from
So for English 102 Felicia has 15 options, for Spanish 102 she has 9 options, for History 102 she has 12 options, and for College Algebra she has 13 options.
By the Fundamental Principle of Counting, the total number of possible schedules is:
15 * 9 * 12 * 13 = 16,920
Therefore, the total number of possible schedules for Felicia is 16,920
What is the most descriptive name for the illustration below?
A. quadrilateral
B. polygon
C. rhombus
D. trapezoid
Answers
The most descriptive name for the illustration below is C. rhombus.
Define term Rhombus?Because it has four sides, a rhombus is a geometric shape that is a quadrilateral. Because its opposite sides are parallel and their angles are equal, it is a unique kind of parallelogram.
The most descriptive name for the illustration above is C. rhombus.
A rhombus is a type of quadrilateral with four sides of equal length and opposite angles that are equal to each other. It can also be defined as a parallelogram with equal diagonals. The illustration appears to have all sides of equal length and opposite angles that are equal, thus making it a rhombus.
While the other options, such as quadrilateral, polygon, and trapezoid, also apply to the illustration, they are less specific than the term rhombus.
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whats the answer for log b + z log c in condense form
Answers
Answer:
log b.c^z
Step-by-step explanation:
from the theorem a log b=log b^a
z log c= log c^z
from the theorem log a + log b= log ab
log b+ log c^z= logb. c^z will be the final answer
Two partner a and b agree to divide 30% of the total profits equally between than a and the balance in the ratio 3:4.if the profits is rs 30000 find a,s share of the profits.
Answers
well, let's first find out how much is 30% of 30000
[tex]\begin{array}{|c|ll} \cline{1-1} \textit{\textit{\LARGE a}\% of \textit{\LARGE b}}\\ \cline{1-1} \\ \left( \cfrac{\textit{\LARGE a}}{100} \right)\cdot \textit{\LARGE b} \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{30\% of 30000}}{\left( \cfrac{30}{100} \right)30000}\implies 9000[/tex]
ok, that means if we split that down the middle, each will get 4500.
so the remaining is 21000, and that's split on a 3:4 ratio, assuming from A to B, so let's simply divide the total 21000 by (3 + 4) and distribute accordingly.
[tex]\stackrel{A}{3}~~ : ~~\stackrel{B}{4} ~~ \implies ~~ \stackrel{A}{3\cdot \frac{21000}{3+4}}~~ : ~~\stackrel{B}{4\cdot \frac{21000}{3+4}}\implies \stackrel{A}{3\cdot 3000}~~ : ~~\stackrel{B}{4\cdot 3000} \\\\\\ \stackrel{A}{9000}~~ : ~~\stackrel{B}{12000}\hspace{9em}\underset{ \textit{share for A} }{\stackrel{ 4500~~ + ~~9000 }{\text{\LARGE 13500}}}[/tex]
If the first 4 terms of an infinite geometric sequence are 150, 120, 96, and 76.8, then the sum of all the terms in the sequence is ______.
Answers
Answer:
The common ratio of this sequence is 120/150 = 4/5, so we have:
[tex] \frac{150}{1 - \frac{4}{5} } = \frac{150}{ \frac{1}{5} } = 150 \times 5 = 750[/tex]
So the sum of all the terms in this sequence is 750.
Does each of these graphs have at least one Hamiltonian circuit? If so, find one.
Answers
The first shape does NOT have a Hamilton circuit
The second shape has a Hamilton circuit
The third shape does NOT have a Hamilton circuit
What is a Hamiltonian circuit?In mathematical graph theory, a Hamiltonian circuit- named after its originator William Rowan Hamilton, an acclaimed Irish mathematician of the 19th century– is a path within a graph that covers every vertex only once and begins and ends at the same vertex.
To put it differently, it's an enclosed movement that goes through each point in the graph exactly once. When the graph consists of a Hamiltonian circuit, this indicates to be a Hamiltonian graph. One can see many practical uses of Hamiltonian circuits such as computer science where they play a crucial role in algorithm analysis and design.
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Kenji walks 44 feet in 10 seconds. At this rate, how many miles does Kenji walk in an hour? Show your work. (1 mile= 5,280 feet)
Answers
Answer:
44 feet in 10 seconds = 44 x 6 (since there are 60 seconds in a minute) = 264 feet per minute
264 feet per minute x 60 minutes (since there are 60 minutes in one hour) = 15,840 feet per hour
15,840 feet per hour / 5,280 feet (since there are 5,280 feet in 1 mile) = 3 miles per hour
Giving a test to a group of students, the grades and gender are summarized below
A B C Total
Male 12 15 4 31
Female 9 8 16 33
Total 21 23 20 64
If one student is chosen at random, find the probability that the student was male GIVEN they got a 'C':
Answers
Answer: [tex]\frac{1}{5}[/tex]
Step-by-step explanation:
Males who got a C = 4
Total amount of people who got a C = 20
[tex]\frac{4}{20}[/tex]
[tex]\frac{4}{20}[/tex] simplifies to [tex]\frac{1}{5}[/tex]
P(A|B) = P(B|A) * P(A) / P(B)
where P(A|B) is the probability of event A given event B, P(B|A) is the probability of event B given event A, P(A) is the prior probability of event A, and P(B) is the prior probability of event B.
In this case, event A is "the student is male" and event B is "the student got a 'C'". We are given the following information:
- P(A) = 31/64 (the prior probability of a student being male)
- P(B|A) = 4/31 (the probability of getting a 'C' given the student is male)
- P(B) = 20/64 (the prior probability of getting a 'C')
Using Bayes' theorem, we can calculate P(A|B) as:
P(A|B) = P(B|A) * P(A) / P(B)
P(A|C) = (4/31) * (31/64) / (20/64)
P(A|C) = 4/20
P(A|C) = 0.2
Therefore, the probability that the student was male given they got a 'C' is 0.2 or 20%.
Can someone please help me
Answers
The height of point A is 0 feet. However, this is not the actual height of point A, as it is at the highest point on the Ferris Wheel, which is 185 feet as calculated previously.
How to solveThe height of point A on the Ferris Wheel can be calculated using the sine function, as it is the vertical component of the point's position. The center of the Ferris Wheel is 100 feet off the ground, and its radius is 85 feet.
Let's denote the height of point A as h_A, which is the vertical distance from the center of the Ferris Wheel to point A.
Since point A is at the 0 radian position, it is at the highest point on the Ferris Wheel. At this point, the height of point A is equal to the sum of the radius and the center's height:
h_A = radius + center's height
h_A = 85 + 100
h_A = 185 feet
So, the height of point A off the ground is 185 feet.
To estimate the value of h_A using the sine function, we can use the fact that at any given angle θ on the unit circle, the height of the point on the circle is given by the sine of that angle multiplied by the radius. In this case, since point A is at the highest point on the Ferris Wheel, we can use the sine of 0 radians (which is 0) to estimate the height of point A:
h_A = radius * sin(0)
h_A = 85 * 0
h_A = 0
So, using the sine function, we estimate that the height of point A is 0 feet. However, this is not the actual height of point A, as it is at the highest point on the Ferris Wheel, which is 185 feet as calculated previously.
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When Caroline runs the 400 meter dash, her finishing times are normally distributed with a mean of 68 seconds and a standard deviation of 1.5 seconds. Using the empirical rule, determine the interval of times that represents the middle 99.7% of her finishing times in the 400 meter race.
Answers
The interval of times that represents the middle 99.7% of Caroline's finishing times is (63.5, 72.5) seconds
How to calculate the intervalIn this case, we are looking for the interval of times that represents the middle 99.7% of Caroline's finishing times.
The middle 99.7% of the data corresponds to three standard deviations from the mean in both directions
Three standard deviations from the mean is 3 x 1.5 = 4.5 seconds.
Therefore, the interval of times that represents the middle 99.7% of Caroline's finishing times is (68 - 4.5, 68 + 4.5) = (63.5, 72.5) seconds
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Suppose you physically simulate the random process of rolling a single die. (a) After 1010 rolls of the die, you observe a "one" 44 times. What proportion of the rolls resulted in a "one"? (b) After 2020 rolls of the die, you observe a "one" 22 times. What proportion of the rolls resulted in a "one"?
Answers
Answer:
(a) 4.356% (b) 1.089%
Step-by-step explanation:
The rolling of a single die is an example of a discrete uniform distribution, where each of the six possible outcomes has an equal probability of 1/6.
(a) After 1010 rolls of the die, observing a "one" 44 times, the proportion of rolls resulting in a "one" is:
Proportion = Number of "ones" / Total number of rolls
= 44 / 1010
= 0.04356
Therefore, approximately 4.356% of the rolls resulted in a "one".
(b) After 2020 rolls of the die, observing a "one" 22 times, the proportion of rolls resulting in a "one" is:
Proportion = Number of "ones" / Total number of rolls
= 22 / 2020
= 0.01089
Therefore, approximately 1.089% of the rolls resulted in a "one".
Answer:
A) 40%
B) 10%
Step-by-step explanation:
A) p = 4/10 = 40%
B) p = 2/20 = 10%
Exercise
find the truth set of the
1.5(3x-5)-2(x-7)=11
Answers
a random sample of 50 employees at a large firm has 22 females and 28 males. if the company has 1300 workers, what is the expected number of male workers?
Answers
Step-by-step explanation:
We can use the proportion of males in the sample to estimate the proportion of males in the population, and then use that proportion to find the expected number of male workers.
The proportion of males in the sample is:
p = 28/50 = 0.56
We can use this proportion to estimate the proportion of males in the population:
p' = 0.56
The expected number of male workers is then:
E(X) = n * p'
where n is the total number of workers in the population:
E(X) = 1300 * 0.56
E(X) = 728
Therefore, we would expect there to be approximately 728 male workers in the company.
please help me!!!!!!
Answers
Answer:2nd one
Step-by-step explanation:
Elnaz rewrote the equation x² - 6x - 18 = 0 so that she can solve by completing the square. Which equation did she write?
A. (x-3)² = -9
B. (x-3)² = 18
C. (x-3)² = 27
D. (x-3)² = 54
Answers
Answer:
C
Step-by-step explanation:
(x-3)(x-3)=27
x^2-3x-3x+9=27
x^2-6x+9=27
-27. -27
x^2-6x-18=0
Answer:
(x-3)²=27
Step-by-step explanation:
x²+6x-18=0
add 18 to both sides of the equation
To complete the square add 3² to both sides and factor what is on the left side. You will be getting
(x-3)² = 27 is the answer to this problem
Answer:
C
Step-by-step explanation:
(x-3)(x-3)=27
x^2-3x-3x+9=27
x^2-6x+9=27
-27. -27
x^2-6x-18=0
Answer:
(x-3)²=27
Step-by-step explanation:
x²+6x-18=0
add 18 to both sides of the equation
To complete the square add 3² to both sides and factor what is on the left side. You will be getting
(x-3)² = 27 is the answer to this problem
What is the length of side b? (Hint: There are 3 sides and 1 angle)
Answers
The length of side b is given as follows:
b = 2.86 m.
What is the law of sines?Suppose we have a triangle in which:
Side with a length of a is opposite to angle A.Side with a length of b is opposite to angle B.Side with a length of c is opposite to angle C.The lengths and the sine of the angles are related as follows:
[tex]\frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c}[/tex]
The sum of the measures of the internal angles of a triangle is of 180º, hence the missing angle is obtained as follows:
B + 30 + 80 = 180
B = 70º.
Then the relation, by the law of sines, is given as follows:
sin(70º)/b = sin(80º)/3
Meaning that the length of side b is given as follows:
b = 3 x sine of 70 degrees/sine of 80 degrees
b = 2.86 m.
Missing InformationThe triangle is given by the image presented at the end of the answer.
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Supplementary angles
Answers
Answer: ∠EAI and/or ∠EDG (More listed below lol)
Step-by-step explanation:
Starting Angle: ∠ADE
Possible Supplements: ∠EAI, ∠EDG, ∠DAH, ∠DEC, ∠ADF, and ∠AEB
Practice: Tell and Write Time-Practice-Level C
Determine if each statement describes the time shown on the clock.
i-Ready
......................
10
9
8
12
11
76
1
5
2
4
3:
(16 minutes before 10:00
»44 minutes after 9:00
Yes
(47 minutes after 8:00
No
» 13 minutes before 10:00 O
13 minutes before 9:00
O
O
00
O
O
O
O
Answers
7:76 - This is not a valid time. There are only 60 minutes in an hour, so the minutes place cannot be 76.
What is time?Time is a measurable and non-spatial quantity used to describe or quantify the sequence, duration, or occurrence of events. It is a concept used to understand the order in which events occur and to measure the duration of those events.
According to question:10:44 - Yes, the statement is true. The time is 44 minutes after 10:00.
9:47 - No, the statement is false. The time is 13 minutes before 10:00.
12:13 - Yes, the statement is true. The time is 13 minutes after 12:00.
11:00 - Yes, the statement is true. The time is exactly 11:00.
7:76 - This is not a valid time. There are only 60 minutes in an hour, so the minutes place cannot be 76.
1:05 - Yes, the statement is true. The time is 5 minutes after 1:00.
5:02 - No information is provided about this time on the clock.
2:00 - Yes, the statement is true. The time is exactly 2:00.
4:00 - Yes, the statement is true. The time is exactly 4:00.
3:00 - Yes, the statement is true. The time is exactly 3:00.
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A pool measuring 18 meters by 30 meters is surrounded by a path of uniform width, as shown in the
figure. If the area of the pool and the path combined is 1900 square meters, what is the width of
the path?
Answers
The width of the path is approximately 7.45 meters.
What is meant by width?
The width refers to the measurement of something from side to side, typically in a direction perpendicular to its length or height, such as the width of a door or the width of a piece of paper.
What is meant by meters?
Meters are a unit of measurement used to measure the length in the metric system. One meter is equal to 100 centimetres or approximately 3.28 feet. It is commonly used to measure distances, heights, and lengths.
According to the given information
The area of the pool is given by the product of its length and width:
18 × 30 = 540 square meters
The area of the entire region, including the pool and the path, is given by the product of the length and width of the outer rectangle:
(18 + 2x) × (30 + 2x)
Expanding this expression, we get:
540 + 36x + 60x + 4x²
Simplifying, we get:
4x² + 96x + 540
We are given that the area of the entire region is 1900 square meters:
4x² + 96x + 540 = 1900
Subtracting 1900 from both sides, we get:
4x² + 96x - 1360 = 0
Dividing both sides by 4, we get:
x² + 24x - 340 = 0
We can use the quadratic formula to solve for x:
x = (-24 ± √(24² - 4 × 1 × (-340))) / (2 × 1)
Simplifying, we get:
x = (-24 ± √(1216)) / 2
x ≈ -17.45 or x ≈ 7.45
Since the width of the path cannot be negative, we can ignore the negative solution.
Therefore, the width of the path is approximately 7.45 meters.
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Please see the attached
Answers
a. Monthly payment for the bank's car loan is $407.67
b. Monthly payment for the savings and loan association's car loan is $315.99
c. Total amount paid would be $2,035 less for the bank's car loan than for the savings and loan association's car loan.
What is interest rate?The cost of borrowing money, usually expressed as a percentage of the amount borrowed, is what a lender charges a borrower to use their money. This cost is known as an interest rate.
(a) To find the monthly payment for the bank's car loan, we can use the formula for the present value of an annuity:
[tex]PV = PMT * \frac{1 - (1 + \frac{r}{n})^{(-n*t)}}{\frac{r}{n} }[/tex]
putting the given values,
⇒ [tex]21000 = PMT * \frac{1 - (1 + \frac{0.065}{12})^{(-12*5)}}{\frac{0.065}{12} }[/tex]
Solving for PMT, we get:
PMT = $407.67
Therefore, the monthly payment for the bank's car loan is $407.67
(b) To find the monthly payment for the savings and loan association's car loan, we can use the same above formula:
where PV is still $21,000, PMT is the monthly payment, r is still 0.065, n is still 12, but t is now 7 years x 12 months/year = 84 payments.
putting the given values,
[tex]21000 = PMT * \frac{1 - (1 + \frac{0.065}{12})^{(-12*7)}}{\frac{0.065}{12} }[/tex]
Solving for PMT, we get:
PMT = $315.99
Therefore, the monthly payment for the savings and loan association's car loan is $315.99.
(c) Bank's car loan: $407.67 x 60 = $24,460.20
Savings and loan association's car loan: $315.99 x 84 = $26,495.16
Therefore, the bank's car loan would have the lowest total amount to pay off, by: $26,495.16 - $24,460.20 = $2,034.96
Therefore, the total amount paid would be $2,035 less for the bank's car loan than for the savings and loan association's car loan.
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You want to buy a $12,000 car. The company is offering a 2% interest rate for 48 months (4 years). What
will your monthly payments be?
Answers
Answer:
240
Step-by-step explanation:
T
2
√
k
aº
90°
Special Right Triangles
2
If the diagonal of a square produces an isosceles right
triangle, then you should be able to find the measures
of the base angles.
What is the measure of angles a and b?
angle
a
b
measure
check
X
X
Answers
The base angles a and b have their measures to be 45 degrees each
Calculating the base angles a and b?If the diagonal of a square produces an isosceles right triangle, then the two legs of the right triangle are congruent, meaning the two base angles are congruent as well.
Since the triangle is a right triangle, one of the base angles must be 45 degrees (which is half of the right angle).
Therefore, both base angles are 45 degrees.
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find the area of the shaded region round to the nearest hundredth where necessary 23.8 21 15
Answers
First, we need to find the radii of the two semicircles. The larger semicircle has a diameter of 23.8, so its radius is 23.8/2 = 11.9. The smaller semicircle has a diameter of 15, so its radius is 15/2 = 7.5.
The area of the larger semicircle is (π/2)(11.9)^2 ≈ 222.93 square units.
The area of the smaller semicircle is (π/2)(7.5)^2 ≈ 44.18 square units.
Next, we need to find the area of the triangle. We can use the Pythagorean theorem to find the height of the triangle:
h^2 = 11.9^2 - 7.5^2
h^2 ≈ 72.16
h ≈ 8.49
The base of the triangle is 21, so the area of the triangle is (1/2)(21)(8.49) ≈ 89.29 square units.
Finally, we can find the area of the shaded region by subtracting the area of the smaller semicircle from the area of the larger semicircle, and then subtracting the area of the triangle:
222.93 - 44.18 - 89.29 ≈ 89.46
Therefore, the area of the shaded region is approximately 89.46 square units, rounded to the nearest hundredth.
Prepare a statement of cash flows for Business Solutions using the indirect method for the three months ended March 31, 2022. Owner Santana Rey contributed $25,000 to the business in exchange for additional stock in the first quarter of 2022 and has received $4,800 in cash dividends.
Answers
A hiker travels 6 mph, which is 3/11 of the rate of a cyclist.Suppose the hiker and cyclist are traveling toward each other.How fast does the distance between them decrease?
Answers
Answer:
28 mph
Step-by-step explanation:
let x= rate of cyclist
3/11(x)=6
x=66/3=22 mph
22mph+ 6mph=28 mph
they move towards each other with a relative speed of 28 mph
The lateral area of a cone is 473pi cm^2. The radius is 43 cm. Find the height to the nearest tenth
Answers
The slant height of the cone is about 11 cm. Using the Pythagorean theorem, the height is approximately 41.4 cm to the nearest tenth.
Use the formula for the lateral area of a cone: L = πrs, where L is the lateral area, r is the radius of the base, and s is the slant height.
Plug in the given values: L = 473π cm² and r = 43 cm. Then solve for s
L = πrs
473π = π(43)s
s = 473/43 ≈ 11
So the slant height of the cone is approximately 11 cm.
Use the Pythagorean theorem to find the height of the cone. Let h be the height of the cone. Then the Pythagorean theorem gives
h² + s² = r²
Substituting in the values for s and r, we get
h² + 11² = 43²
Simplifying this equation, we get
h² = 43² - 11²
Evaluating the right-hand side, we get
h² = 1764
Taking the square root of both sides, we get
h ≈ 41.4
So the height of the cone is approximately 41.4 cm to the nearest tenth.
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On Friday, three friends shared how much they read during the week
Barbara read the first 100 pages from a 320-page in the last 4 days
Judy read the first 54 pages from a 260-page book in the last 3 days.
Nancy read the first 160 pages from a 480-page book in the last 5 days
Order the friends from the first one who is predicted to finish her book to the third one who is predicted to finish her book(Show all work)
Answers
The friends from the first one who is predicted to finish her book to the third one who is predicted to finish her book is given by the order Nancy > Barbara > Judy
Given data ,
The total number of pages in each friend's book as follows:
Barbara's book: 320 pages
Judy's book: 260 pages
Nancy's book: 480 pages
Now, we can calculate their reading rates as pages read per day:
Barbara's reading rate: 100 pages / 4 days = 25 pages/day
Judy's reading rate: 54 pages / 3 days = 18 pages/day
Nancy's reading rate: 160 pages / 5 days = 32 pages/day
Hence , the descending order is Nancy > Barbara > Judy
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1. (i) Find the first 3 terms in the expansion of (2 - y)^5 in ascending powers of y.
(ii) Use the result in part (i) to find the coefficient of x² in the expansion of (2 − (2x - x²))^5.
Answers
1) The first three terms in the expansion of [tex](2 - y)^{5}[/tex] in ascending powers of y are as follows:
32, -80y, and [tex]80y^2[/tex]
2)The [tex]x^{2}[/tex] coefficient in the expansion of (2 (2x - [tex]x^{2}[/tex]))5 is 80.
What is the binomial theorem?As the power increases, the expansion gets more difficult to compute. The Binomial Theorem can be used to easily calculate a binomial statement that has been raised to very big power.
(i) Using the binomial theorem, we can expand [tex](2 - y)^{5}[/tex] as follows:
[tex](2 - y)^{5}[/tex] = [tex]2^{5} - 5(24)y^{1} + 10(23)y^{2} + 10(23)y^{3} + 5(2)y^{4} - y^{5}[/tex]
By combining the powers of two and simplifying, we get:
[tex]32 - 80y + 80y^{2} - 40y^3 + 10y^4 - y^5[/tex]
As a result, the first three terms in the expansion of [tex](2 - y)^{5}[/tex] in ascending powers of y are as follows:
32, -80y, and [tex]80y^2[/tex]
(ii) Using the result of section (i), we can expand [tex](2 - (2x - x^{2} ))^{5}[/tex] by substituting y with 2x - x2:
[tex](2 - (2x - x^2))^5 = 2^5 - 5(2^4)(2x - x^2) + 10(2^3)(2x - x^2)^2 - 10(2^2)(2x - x^2)^3 + 5(2)(2x - x^2)^4 - (2x - x^2)^5[/tex]
By combining the powers of two and simplifying, we get:
[tex]32 - 80x + 80x^2 - 40x^3 + 10x^4 - x^5[/tex]
As a result, the [tex]x^{2}[/tex] coefficient in the expansion of (2 (2x - [tex]x^{2}[/tex]))5 is 80.
Learn more about the expansion of power here:
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