# Thread: Short Calculus Word Problem

1. ## Short Calculus Word Problem

A closed rectangular box is to be constructed with one side 1 meter long. The material for the top of the box cost $20 per square meter, and the material for the sides and bottom costs$10 per square meter. Find the dimensions of the box with the largest possible volume that can be built at a cost of $240 in materials. (Thanks in advance!) 2. Originally Posted by Affinity A closed rectangular box is to be constructed with one side 1 meter long. The material for the top of the box cost$20 per square meter, and the material for the sides and bottom costs $10 per square meter. Find the dimensions of the box with the largest possible volume that can be built at a cost of$240 in materials. (Thanks in advance!)
hello,

1. main condition (contains the value which will become extreme): $\displaystyle V=w \cdot h \cdot 1$

$\displaystyle A_{\text{top}}=w \cdot 1$

$\displaystyle A_{\text{side and bottom}}=w \cdot 1+2 \cdot w \cdot h+2\cdot h\cdot 1$

costs:
$\displaystyle \text{costs}=20w + 10\cdot (w \cdot 1+2 \cdot w \cdot h+2\cdot h\cdot 1)=240$ . Solve for h and plug this term into the equation of the main condition:

$\displaystyle h=\frac{24-3w}{2w+1}$. You'll get the characteristic function of this problem:

$\displaystyle V(w)=\frac{24w-3w^2}{2w+1}$

Now calculate the derivative of V: $\displaystyle V'(w)=\frac{-6w^2-6w+24}{(2w+1)^2}$

V has an extreme value (maximum or minimum) if V'(w) = 0. Calculate for w. (V'(w) = 0 if the numerator is zero and the denominator is not zero). You'll get two values for w. Only the positive value makes some sense.
I've got $\displaystyle w=\frac{-1+\sqrt{17}}{2}\approx 1.56\ m$

My result for h = 4.68 m

EB

3. Hi, I tried to redo the problem for myself, but for the denominator it came out as 2w+2 as opposed to your 2w+1, is there something I'm doing wrong? Thanks!

4. Originally Posted by Affinity
Hi, I tried to redo the problem for myself, but for the denominator it came out as 2w+2 as opposed to your 2w+1, is there something I'm doing wrong? Thanks!
Hello,

no, you have done the calculation perfectly.

I've checked my calculations again and I got now: w = 2 m and h = 3 m

(But as you kow now: I am the master of desaster if it comes to calculation, so if your results differ from these values - don't trust me!)

EB

5. Hi!

I got the same answer as you did! Yay! Thank you very much for your help !