How to convert 5/8 to a decimal

A rectangle is constructed with its base on the diameter of a semicircle with radius 2 and with its two other vertices on the se

stealth61 [152]

Answer:

L = 2*√2

w = √2

Step-by-step explanation:

Given:

A rectangle is constructed with its base on the diameter of a semicircle with radius 2 and with its two other vertices on the semicircle.

Find:

What are the dimensions of the rectangle with maximum area?

Solution:

- Let the length and width of the rectangle be L and w respectively.

- We know that Length L lie on the diameter base. So , L < 4 and the width w is less than 2 . w < 2.

- Using the Pythagorean Theorem, we relate the L with w using the radius r = 2 of the semicircle.

r^2 = (L/2)^2 + (w)^2

sqrt (4 - w^2 ) = L / 2

L = 2*sqrt (4 - w^2 ) L < 4 , w < 2

- The relation derived above is the constraint equation and the function is Area A which is function of both L and w as follows:

A ( L , w ) = L*w

- We substitute the constraint into our function A:

A ( w ) = 2*w*sqrt (4 - w^2 )

- Now we will find the critical points for width w for which A'(w) = 0

A'(w) = 2*sqrt (4 - w^2 ) - 2*w^2 / sqrt (4 - w^2 )

0 = [2*sqrt (4 - w^2 )*sqrt (4 - w^2 ) - 2*w^2] / sqrt (4 - w^2 )

0 = 2*(4 - w^2 ) - 2*w^2

0 = -4*w^2 + 8

8/4 = w^2

w = + sqrt ( 2 ) ..... 0 < w < 2

- From constraint equation we have:

L = 2*sqrt (4 - 2 )

L = 2*sqrt(2)

7 0

4 years ago

Solve for x<br>a.81<br>b.41<br>c.1681

Jlenok [28]

A=9 , b=40

Pythagorean theorem
$a^2+b^2=x^2 \\\\ x=\sqrt{a^2+b^2}=\sqrt{9^2+40^2}=\sqrt{81+1600}=\sqrt{1681}=\sqrt{41^2}=41$

5 0

4 years ago

What is the inverse function for f(x)= square root of 1/2x+4

Zepler [3.9K]

The inverse of f(x) would be f(x) = 2 $x^{2}$ - 8.

We can find this by taking the original equation and switching the x and f(x). Once we do this, we solve for the f(x) in its new place and that is the inverse equation. It is taken step by step for you below.

Take the original.

f(x) = $\sqrt{\frac{1}{2}x + 4}$

Now switch the x and f(x)

x = $\sqrt{\frac{1}{2}f(x) + 4}$

Now square both sides.

$x^{2}$ = $\frac{1}{2}$ f(x) + 4

Now subtract 4 from both sides.

$x^{2}$ - 4 = $\frac{1}{2}$ f(x)

Now multiply both sides by 2.

2 $x^{2}$ - 8 = f(x)

And once you've returned to f(x) =, then you have your inverse.

6 0

3 years ago

Read 2 more answers

Matika knows that 230 - 39 = 191. How can she use that, to help her solve 233 - 42? Use the lines below to answer.

amid [387]

Answer:

Step-by-step explanation:

8 0

3 years ago

The number of users of a certain website (in millions) from 2004 through 2011 follows:

leonid [27]

Answer:

y = 26.3x² - 116.9x + 109.6

Step-by-step explanation:

Given the data ;

Year Period Users (Millions)

2004 1 1

2005 2 5

2006 3 11

2007 4 58

2008 5 145

2009 6 359

2010 7 607

2011 8 846

A quadratic regression model can be obtained using a quadratic regression calculator ; The quadratic regression modeled obtained is in the form :

y = Ax² + Bx + C

y = 26.3x² - 116.9x + 109.6

3 0

3 years ago