Pre-Calc - Follow-up for Reiny - MUFFY, Saturday, October 3, 2009 at 10:27pm
I wasn't sure if you would notice my question since it is so much later so I re-sent this as a new question. Please let me know if I shouldn't do that.
I 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
Question:
Here I am again stuck with another geometry type question. Here goes:
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 11:34am
Have 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 = �} �ã12
so the max area occurs when x = �ã12
and it is
A(�ã12) = 36�ã12 - 12�ã12
= 24�ã12 or appr. 41.57
Using a graphing calculator you are on your own, I don't have one, but 24�ã12 is the 'exact' answer.
OOOPS - Pre-Calc - Reiny, Saturday, October 3, 2009 at 11:41am
silly 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 = �ã12
and it is
A(�ã12) = 72�ã12 - 24�ã12
= 48�ã12 or appr. 166.277
To find the maximum area of the rectangle, we need to find the value of x that maximizes the area function A(x). In order to do this, we differentiate the area function A(x) with respect to x, which gives us the derivative dA/dx.
In this case, the area function is given by A(x) = 72x - 6x^3 (given in the corrected version). To differentiate this function, we apply the power rule, which states that the derivative of x^n is n*x^(n-1).
For the first term 72x, the derivative is (72)(1)x^(1-1) = 72.
For the second term -6x^3, the derivative is (-6)(3)x^(3-1) = -18x^2.
Therefore, the derivative of the area function is dA/dx = 72 - 18x^2.
To find the maximum area, we set the derivative equal to zero and solve for x:
72 - 18x^2 = 0
18x^2 = 72
x^2 = 4
x = ±2
Since the rectangle is inscribed between the x-axis and the parabola, the value of x must be between -6 and 6. Therefore, the solution x = -2 is not valid in this case.
So the maximum area occurs when x = 2.
Plugging this value into the area function, we get:
A(2) = 72(2) - 6(2^3) = 144 - 48 = 96.
Hence, the maximum area that the rectangle can have is 96 square units.
In part c of the problem, Reiny differentiated the equation for the area of the rectangle, A(x) = 72x - 2x^3, with respect to x.
To differentiate a function means to find its derivative, which represents the rate at which the function is changing with respect to its independent variable, in this case, x.
The derivative dA/dx of the function A(x) is found by applying the power rule of differentiation. The power rule states that if a function has the form f(x) = x^n, where n is a constant, then its derivative is given by f'(x) = nx^(n-1).
In this case, the function A(x) has the form f(x) = 72x - 2x^3.
Using the power rule, we differentiate each term of the function:
The derivative of the first term, 72x, is 72.
The derivative of the second term, -2x^3, is -(3)(2)x^(3-1) = -6x^2.
Therefore, the derivative of A(x) is dA/dx = 72 - 6x^2.
To find the maximum area of the rectangle, Reiny set dA/dx equal to zero and solved for x.
Setting dA/dx = 72 - 6x^2 = 0,
Reiny divided both sides of the equation by 6 to isolate x^2:
6x^2 = 72
Then, Reiny divided both sides by 6 again to solve for x:
x^2 = 12
Finally, Reiny took the square root of both sides to find the value of x:
x = √12
Substituting this value of x back into the equation for A(x),
A(√12) = 72√12 - 24√12
Simplifying,
A(√12) = 48√12 or approximately 166.277.
Therefore, the maximum area of the rectangle is approximately 166.277.