An triangle has vertices
as shown in the diagram. It has been translated vertically and/or horizontally to the triangle with vertices labelled
and
using a translation matrix defined as
.

A coordinate system has an unlabeled horizontal x-axis and an unlabeled vertical y-axis. There are 2 right triangles. The first triangle has vertices labeled A, B and C. Point A is 2 units to the left of the origin, point B is at the origin, and point C is 1 unit above the origin. The second triangle has vertices labeled A’, B’, and C’. Point A’ is 1 unit below the origin. Point B’ is 2 units to the right and one unit below the origin. Point C’ is 2 units right of the origin.
(The choices and pic are attached.)

An Triangle Has Vertices As Shown In The Diagram. It Has Been Translated Vertically And/or Horizontally
An Triangle Has Vertices As Shown In The Diagram. It Has Been Translated Vertically And/or Horizontally

Answers

Answer 1

The translation matrix is,

[tex]\left[\begin{array}{ccc}2&2&2\\-1&-1&-1\end{array}\right][/tex]

How to solve

The coordinates of the triangle as a coordinate matrix is

-2 0 0

0 0 1.

Here the triangle has been translated 2 units right and 1 unit down.

Translating the triangle 2 units to the right means that each x -coordinate increases by 2.

Translating the triangle 1 unit down means that each y -coordinate decreases by 1.

So the translation matrix is,

[tex]\left[\begin{array}{ccc}2&2&2\\-1&-1&-1\end{array}\right][/tex]

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Related Questions

Jorge bought a crate of floor tiles for $95.94. The crate had 6 boxes of floor tiles. Each box contained 20 floor tiles.
Write and solve an equation to determine the cost per box, b. Then write and solve a second equation to determine the cost per tile, t, to the nearest cent. Show your work.

HELP!!

Answers

The solution is:

⇒ 6b = 95.94   (equation to determine cost of one box)

cost of one box 'b' = $`15.99

⇒ 12t = 95.94   (equation to determine cost of per tile)

cost of one tile t = $0.7995.

Given :

Jorge bought a crate of floor tiles for $95.94.

The crate had 6 boxes of floor tiles.

Each box contained 20 floor tiles .

To Find :

Write and solve equation to determine the cost per box'b'.

Write and solve a second equation to determine the cost per tile't'

Solution :

Cost of one box = b

There are 6 boxes

So, cost of 6 boxes = $  6b

Since Jorge bought 1 crate( = 6 boxes) of cost $95.94

⇒     (equation to determine cost of one box)

⇒6b = 95.94

⇒b =15.99

Thus cost of one box = $`15.99

Since 1 box 20 floor tiles

So, 6 boxes (=1 crate) contain tiles = 6*20 = 120 tiles

We are given that cost of 1 crate( = 6 boxes = 120 tiles) is $95.94

Cost of one tile = t

Cost of 120 tiles = $120t

⇒ (equation to determine cost of per tile)

⇒12t = 95.94

⇒t = 0.7995.

Thus cost of one tile t = $0.7995.

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Nvm I found the answer

Answers

I’m just writing this to get the points

the state of ohio conducted covid tests on 64585 people, and 9459 tested positive, what is the numaber that tested positive

Answers

1402800 is the number of people who would test positive for the virus

To find the probability for the number who tested positive we divide the positive cases between the total of the covid tests and then mutiply by 100.

P = (8,533/68,811) × 100

P = 12.400, the nearest hundredth of a percent is 12.

11690000 people who live in ohio and the testing results is that 12% of this people would test positive for the virus.

so we need to find percentage 12 for 11690000.

(11690000×12) / 100

Hence, 1402800 is the number of people who would test positive for the virus

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The state of Ohio conducted COVID 19 tests on 68,811 people, and 8,533 tested positive. (6 points)A. What is p for the number who tested positive (round to the nearest hundredth of a percent)?B. Based upon the testing results above, of the 11.69 million people who live in Ohio, what is the bestestimate for the number of people who would test positive for the virus?

Some steps to construct an angle MNT congruent to angle PQR are listed below.

Step 1: Use a compass to draw an arc from point Q that intersects the side PQ at point A and the side QR at point B.
Step 2: Draw a segment NT, and use the same width of the compass to draw an arc from point N that intersects the segment NT at a point X.
Step 3:
Step 4: Join points N and Y using a straightedge.

Which statement describes step 3 correctly?

Adjust the width of the compass to AB, and draw an arc from point X such that it intersects the arc drawn from N in a point Y.
Adjust the width of the compass to AQ, and draw an arc from point X such that it intersects the arc drawn from N in a point Y.
Adjust the width of the compass to BQ, and draw an arc from point X such that it intersects the arc drawn from N in a point Y.
Adjust the width of the compass to NX, and draw an arc from point X such that it intersects the arc drawn from N in a point Y.

Answers

Option (A) "Adjust the width of the compass to AB, and draw an arc from point X such that it intersects the arc drawn from N in a point Y" is correct.

We have,

When two lines or rays converge at the same point, the measurement between them is called a "Angle."

We have a statement:

Steps to construct an angle MNT congruent to angle PQR.

To construct an angle MNT congruent to angle PQR, the steps are as follows:

Step 1: Use a compass to draw an arc from point Q that intersects the side PQ at point A and the side QR at point B.

Step 2: Draw a segment NT, and use the same width of the compass to draw an arc from point N that intersects the segment NT at a point X.

Step 3: Adjust the width of the compass to AB, and draw an arc from point X such that it intersects the arc drawn from N in a point Y.

Step 4: Join points N and Y using a straightedge.

Thus, option (A) "Adjust the width of the compass to AB, and draw an arc from point X such that it intersects the arc drawn from N in a point Y" is correct.

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Fill in the blanks in the equation
Pls

Answers

The blanks are filled using the concept of translation transformation  as follows

Parabola            Equation

Red                       y = x²

Blue                     y = x² + 3

Green                  y = (x - 3)²

Orange                y = x² - 3

Purple                  y = (x + 3)²

How to determine the graphs

Applying translation the graphs are interpreted as follows

y = x²: the parent function

y = x² + 3: shift 3 units up

y = (x - 3)²: shift to the right 3 units

y = x² - 3: shift 3 units down

y = (x + 3)²: a shift 3 units to the left

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There are 6 acts in a talent show.
An acrobat, a dancer, a guitarist, a singer, a violinist, and a whistler.
A talent show host randomly schedules the 6 acts.
Compute the probability of each of the following events.
Event A: The acrobat is first, the singer is second, the violinist is third, and the whistler is fourth.
Event B: The first four acts are the whistler, the acrobat, the guitarist, and the dancer, In any order.
Write your answers as fractions in simplest form.
P(A) =
P (B) =

Answers

The probability of Event A is 1/360 and the probability of Event B is 1/15.

For Event A, we can calculate the probability by considering the order in which the acts are scheduled.

There are 6 acts, so there are 6 possible choices for the first act. After the first act is scheduled, there are 5 remaining acts, so there are 5 possible choices for the second act.

Similarly, there are 4 possible choices for the third act, and 3 possible choices for the fourth act.

Finally, there are 2 possible choices for the fifth act, and only 1 choice remains for the last act.

Therefore, the total number of possible ways to schedule all 6 acts is:

6 x 5 x 4 x 3 x 2 x 1 = 720

Since we are only interested in one specific order of the first four acts, we need to count the number of ways to schedule the remaining two acts after the first four have been scheduled in the desired order.

There are 2 remaining acts, so there are 2 possible choices for the fifth act.

Only one act remains for the last slot.

Therefore, the total number of ways to schedule all 6 acts such that the acrobat is first, the singer is second, the violinist is third, and the whistler is fourth is:

1 x 1 x 1 x 1 x 2 x 1 = 2

Thus, the probability of Event A is:

P(A) = 2/720 = 1/360

For Event B, we want to count the number of ways to schedule the first four acts as the whistler, the acrobat, the guitarist, and the dancer, in any order.

There are 4 acts, so there are 4 possible choices for the first act. After the first act is scheduled, there are 3 remaining acts, so there are 3 possible choices for the second act.

Similarly, there are 2 possible choices for the third act, and only one choice remains for the fourth act.

Therefore, the total number of possible ways to schedule the first four acts is:

4 x 3 x 2 x 1 = 24

Since we do not care about the order in which the last two acts are scheduled, we can simply choose any two acts from the remaining 2. There are 2 ways to do this.

Therefore, the total number of ways to schedule all 6 acts such that the first four acts are the whistler, the acrobat, the guitarist, and the dancer, in any order is:

24 x 2 = 48

Thus, the probability of Event B is:

P(B) = 48/720 = 1/15

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A carnival game is designed so that approximately 10% of players will win a large prize. If there is evidence that the percentage differs significantly from this target, then adjustments will be made to the game. To investigate, a random sample of 100 players is selected from the large population of all players. Of these players, 19 win a large prize. The question of interest is whether the data provide convincing evidence that the true proportion of players who win this game differs from 0.10. Are the conditions for inference met for conducting a z-test for one proportion?

A) Yes, the random, 10%, and large counts conditions are all met.
B) No, the random condition is not met.
C) No, the 10% condition is not met.
D) No, the large counts condition is not met.

Answers

Yes, the random, 10%, and large counts conditions are all met.

Here, the expected count of players who win a large prize is

np = 100 x 0.10

np = 10

and, the expected count of players who do not win a large prize is

n(1-p) = 100 x 0.90 = 90.

The second prerequisite is also satisfied because both of these anticipated counts are higher than or equal to 10.

The independence criteria is also satisfied if each player's outcome is independent of the results of the other players.

Therefore, Yes, the random, 10%, and large counts conditions are all met.

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The owner of a sporting goods store buys pairs of rollerblades for $60 and marks them up 25%. Several months later, he decides to clear his inventory and sells each pair of rollerblades at a discount of 20%. What is the total price of a pair of these rollerblades with the discount and a 6% sales tax?
Show your work.

Answers

The total price of the rollerblades with the discount and a 6% sales tax is $59.63.

Given that a sporting goods store buys pairs of rollerblades for $60 and marks them up 25%.

Then he decides to clear his inventory and sells each pair of rollerblades at a discount of 20%.

Initial Price = $60

Markup = 25% x $60 = 0.25 x 60 = $15

Price after Markup = $60 + $15 = $75

Discount = 20% x $75 = 0.25 x 70 = $18.75

Price after Discount = $75 - $18.75 = $56.25

Sales Tax = 6% x $56.25 - 0.06 x 56.25 = $3.38

Price = $56.25 + $3.38 = $59.63

Hence the total price of the rollerblades with the discount and a 6% sales tax is $59.63.

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the sum of four number is 20500.three of the numbers are 2341.578 and 10690.What is the fourth number?​

Answers

Answer:

5216.844

Step-by-step explanation:

Let x be the fourth number.

We know that the sum of the four numbers is 20500:

2341.578 + 10690 + x + 2341.578 = 20500

Simplifying and solving for x:

x = 20500 - 2341.578 - 10690 - 2341.578

x = 5216.844

The base of a triangle is 1 cm more than the height. If the area is 10 cm², find the base and the height.
Part 1 of 2
What is the base of the triangle?

What is the height of the triangle?

Answers

Answer:

Base of triangle = 5 cmHeight of triangle = 4 cm.

Step-by-step explanation:

It's given that, The base of a triangle is 1 cm more than the height

Let's assume height of the triangle be x and base of the triangle be 1 + x

Also the area of the triangle is 10 cm².

We know that area of triangle is calculated by,

Area = 1/2 × Base × height

Putting the values we get ,

[tex] \frac{1}{2} (1 + x) x = 10 \\ \\ (1 + x)x = 10 \times 2 \\ \\ {x}^{2} + x = 20 \\ \\ {x}^{2} + x - 20 = 0[/tex]

Further solving by factorisation,

[tex]x^2 + x-20 = 0 \\ \\ x^2 +5x-4x - 20 = 0 \\ \\ x(x+5)-4(x+5)= 0 \\ \\ (x-4) (x+5) = 0 \\ \\ (x-4) = 0 \: or \: (x+5)=0 \\ \\ { \underline{x= 4 \: or \: x = 5 }}[/tex]

Since value can't be negative.

So, x = 4.

Hence, height of triangle x = 4 cm

base of triangle = x + 1

= 4 + 1

= 5 cm

Therefore the Base of traingle will be 5 cm and Height of the triangle will be 4 cm.

Find all three primary trigonometric ratios for the mentioned angle.
ZR
P
sinR
cos R
tan R
15
12
Q
R
9

Answers

The ratios of the identities are;

sin R = 9/15

cos R = 12/15

tan R = 9/15

How to determine the identities

It is important that we note the different trigonometric identities and their corresponding ratios.

From the information given, we get;

sin θ = opposite/hypotenuse

cos θ = adjacent/hypotenuse

tan θ = opposite/adjacent

From the information given, we have the parameters given as;

Hypotenuse side = 15

Opposite side = 9

Adjacent side = 12

Now, substitute the values, we have;

sin R = 9/15

cos R = 12/15

tan R = 9/15

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Find the value of x for which the lines l and m are parallel.

A) 13

B) 10

C) 4.7

D) 5

Answers

Answer:

A) 13

Step-by-step explanation:

The two given angle measurements are corresponding angles. By definition of corresponding angles, the two given angle measurements are equal. Set the two measurements equal to each other:

[tex]9x + 10 = 127[/tex]

Isolate the variable, x. Note the equal sign, what you do to one side, you do to the other. Do the opposite of PEMDAS.

PEMDAS is the order of operations, and stands for:

Parenthesis

Exponents (& roots)

Multiplication

Division

Addition

Subtraction

~

First, subtract 10 from both sides of the equation:

[tex]9x + 10 = 127 \\9x + 10 (-10) = 127 (-10)\\9x = 127 - 10\\9x = 117[/tex]

Next, divide 9 from both sides of the equation:

[tex]9x = 117\\\frac{(9x)}{9} = \frac{(117)}{9}\\ x = \frac{117}{9}[/tex]

Simplify:

[tex]x = \frac{117}{9}\\ x = 13[/tex]

~

A) 13 is your answer for x.

~

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The question is in the picture. Thank you

Answers

The square root of 158 is around 12.57.
So the answer is A. 12.5 is the closest.

Select the correct answer. Triangle ABC is reflected across the y-axis and then dilated by a factor of 1/2 centered at the origin. Which statement correctly describes the resulting image, triangle ? Image of right triangle ABC in a graph with vertices A (minus 4, minus 2), B (4, minus 2), and C (4, 4) A. Neither the reflection nor the dilation preserves the side lengths and angles of triangle ABC . B. The reflection preserves the side lengths and angles of triangle . The dilation preserves angles but not side lengths. C. Both the reflection and dilation preserve the side lengths and angles of triangle ABC . D. The dilation preserves the side lengths and angles of triangle . The reflection does not preserve side lengths and angles.

Answers

The reflection preserves the side lengths and angles of triangle. The dilation preserves angles but not side lengths.

The triangle underwent the following operations:

The triangle's coordinates shift from (x, y) to (-x, y) when the y-axis is reflected, but the side lengths and angles remain the same.The triangle's side length is increased when it is expanded by a factor and then centred at the origin, but the angles in the triangle stay unchanged because the ratio of the sides before and after expansion is the same.

As a result, while the dilation would only maintain the angles, the reflection would preserve both side lengths and angles.

Thus, Angles are preserved, but side lengths are not.

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A doctor wants to estimate the mean HDL cholesterol of all 20-to 29-year-old females How many subjects are needed to estimate the mean HDL cholesterol within 4 points with 99% confidence assuming s=17.8 based on earlier studes? Suppose the doctor would be content with 90% confidence. How does the decrease in confidence affect the sample size required?

REFER TO PHOTO to view a partial table of critical values
A 99% confidence level requires ___ subjects. (Round up to the nearest subject)


B.) 90% confidence level requires _____ subjects. (Round up to the nearest subject.)

C. How does the decrease in confidence affect the sample size required?

A. The sample size is the same for all levels or confidence .
B. Decreasing the confidence level decreases the sample size needed*
C. Decreasing the confidence level increases the sample size needed

Answers

A) For 99% confidence, the critical value (from the table) is approximately 2.576.

48 subjects are required for a 99% confidence level. (Round up to the nearest subject)

B) For 90% confidence, the critical value (from the table) is approximately 1.645.

19 subjects are required for a 90% confidence level. (Round up to the nearest subject)

C) Decreasing the confidence level decreases the sample size needed (option B).

What is a Confidence Interval?

A confidence interval is a range of estimates for an unknown parameter in frequentist statistics. The 95% confidence level is the most usual, but other levels, such 90% or 99%, are occasionally used to generate a confidence range.

How to solve

z for 99%=2.58

z for 95%=1.96

a)sample size= [tex](z*s/E)^2=(2.58*12.6/4)^2=66[/tex]

b)sample size= [tex](z*s/E)^2=(1.96*12.6/4)^2=38[/tex]

decreasing the confidence level decreases the sample size required

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The hypotenuse of a right angle triangle Is 10 units and one of the acute angles is 40

Answers

The measurement of other two legs of the right triangle is 6.4 and 7.6.

Given is right triangle with hypotenuse equal to 10 units and an acute angle of 40°, we need to find the other two legs,

Here we will use the trigonometric ratios to find the legs,

Let 40° be the reference angle,

So,

Sin 40° = perpendicular / 10

Perpendicular = 6.4

Cos 40° = base / 10

Base = 7.6

Hence the measurement of other two legs of the right triangle is 6.4 and 7.6.

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THE DIMENSIONS OF A RECTANGULAR PRISM ARE 1.5 FEET BY 3.5 FEET BY 2 FEET. WHAT IS THE VOLUME OF THE RECTANGULAR PRIMS IN CUBIC FEET?

Answers

Answer:

10.5 cubic feet

(10.5ft³)

Step-by-step explanation:

1.5 x 3.5 = 5.25

5.25 x 2 = 10.5ft³

The mean of 5 numbers is 9.
The numbers are in the ratio 1 : 1 : 2 : 2 : 3.
Find the numbers.

Answers

The numbers are 5, 5, 10, 10, 15

To find the value of x, we need to use the fact that the mean of the five numbers is 9. The mean is equal to the sum of all the numbers divided by the total number of numbers. In this case, we have:

(mean) = (sum of all numbers) / (total number of numbers)

We know the mean is 9, and there are 5 numbers, so we can write:

9 = (x + x + 2x + 2x + 3x) / 5

Simplifying the right-hand side, we get:

9 = 9x / 5

Multiplying both sides by 5, we get:

45 = 9x

Dividing both sides by 9, we get:

5 = x

So we now know that the first two numbers are both equal to 5. Using the ratio, we can find the values of the next two numbers:

2x = 2(5) = 10

So the third and fourth numbers are both equal to 10.

Finally, we can use the ratio to find the value of the fifth number:

3x = 3(5) = 15

So the five numbers are:

5, 5, 10, 10, 15

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PLEASE PLEASE HELP WILL MARK BRAINLIST!!!!!!!!!!!!

Answers

The function rule for g(x) is h(x) = (x - 0)² - 7.

How to determine the vertex form of a quadratic equation?

In Mathematics, the vertex form of a quadratic function is represented by the following mathematical equation:

f(x) = a(x - h)² + k

Where:

h and k represents the vertex of the graph.a represents the leading coefficient.

Based on the information provided about the vertex (0, -7) and other points (-3, 2), we can determine the value of a as follows:

f(x) = a(x - h)² + k

2 = a(-3 - 0)² - 7

2 + 7 = 9a

9 = 9a

a = 1

Therefore, the required quadratic function in vertex form is given by:

h(x) = a(x - h)² + k

h(x) = (x - 0)² - 7

h(x) = x² - 7

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The square base of the figure shown has a side length of 10 millimeters (mm) where the overall height of the figure is 17.25 mm.
10 mm
10 mm
17.25 mm

Answers

Answer:

1617.5mm^3

Step-by-step explanation:

The square base with a side length of 10 mm and an overall height of 17.25 mm.


Based on your question, it seems that you have a figure with a square base and a height.

The extent or potential of a rectangular pyramid can be calculated the usage of the formula:

V = (1/3)abH

Where "a" and "b" are the size and width of the rectangular base, and "H" is the top of the pyramid.

Similarly, the floor vicinity of a rectangular pyramid can be calculated the use of the formula:

A = ab + (1/2)P(lateral)

where "P(lateral)" is the perimeter of the lateral face, which can be calculated as the sum of the slant heights of the 4 triangular faces.

The slant top can be calculated the use of the Pythagorean theorem, the place the base of the triangle is one facet of the rectangular base, the top is the top of the pyramid, and the hypotenuse is the slant height.

To provide a relevant answer,

I need to know the type of figure you are referring to (e.g., pyramid, prism). Once you provide that information.

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Which of the following represents a possible restricted domain on which

Answers

The domain of f⁻¹(y) is all real numbers greater than or equal to 4. Therefore, option B is the correct answer.

The domain of a function is the set of all possible input values for which the function is defined. In this case, the domain of f(x) is all real numbers, since there are no restrictions given in the problem.

To find the inverse of f(x), we need to solve for x in terms of y.

f(x) = (x-4)^2 + 7

y = (x-4)² + 7

y - 7 = (x-4)²

√(y-7) = x-4

x = √(y-7)+4

The inverse of f(x) is:

f⁻¹(y)=√(y-7)+4

Therefore, option B is the correct answer.

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Answer:

x less than or equal to 4 option c

Step-by-step explanation:

took the test

PLS HELP!!! MARKING AS BRAINLIST PLEASE

Answers

Answer:

P = (1^2)/(2^2) = 1/4 = .25 = 25%

A deli provides sandwich wraps for a school group on a field trip. Each wrap contains one meat, either chicken or turkey, and one vegetable either spinach or tomatoes. What outcomes are possible if a student randomly selects a sandwich wrap?

What outcomes are possible for the meat in a randomly selects wrap? For the vegetable?

Answers

Answer:

There are four possible outcomes for a student who randomly selects a sandwich wrap: chicken and spinach, chicken and tomatoes, turkey and spinach, or turkey and tomatoes. There are two possible outcomes for the meat in a randomly selected wrap: chicken or turkey. There are two possible outcomes for the vegetable in a randomly selected wrap: spinach or tomatoes.

Step-by-step explanation:

help me please Which is an appropriate statement about angles 3 and 4 in the diagram below?
A) Angles 3 and 4 are complementary angles.
B) Angle 3 and 4 are congruent angles.
C) Angle 4 is greater than angle 3.
D) Angle 3 is greater than angle 4.

Answers

Answer:

A) Angles 3 and 4 are complementary angles.

1. Mrs. Hamilton worked for a real estate agency. She sold a house for
$175,000. The agency pays commission of 4% of the sale price. How much
commission would Mrs. Hamilton be paid?

Answers

Answer: $7,000.

Step-by-step explanation:

4% = 0.04. Multiply that by the sales price to calculate commission.

$175,000 x 0.04 = $7,000.

I need to know the domain of the first equation

Answers

Answer:

d: (0, pi)

r: [-1/2, 1/2]

Step-by-step explanation:

how does the fraction of millennials with at least 100,000 in retirement compared to the portion of gen x people who have no retirement savings?

A. A lower percentage of millennials with at least 100.000 saved for retirement when comparing to Gen X

B. A higher percentage of millennials have at least 100.000 saved for retirement when comparing to Gen X.

C. MILLENNIALS and Gen X have the same amount of people who have at-least 100.000 invested for retirement.

Answers

Based on available research and studies, the answer is B: A higher percentage of millennials have at least $100,000 saved for retirement when comparing to Gen X.

How to explain the information

According to a report from Fidelity Investments, 30% of millennials have saved at least $100,000 for retirement. In contrast, a 2019 study from the Transamerica Center for Retirement Studies found that 31% of Gen Xers have no retirement savings at all.

This suggests that the fraction of millennials with at least $100,000 saved for retirement is higher than the portion of Gen X people who have no retirement savings.

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What is the exact area of a regular hexagon with a side length of 8 cm?
Enter your answer in the box.
cm²
URGENTTTT

Answers

The area of the regular hexagon is A = 166.28 cm²

Given data ,

Let the side length of the hexagon be a = 8 cm

Now , area of hexagon is A = ( 3√3/2 )a²

On simplifying , we get

A = ( 1.5 ) ( √3 ) ( 64 )

A = 96√3

The value of A = 96√3 cm²

On further simplification , we get

A = 166.28 cm²

Therefore , the value of A is A = 166.28 cm²

Hence , the area is A = 166.28 cm²

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determine the maximum and minimum values of the function, at what values of x are they achieved? (without using a derivative)
[tex]y=\sin^3x-\sin^6x[/tex]

Answers

The maximum and minimum values of the function is solved

Given data ,

We can find the maximum and minimum values of the function by taking the derivative of y with respect to x and setting it equal to zero.

y = (sin x)³ - (sin x)⁶

y' = 3(sin x)² cos x - 6(sin x)⁵ cos x

Setting y' equal to zero:

0 = 3(sin x)² cos x - 6(sin x)⁵ cos x

0 = 3(sin x)² cos x (1 - 2(sin x)³)

sin x = 0 or (sin x)³ = 1/2

If sin x = 0, then x = kπ for any integer k.

If (sin x)³ = 1/2, then sin x = (1/2)^(1/3) ≈ 0.866. This occurs when x = π/3 + 2kπ/3 or x = 5π/3 + 2kπ/3 for any integer k.

To determine whether these values correspond to a maximum or minimum, we can use the second derivative test.

y'' = 6(sin x)³ cos² x - 15(sin x)⁴ cos² x - 9(sin x)⁴ cos x + 6(sin x)⁵ cos x

y'' = 3(sin x)³ cos x (4(sin x)² - 5(sin x)² - 3cos x + 2)

For x = kπ, y'' = 3(0)(-3cos(kπ) + 2) = 6 or -6, depending on the parity of k. This means that these points correspond to a maximum or minimum, respectively.

For x = π/3 + 2kπ/3 or x = 5π/3 + 2kπ/3, y'' = 3(1/2)^(5/3) cos x (4(1/2)^(2/3) - 5(1/2)^(1/3) - 3cos x + 2). This expression is positive for x = π/3 + 2kπ/3 and negative for x = 5π/3 + 2kπ/3, which means that the former correspond to a minimum and the latter to a maximum.

Hence , the maximum value of the function is y = 27/64, which occurs at x = 5π/3 + 2kπ/3, and the minimum value is y = -1/64, which occurs at x = π/3 + 2kπ/3

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Answer:

maximum: 0.25minimum: -2

Step-by-step explanation:

You want the maximum and minimum values of the function ...

  y = sin³(x) -sin⁶(x)

Solution

When we substitute sin³(x) = z, we have the quadratic expression ...

  y = z -z² . . . . . a quadratic function

Adding and subtracting 1/4, we can put this in vertex form:

  y = -(z -1/2)² +1/4

Maximum

This version of the function describes a parabola that opens downward and has a vertex at (z, y) = (1/2, 1/4). The y-value of the vertex represents the maximum value of the function.

The maximum value of y is 1/4.

Minimum

The sine function is a continuous function with a range of [-1, 1]. Then z will be a continuous function of x, with a similar range. We already know that y describes a function of z that is a parabola opening downward with a line of symmetry at z = 1/2. This means the most negative value of y will be found at z = -1 (the value of z farthest from the line of symmetry). That value of y is ...

  y = (-1) -(-1)² = -1 -1 = -2

The minimum value of y is -2.

__

Additional comment

The range of y is confirmed by a graphing calculator.

<95141404393>

Which numerical expression represents the following calculation?

Add 15.35 to the product of 2.25 and 3.

A (15.35÷ 3) + (2.25 x 3)

B (15.353) +2.25

C 15.35+ (2.25 x 3)

D (2.25 +15.35 + 3)

Answers

it would be b the only right one
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