Question

You need to construct an open-top rectangular box with a square base that must hold a volume of exactly 800 cm3. The material for the base of the box costs 5 cents/cm2 and the material for the sides of the box costs 7 cents/cm2. Find the dimensions for a box that will minimize the cost of the materials used to construct box.

Answer 1

Answer:

Step-by-step explanation:

Given

Volume of box is 800 cm^3

$V=800 cm^3$

Let the dimension of base is $L\times L$ and height of box be h

$V=L^2\times h$

cost base $c_1=L^2\times 5=5L^2$

cost of sides $c_2=(4Lh)\cdot 7$

$c_2=28Lh$

Total Cost$=c_1+c_2$

$C=5L^2+28Lh$

$C=5L^2+28L\cdot \frac{800}{L^2}$

$C=5L^2+\frac{800\times 28}{L}$

differentiate C w.r.t L to get maximum/minimum Value

$\frac{\mathrm{d} C}{\mathrm{d} L}=10L-\frac{800\times 28}{L^2}$

$L=13.05 cm$

$h=4.69 cm$