Find the dimensions of the rectangle of maximum area with sides parallel to the coordinate axes that can be inscribed in the ellipse 4x^2 16y^2

Answers

Answer 1

Answer:

The answer is below

Step-by-step explanation:

Find the dimensions of the rectangle of maximum area with sides parallel to the coordinate axes that can be inscribed in the ellipse 4x² + 16y² = 16

Solution:

Given that the ellipse has the equation: 4x² + 16y² = 16

let us make x the subject of the formula, hence:

4x² + 16y² = 16

4x² = 16 - 16y²

Dividing through by 4:

x² = (16 - 16y²)/4

x² = 4 - 4y²

Taking square root of both sides:

[tex]x=\sqrt{4-4y^2 }\\[/tex]

The points of the rectangle vertices is at (x,y), (-x,y), (x,-y), (-x,-y). Hence the rectangle has length and width of 2x and 2y.

The area of a rectangle inscribed inside an ellipse is given by:

Area (A) = 4xy

A = 4xy

[tex]A=4(\sqrt{4-4y^2} )y\\\\A=4y\sqrt{4-4y^2}=4\sqrt{4y^2-4y^4} \\\\The\ maximum\ area\ of\ the\ rectangle\ is\ at\ \frac{dA}{dy}=0\\\\ \frac{dA}{dy}=4(\frac{4-8y^2}{\sqrt{4-4y^2} } )\\\\4(\frac{4-8y^2}{\sqrt{4-4y^2} } )=0\\\\4-8y^2=0\\\\8y^2=4\\\\y^2=1/2\\\\y=\frac{1}{\sqrt{2} }\\\\x=\sqrt{4-4(\frac{1}{\sqrt{2} })^2}=\sqrt{2}[/tex]

Therefore the length = 2x = 2√2, the width = 2y = 2/√2


Related Questions

Select the correct answer.
de.
Currently there are 691,255 trees in a national forest. A park ranger discovered that an
untreatable disease is killing the trees at a rate of 1.5% every year.
Let t represent the number of years after the current date. Determine the inequality that can be
used to model when there will be no more than 621,859 trees, and then solve the inequality for t.
CHECK ATTACHMENT FOR OPTIONS

Answers

Answer:

691,255(0.985)t ≤ 621,859; t ≥ 7

Step-by-step explanation:

i just did it. it was C

Answer:

Yes it is corect,

C

Step-by-step explanation:

You are given that angle RST and angle TUR are right angles. What additional piece of info allows you to prove they are congruent

Answers

Answer:

the additional info required is - the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle.

Step-by-step explanation:

As given , RST and angle TUR are right angles.

So the triangles are right angle triangles.

If the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, the two triangles are congruent.

So, the additional info required is - the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle.

Gary’s cat weighs 11 pounds. How many ounces is Gary’s cat?

Answers

Answer:

176 ounces :)

Step-by-step explanation:

What is the solution to this system of equations?

Answers

Answer:

The fourth option i.e. 'no solution' is correct.

Step-by-step explanation:

Given the system of equations

[tex]\begin{bmatrix}\frac{2}{3}x+y=6\\ -\frac{2}{3}x-y=2\end{bmatrix}[/tex]

Adding both equations

[tex]-\frac{2}{3}x-y=2[/tex]

[tex]+[/tex]

[tex]\underline{\frac{2}{3}x+y=6}[/tex]

[tex]0=8[/tex]

so the system of equations becomes

[tex]\begin{bmatrix}\frac{2}{3}x+y=6\\ 0=8\end{bmatrix}[/tex]

0 = 8 is FALSE!

Therefore, the system of equations has no solution.

Thus,

No Solution!

Hence, the fourth option i.e. 'no solution' is correct.

How would you write twenty-one thousand, six hundred fifty-sixin standard form?
2165
21666
21566
21656

Answers

option 4 because all the numbers line up and i don’t know i just know that’s right

Answer:

21,656

Step-by-step explanation:

can someone give me 500 pts and 5 brainliest

Answers

Answer:

No -_-

Step-by-step explanation:

wot why??- 0,0

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You have a loan for $200,000 at 8.5% annual simple interest.what is your monthly interest payment

Answers

Answer:

~1467

Step-by-step explanation:

$200,000 x 8.5% or .085 = $17,000 Annualy

$17,000 / 12 Months out of the year = $1466 2/3 Monthly Payments

Answer is $1467 Rounded

The monthly interest payment is $1,467.

Given that,

The loan amount is $200,000.The rate of interest is 8.5%.We need to find out the monthly interest payment.

Based on the above information, the calculation is as follows:

[tex]= (\$200,000 \times 8.5\%) \div 12[/tex]

= $1,467

Therefore we can conclude that the monthly interest payment is $1,467.

Learn more: brainly.com/question/13324776

Write the phrase as an expression.

5 less than 8

An expression is

Answers

Answer:

the expression is 5 - 8

which is -3

or

the expression is 5 < 8

Which values from the set {2,4,6,8,10} make this inequality n + 5 > 10 true?​

Answers

Answer:

6,8,and 10

Step-by-step explanation:

Answer:

6,8,and 10

Step-by-step explanation:

when you plug in each number from the set into the inequality you will see which values from the set will make the inequality true.

1. 2+5>10 = 7>10 FALSE

2. 4+5>10 = 9>10 FALSE

3. 6+5>10 =11>10 TRUE

4. 8+5>10 = 13>10 TRUE

5. 10+5>10 = 15>10 TRUE

HELP PLEASE IM BEGGING PLEASE PLEASE PLEASE

Answers

Answer:

h - 18 = 67

Step-by-step explanation:

If h is her height, then her height minus 18 is 67, or h - 18 = 67

Answer:

67 = h - 18

Step-by-step explanation:

Sry I meant h-18 = 67

The value of the ratio y/x of two proportional quantities x and y is called..

either constant of proportionality, ratio, unit rate, proportion, rate, or proportional relationship

Answers

Answer:

constant of proportionality

Step-by-step explanation:

Assuming that we have two proportional quantities; one is x and one is y.

If we know whether the relationship between them is direct or inverse proportion, then the ratio of the two quantities y/x gives us a constant generally known as the constant of proportionality.

HELP PLEASSEEEE I NEED TO FIND THE MISSING SIDE HYPOTENUSE

Answers

Answer:

B. 6.5

Step-by-step explanation:

You can use the 5 against the unknown number and  add a few more moves which makes it 6.5.. It's easier in my head man.

Identify the constant of proportionality from the graph. 11 10 9 00 7 6 5 1 2 3 4 5 6 7 8 9 10 11 O A. 4 B. 8 O c. D. 3​

Answers

Answer:

A. 4

Step-by-step explanation:

Constant of proportionality (k) = y/x

We can use the coordinates of any point on the line to find k.

Let's use (2, 8)

Constant of proportionality (k) = 8/2

Constant of proportionality (k) = 4

List three examples of words/phrases that describe the mathematical operation of addition. (Example "Increased by") *

Answers

Answer:

Increased by, added, brought up (All I know of haha)

Step-by-step explanation:

Theres really no explanation to this other than learning your mathematical terms.

At a party there are 30 students over age 21 and 20 students under age 21. You choose at random 3 of those over 21 and separately choose at random 2 of those under 21 to interview about attitudes toward alcohol. You have given every student at the party the same chance to be interviewed: what is that chance

Answers

Answer:

10%

Step-by-step explanation:

According to the scenario, given data are as follows:

Students over age 21 = 30

Students under age 21 = 20

Choose student over age 21 = 3

Choose student under age 21 = 2

So, we can calculate chances by using following method,

Chances of student over age 21 = Number of choose student ÷ Total students

= 3 ÷ 30  = 0.10 = 10%

Similarly, Chances of student under age 21 = Number of choose student ÷ Total students

= 2 ÷ 20  = 0.10 = 10%

Hence, in both condition, chances are 10%.

The expression 10s+5m represents a student’s score on a social studies test, where s = the number of correct short answer items and m = the number of correct multiple choice items. Noor answers 16 multiple choice items correctly. She does no answer any short answer items correctly. What is Noor’s score?

A. 31
B. 15
C. 80
D. 160

Answers

it’s A. 31 hope this helps

Can you explain how u got 31?

Write the equation in slope intercept form of the line perpendicular to the given line that passes through the-given points (8,-5);y=2x+3

Answers

Answer:

y -y1= m(x-x1)  is in point slope form

y --9 = -2 (x-10)

y+9 = -2(x-10)

Step-by-step explanation:

HOPE THIS HELPS!!!

This list shows the numbers of stuffed animals nine children own.

6, 12, 2, 5, 1, 6, 15, 10, 13

What is the range of the number of stuff animals the children own?


7

13

14

15

Answers

Answer:

I believe it's 14

Step-by-step explanation:

To find the range you subtract to highest and lowest values, (15 and 1)

15 - 1 = 14

I took the test and its 13.

please help me, yall :))))

Answers

Choose A or C that’s all on u

there is a college student who wants to buy apples from the local store. the store generally sells about 5 apples for every 2 coustomers they serve. About how many apples will they sell in 6 hours if they serve 10 coustomers every hour?

Answers

Theirs a total of 60 customers divide it by 2 then times it by 5, you would get 150apples

Can someone help me? I don’t know how to solve it. Thank you!

Answers

Answer:

C. 5x^2 - 8x + 13

Step-by-step explanation:

If r * s = s to the power of r, then 3 * 2 =​

Answers

Answer:

[tex]3*2=8[/tex]

Step-by-step explanation:

We know that [tex]r*s=s^{r}[/tex], and we want to find [tex]3*2[/tex] using this formula. To do this, notice that [tex]r=3[/tex] and [tex]s=2[/tex], so all we have to do is substitute the given values into the formula. Therefore:

[tex]r*s=s^{r}[/tex]

[tex]3*2=2^{3}[/tex] (Substitute [tex]r=3[/tex] and [tex]s=2[/tex] into [tex]r*s=s^{r}[/tex])

        [tex]=8[/tex] (Simplify)

Hope this helps!

How to find the domain of a composite function

Answers

Answer:

The domain of a composite function f(g(x)) f ( g ( x ) ) is the set of those inputs x in the domain of g for which g(x) is in the domain of f .

Each school bus going on the field trip holds 36 students and 4 adults. There are 6 field buses on the field trip. How many people are going on the field trip?

Answers

Answer:

240 people are going on the trip

Step-by-step explanation:

216 students and 24 parents

Can someone please help me out?

Answers

9514 1404 393

Answer:

  64°

Step-by-step explanation:

y° and the angle marked 64° are "alternate interior angles", so are congruent.

  y° = 64°

According to the table, if a married couple filing jointly earns $67,400, what will their taxes be to the nearest dollar?

A): $10,470
B): $10,590
C): $11,150
D): $11,320

Answers

Answer:

c

Step-by-step explanation:

Answer:

C

Step-by-step explanation:

Dont worry about what I type hear the answer is C

PLEASE help!!!!


Im timed!

Answers

Answer:

it would be B

Step-by-step explanation:

can anyone help me please ?

Answers

Answer: 84 dollars

Step-by-step explanation:

The answer is 84....

What accounted for the high mortality rate of the French Army in Russia?

Answers

Answer:the lost the war

Step-by-step explanation:

Suppose Anthony Ianni attempted 18 baskets in a basketball game. If the ratio of baskets made to baskets missed is 1:5, how many baskets did Ianni make?

Answers

Answer:

Number of baskets Lanni make = 3 basket

Step-by-step explanation:

Given:

Total attempt = 18 times

Ratio(Basket make : Basket missed) = 1:5

Find:

Number of baskets Lanni make

Computation:

Number of baskets Lanni make = Total attempt x [Basket make ratio]

Number of baskets Lanni make = 18 x [1/(1+5)]

Number of baskets Lanni make = 3 basket

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