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July 4, 2015

July 4, 2015

Posted by **MUFFY** on Saturday, October 3, 2009 at 11:19am.

A rectangle is inscribed between the x-axis and the parabola y=36-x^2

with one side along the x-axis.

He drew the picture with the parabola at points (-6,0) (6,0) (0,6)

the rectanle is drawn inside the parabola along the x axis.

Not sure if you can help me without seeing the picture.

a. Let x denote the x-coordinate of the point shown in the figure. Write the area A of the rectangle as a function of x.

b. What values of x are in the domain of A?

c. Determine the maximum area that the rectangle may have. (hint, use a graphing calculator).

I'm already confused just looking at a.

if you can guide me it would be great!

- Pre-Calc -
**Reiny**, Saturday, October 3, 2009 at 11:34amHave seen this type of questions many many times.

Suppose we label the point of contact P(x,y). I bet P is in the first quadrant.

but we have the equation for y, so we could call the point P(x,36-x^2)

Isn't the contact point on the x-axis (x,0) ?

And isn't the base of the rectangle 2x (The distance from the origin to the right is the same as the distance to the left)

a)

so the area is

A(x) = x(36 - x^2) or

A(x) = 36x - x^3

b) wouldn't the domain be -6 < x < +6 or else the height 36-x^2 wouln't make any sense.

c) I would now differentiate to get

dA/dx = 36 - 3x^2

set that equal to zero for a max of A

3x^2 = 36

x^2 = 12

x = ± √12

so the max area occurs when x = √12

and it is

A(√12) = 36√12 - 12√12

= 24√12 or appr. 41.57

Using a graphing calculator you are on your own, I don't have one, but 24√12 is the 'exact' answer.

- Pre-Calc -
**MathMate**, Saturday, October 3, 2009 at 11:37am"He drew the picture with the parabola at points (-6,0) (6,0) (0,

**3**6)"

The point (0,6) is not on the parabola, (0,36) is.

The parabola is given as:

y(x)=36-x²

Given the rectangle is inside the parabola and above (I think) the x-axis, we define the four corners of the rectangle as

(-x,0), (x,0), (x,y(x)), (-x, y(x))

from which the area A can be calculated as the product of the width (x-(-x) and the height (y(x)).

A(x) = 2x(y(x)

domain A(x) [-6,6]

range A(x) [0,36].

For the maximum, you can use the graphics calculator, or you can tabulate the values, refining the grid where the maximum is located. Since this is a pre-calc course, I do not assume you are allowed to find the minimum by derivatives.

All this is assuming that I interpreted the "diagram" correctly. Check my steps.

- OOOPS - Pre-Calc -
**Reiny**, Saturday, October 3, 2009 at 11:41amsilly me, right after telling you the base is 2x, I use only 1x in my calculation.

HERE IS THE NEW VERSION :

so the area is

A(x) = 2x(36 - x^2) or

A(x) = 72x - 2x^3

b) wouldn't the domain be -6 < x < +6 or else the height 36-x^2 wouldn't make any sense.

c) I would now differentiate to get

dA/dx = 72 - 6x^2

set that equal to zero for a max of A

6x^2 = 72

x^2 = 12

so the max area occurs when x = √12

and it is

A(√12) = 72√12 - 24√12

= 48√12 or appr. 166.277

- Pre-Calc - Follow-up for Reiny -
**MUFFY**, Saturday, October 3, 2009 at 10:27pmI understand how you got a and b, but I am confused with c. What is differentiate and how did you get the equation dA/dx = 72 - 6x^2

- Pre-Calc -
**Anonymous**, Monday, October 26, 2009 at 7:55pmpre caculates is the ansewer to above is 25545872