Talk:Seddie/@comment-5540817-20121105003051/@comment-3247345-20121105014417

Okay socks. The first two are optimize problems and the 3rd one is a basic integral problem. I'm gonna write what to do here then I'll scan in my handwritten work so you can see that.

1.

A. Label your variables. I'll be using x, y, and a. x is length, y is width, and a is area.

B. Find the equations you'll need for your optimization and to reduce everything to one variable excluding the one you want to optimize. Our equations are a=xy and 2x+2y=100.

C. Write y in terms of x. This will give you y=50-x. Plus this into y in a=xy. When you multiply it out you'll get 50x-x^2.

D. Take the derivative of that and you'll get 50-2x. Set that equal to 0 and solve for it to get your maximum. x=25 here. Plug that into 50x-x^2 and you a=625. With a and x you can solve for y in a=xy. y=25 here.

Your answer here would be the rectangle is 25mx25m making it 625 square meters.

2.

A. The variables here are a, x, and y. x is the horizontal part of the rectangle and y is the vertical. a is the area.

B. a=xy.

C. x is identical to the X axis here. Since your other 2 points (the ones not on the Y axis) are determined directly by x for the x coordinate that's simple enough. But the difference in the y coordinates gives you y. Since that's determined by plugging x into the equations y=g(x)-f(x). y=3x+8. Plug this into a=xy and multiply it out. You get -3x^2 +8x.

D. Take the derivative of this to get -6x+8 and set that to equal 0 for maximize the area. You end up with x=4/3. Plus this into the equation at the end of step C and you'll find the area is 16/3. Plug a and x into a=xy and solve for y and y=4. So the rectangle will be (4/3)x4 and the area is 16/3.

3.

You'll be using antiderivatives like the hint said. Since this is extra credit and very basic I'm guessing you haven't covered it yet. Basically you take the antiderivative of the equation and plug in the 1 and you get 8/3. Then plug the -5 into that same equation and you get -100/3. Subract the second number from the first. 8/3-(-100/3)=36

Your answer is 36. My written work should help you with this.