Calculus Optimization Problem

[missjina]

Design a rectangular water tank with a square base and a lid. The box will hold 2000 m^3 of water, and the lid will come down 2 meters to overlap the rest of the box.

Part 1. Find the dimensions of the box which minimize the amount of materials used to construct the box.

For this part I used V = x^2 * h ...I got h = 2000 / x^2 .... then by using the formula S = x ^2 + 4xh ...I plugged in h...took the derivative...set it to 0..etc.etc..and got these dimensions x = 15.87401052 and h = 7.93700526....

I don't know how to start this problem....
Part 2. The material used for the lid costs twice as much per square meter as the material used to construct the other piece of the box. Find the dimensions of the box which minimizes the cost of production.

Please helpp!!!

 

[galactus]

Design a rectangular water tank with a square base and a lid. The box will hold 2000 m^3 of water, and the lid will come down 2 meters to overlap the rest of the box.

Part 1. Find the dimensions of the box which minimize the amount of materials used to construct the box.

For this part I used V = x^2 * h ...I got h = 2000 / x^2 .... then by using the formula S = x ^2 + 4xh ...I plugged in h...took the derivative...set it to 0..etc.etc..and got these dimensions x = 15.87401052 and h = 7.93700526....

Don't forget the lid that comes down 2 meters on each side. The base has area \(\displaystyle x^{2}\). The four sides have area \(\displaystyle 4xy\)

The lid has area: the top is the same as the base, \(\displaystyle x^{2}\). But, the four edges that come down 2 m on each side have total area \(\displaystyle 8x\)

Thus, the total surface area of the box is \(\displaystyle S=x^{2}+4xy+x^{2}+8x\Rightarrow 2x^{2}+4xy+8x\)

Now, use your \(\displaystyle y=\frac{2000}{x^{2}}\) and sub into S. Then, differentiate, set to 0, and all that stuff.

I don't know how to start this problem....
Part 2. The material used for the lid costs twice as much per square meter as the material used to construct the other piece of the box. Find the dimensions of the box which minimizes the cost of production.

Please helpp!!!

Just let some variable represent the cost. Call it C. Then, the lid has cost 2C. Just incorporate this into the formula above.