# Find the greatest area within a curve.

• Apr 30th 2006, 03:25 PM
Find the greatest area within a curve.
Q: A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y= 3-x^2. What are the dimensions of such a rectangle with the greatest possible area?

My Solution:

Okay, I understand I have to find a rectangle with the greatest size that has to corners in f(x) = 3-x^2. But I really do not know how to even start, would someone please give me a hint?
• Apr 30th 2006, 03:40 PM
Jameson
Start at the origin. Call the length from the origin going directly along the x-axis to the end of the rectangle "x". Thus if you went the other way along the x-axis you would go another distance of "x", and the total length of the base is "2x". Now since the upper corners are on the graph of \$\displaystyle y=3-x^2\$, that's a give away that your height can now be given by \$\displaystyle y=3-x^2\$. So, area of a triangle is \$\displaystyle B*H\$, which gives you an area function of \$\displaystyle 2x(3-x^2)\$. Now differentiate and set the derivative equal to zero to find the critical point(s). And once you get you critical x-value, plug that into your area function and you should have your answer.
• May 1st 2006, 08:44 AM