Chapter 2.7, Problem 37E

Solution

a.

Express the surface area S as a function of x.

  $S=2\pi x^2+\frac{1000}{x}$

Given:

Cylindrical cans of volume $0.5-L$ . Let x be the radius of the can in cm.

Calculation:

Volume of cylindrical can is $0.5-L=500{\text{cm}}^{\text{3}}$ .

Volume of a cylinder is given by

  $\begin{array}{c}V=\pi r^{2}h\\ 500=\pi x^{2}h\\ h=\frac{500}{\pi x^2}\end{array}$

Hence, surface area S is given by

  $\begin{array}{c}S=2\pi r\left(r+h\right)\\ =2\pi x\left(x+\frac{500}{\pi x^2}\right)\\ =2\pi x^2+\frac{1000}{x}\end{array}$

b.

Find the radius and height of the can.

  $x=1.12\text{ cm},h=126.88\text{ cm}$ and $x=11.37\text{ cm},h=1.23\text{ cm}$ .

Given:

  $S=2\pi x^2+\frac{1000}{x}$ .

  $h=\frac{500}{\pi x^2}$

The surface area is $900{\text{ cm}}^2$ .

Calculation:

Surface area is given by $S=2\pi x^2+\frac{1000}{x}$ .

Hence,

  $2\pi x^2+\frac{1000}{x}=900$

On solving this equation

  $x=-12.49,x=1.12,x=11.37$

Since, radius can’t be negative. Hence,

  $x=1.12,x=11.37$

For $x=1.12$

  $h=\frac{500}{\pi {\left(1.12\right)}^2}=126.88$

For $x=11.37$

  $h=\frac{500}{\pi {\left(11.37\right)}^2}=1.23$

Therefore,

For $x=1.12\text{ cm},h=126.88\text{ cm}$ and $x=11.37\text{ cm},h=1.23\text{ cm}$ .