Question

Topic: Optimization

1. A close right circular cylinder is to be constructed to hold a 1 liter oil can shape.

What dimensions will minimize the amount of material, assuming that the

thickness of the material is uniform?

2. Find two positive numbers whose sum is 9 and whose product is a maximum.

Answer 1

1. Let $r=$ the radius of the base of the cylinder, let $h=$ the height of the cylinder. Then

$$V=\pi r^2h=1000 {cm}^3$$

$$h=\dfrac{1000}{\pi r^2}$$

The total surface area of cylinder is

$$A=2(\pi r^2)+2\pi rh$$

Substitute

$$A=A(r)=2\pi r^2+2\pi r(\dfrac{1000}{\pi r^2})$$

$$=2\pi r^2+\dfrac{2000}{ r}$$

Differentiate with respect to $r$

$$A'(r)=4\pi r-\dfrac{2000}{ r^2}$$

Find the critical number(s)

$$A'(r)=0=>4\pi r-\dfrac{2000}{ r^2}=0$$

$$r=\sqrt[3]{\dfrac{2000}{4\pi}}$$

$$r=5\sqrt[3]{\dfrac{4}{\pi}}$$

If $0<r<5\sqrt[3]{\dfrac{4}{\pi}}, A'(r)<0, A(r)$ decreases.

If $r>5\sqrt[3]{\dfrac{4}{\pi}}, A'(r)>0, A(r)$ increases.

The function $A(r)$ has the local minimum at $r=5\sqrt[3]{\dfrac{4}{\pi}}.$

Since the function $A(r)$ has the only extremum, then the function $A(r)$ has the absolute minimum at $r=5\sqrt[3]{\dfrac{4}{\pi}}.$

$$h=\dfrac{1000}{\pi (5\sqrt[3]{\dfrac{4}{\pi}})^2}=10\sqrt[3]{\dfrac{4}{\pi}}$$

$r=5\sqrt[3]{\dfrac{4}{\pi}}\ cm, h=10\sqrt[3]{\dfrac{4}{\pi}}\ cm$

2.

Let $x=$ the first number. Then the second number will be $9-x.$

Since two numbers are positive, then $0<x<9.$

The product of two numbers is $f(x)=x(9-x).$

$$f(x)=-x^2+9x$$

This quadratic function has the absolute maximum at

$$x=-\dfrac{9}{2(-1)}=\dfrac{9}{2}$$

The first number is $\dfrac{9}{2}.$ The second number is $\dfrac{9}{2}.$