Maximum area of semicircle inscribed under y = 12 - 3x²

[lp27]

Homework Statement

Given the function
y = 12- 3x^2,
find the maximum semi-circular area bounded by the curve and the x-axis.

Homework Equations

A= Pi(r^2)

The Attempt at a Solution

I found my zeros, 2 and -2, and my maximum height of 12 from the y'.

A' = 2Pi(r)

 

[vela]

Plot the function. The answer will be clear.

 

[Dick]

Well, no. r isn't equal to 2. If it's an inscribed semicircle bounded by the x-axis the circle must have center (0,0), agree? Draw a picture. It's a wee bit less than 2. So the equation of the circle is r^2=x^2+y^2. If it's inscribed the semicircle must be tangent to the parabola y=12-3*x^2 at the point of intersection. Find dy/dx for each and set them equal. It works out particularly easy if you find dy/dx for the circle using implicit differentiation.

 

[vela]

You are, of course, correct. I should have thought about it more carefully.

 

[Dick]

You are, of course, correct. I should have thought about it more carefully.

You didn't say anything wrong. I wasn't responding to your suggestion. I was responding to the original post. Plotting is always a great idea!