Calculus 1 Help

[Viper0329]

Could someone help me to solve this?

An isoceles triangle has base b and equal siades of length a. Find the dimesions of the rectangle of maximum area that can be inscribed in the triangle if one side of the rectangle lies on the base of the triangle.

I have the picture drawn, but I can't figure out what to call the area. I know I need to set it up so I can substitute in the variable, then take the derivative and set it to 0, but the first part has me scratching my head.

Any ideas?

(This isn't homework. I'm reviewing for my test that's soon.)

 

[LostHiWay]

I feel your pain..wish I could help but I'm in calc 1..with just barely a C

 

[crt1530]

Google is your friend...very similar problem explained in detail

Not the exact same problem, but if I told you exactly how to do it, you wouldn't have as much fun.

 

[Mday]

assumptions: the side of your rectangle is colinear with the base of your triangle.

what you do is place a coordinate axis on the triangle. make the base on the x-axis, and the midpoint of the triangle base as the origin. then you have 2 right triangles to worry about. since the area of the rectangle INSCRIBED will be distributed equally across both triangles, what you have to do is worry about a rectangle inscribed in a right triangle.

back to the right triangle. choose the triangle on the 1st quadrant, as it uses positive x and y coordinates. all the hypotenuse of the triangle is, is a line. the equation of the line can be obtained from using the formula of your choice, slope intercept is easier. call that equation f(x). the area of the inscribed rectangle, within the right triangle can be obtained by multiplying the x and y coordinates of some point which lies on the hypotenuse of the triangle. how? because 3 of the corners of the rectangle are actually on the x and y axes. the 4th point lies on the hypotenuse. so, the area of the rectangle is merely x*y, where y is the equation of the line of the hypotenuse, ie y=f(x).

back to the isoceles triangle. you now have the area of the rectangle within the right triangle, x*y. To get the area of the FULL rectangle within the isoceles triangle, multiply it by 2, so A(rect)=2*x*y. Take the derivative of this equation. Set the derivative = to 0. solve for x. then plug x back into A(rect)=2*x*y to get the value of the rectangle.

 

[her209]

Easy.

1. You know the top of the triangle is at point b/2.
2. You know the slope. Get the equation.
3. Setup another equation that calculates the area of the square with one point on the equation above.
4. Derive and find where the maximum is, assuming its a parabolic function you got in 3.