Question

The base and sides of a container is made of wood panels. The container does not have a lid. The base and sides are rectangular. The width of the container is x cm . The length is double the width. The volume of the container is 54cm3 . Determine the minimum surface area that this container will have.

Answer 1

Answer:

Minimum surface area =$70.77 cm^2$

Step-by-step explanation:

We are given that

Width of container=x cm

Length of container=2x cm

Volume of container=$54 cm^3$

We have to find the minimum surface areas that this container will have.

Volume of container=$l\times b\times h$

$x\times 2x\times h=54$

$2x^2h=54$

$h=\frac{54}{2x^2}=\frac{27}{x^2}$

Surface area of container=$2(b+l)h+lb$

Because the container does not have lid

Surface area of container=$S=2(2x+x)\times \frac{27}{x^2}+2x\times x$

$S=\frac{162}{x}+2x^2$

Differentiate w.r.t x

$\frac{dS}{dx}=-\frac{162}{x^2}+4x$

$\frac{dx^n}{dx}=nx^{n-1}$

Substitute $\frac{dS}{dx}=0$

$-\frac{162}{x^2}+4x=0$

$4x=\frac{162}{x^2}$

$x^3=\frac{162}{4}=40.5$

$x^3=40.5$

$x=(40.5)^{\frac{1}{3}}$

$x=3.4$

Again differentiate w.r.t x

$\frac{d^2S}{dx^2}=\frac{324}{x^3}+4$

Substitute x=3.4

$\frac{d^2S}{dx^2}=\frac{324}{(3.4)^3}+4=12.24>0$

Hence, function is minimum at x=3.4

Substitute x=3.4

Then, we get

Minimum surface area =$\frac{162}{(3.4)}+2(3.4)^2=70.77 cm^2$