volume of a rectangular box
We are tasked with constructing a rectangular box with a volume of $15$ cubic feet:
- The material for the top costs $12$ dollars per square foot.
- The material for the $4$ sides costs $3$ dollars per square foot.
- The material for the bottom costs $11$ dollars per square foot.
To the nearest cent, what is the minimum cost for such a box ?.
I tried to solve this problem but I just dont know how to set it up, I tried to do
$$
15 = 12x^{2} + 11x^{2} + \left(4 \times 3 \times x\times {15 \over x^{2}}\right)
$$
but I couldn't get the right answer from there.
Can anyone help walk me through whether I messed up setting up the problem or am I just solving it wrong ?.
I got $1.58$ so can someone show me what they got ?.
1 Answer
$\begingroup$Basic approach. You wrote down the cost, but then you equated it with $15$. The volume is $15$; you should simply write the cost $C(x)$ as a function of $x$:
$$ C(x) = 12x^2+11x^2+\frac{4 \cdot 3 \cdot x \cdot 15}{x^2} = 23x^2+\frac{180}{x} $$
and then find the value of $x$ that minimizes $C$. Find an expression for $C'(x)$ and then set $C'(x) = 0$. This will be an equation in $x$; you should then find a solution that is a minimum of $C(x)$ (rather than a maximum).
$\endgroup$ 4
