Math Help - Optimization

1. Optimization

I've tried to figure these out, but I'm really confused. Sorry there are so many, but I would really appreciate it if someone could help me to understand them. Thanks.

Which of the following should be used to minimize the lateral surface area of a cone, where the volume of the cone is 125 cm^3? V = pi*r^2h/3 Surface area of a cone is: S = pi*r(sqrt(r^2+h^2))
-This was my attempt: 125cm3=pir2h/3 ==> 375 cm^3=pir^2h ==>?

The legs of a triangle are x and y. Find the equation that will maximize the area of the triangle given that 2x + y = 16.
-I don't understand the relation between 2x+y=16 and A=1/2bh

Given the area of a rectangle is A = bh. If perimeter of the rectangle is 2b + 2h = 20, maximize the area of the rectangle.
-Which do I solve for? B or h? Because I thought I needed to solve for one to plug into the perimeter...?

Sue wants to build an enclosed area behind her house for her pets. One wall of the enclosed area will be the back of her house. She needs to the total to be 120 sq feet. She wants to minimize the cost of the fence materials. For the sides (W) fence materials cost $3 /ft, for the length (L) they cost$5 /ft. What dimensions should her fence be?
-120=4xh+x^2 ==> 120-x^2 = 4xh ==> (120-x^2)/4x=h ==> is it correct so far? I'm not sure what to do after that.

Find the largest possible area for a rectangle inscribed in a circle of radius 4.
-For this one, I realize that the radius is 4, therefore the diameter and dimensions of the rectangle are 4*4.

A student wishes to maximize the amount of poster space for an art exhibit. The requirements are that the height and width must sum to 50. What should the dimensions of the poster be?

A piece of sheet metal is rectangular, 5 feet wide and 8 feet long. Congruent squares are to be cut from its four corners. The resulting piece of metal is to be folded and welded to form a box with an open top. How should this be done to get a box of largest possible volume?

I'm not looking for the answers, but I'm looking to understand optimization. I don't necessarily need all of them answered, and I would be grateful if anyone would help me with these.

2. Originally Posted by iyppxstahh
I've tried to figure these out, but I'm really confused. Sorry there are so many, but I would really appreciate it if someone could help me to understand them. Thanks.

Which of the following should be used to minimize the lateral surface area of a cone, where the volume of the cone is 125 cm^3? V = pi*r^2h/3 Surface area of a cone is: S = pi*r(sqrt(r^2+h^2))
-This was my attempt: 125cm3=pir2h/3 ==> 375 cm^3=pir^2h ==>?
So $h = \frac{375}{\pi r^2}$.

Substitute into the formula for surface area:

$S = \pi r\sqrt{r^2 + \left(\frac{375}{\pi r^2}\right)^2}$

$= \pi r \sqrt{r^2 + \frac{140625}{\pi^2 r^4}}$

$= \pi r \sqrt{\frac{\pi^2 r^6 + 140625}{\pi^2 r^4}}$

$= \frac{\pi r}{\pi r^2}\sqrt{\pi^2 r^6 + 140625}$

$= \frac{\sqrt{\pi^2 r^6 + 140625}}{r}$.

Now differentiate this function, set the derivative equal to 0, solve for r.

3. ahhh, okay, thank you!
I don't know why I didn't think to solve for the h.