Power functions

Exercise 1

Given the power function $f (x) = 120 x^{5}$ .

a

Determine $f (4)$ .

b

What value of $x$ gives you $f (x) = 20000$ ?

c

If the value of $x$ is multiplied by four, by which factor do you have to multiply the function value?

Exercise 2

There is a relationship between the speed of a car $s$ and the corresponding braking distance $r$ . The braking is the distance is the distance the car will still travel if you brake as hard as possible. A rule of thumb for this relationship is: $r = \frac{s^{2}}{100}$ .

a

$r$ is directly proportional to a power of $s$ . What is the constant of proportionality?

b

At a sharp bend in the road you can only see $10$ meters ahead. Safe driving requires you to be able to stop within the distance you can see ahead. According to this requirement, what should your maximum speed be through this bend?

c

Write down the formula that expresses speed as a function of braking distance. Describe in words what sort of a relationship this is.

d

Comment on the following statement: "When you can see ahead for $100$ metres, you can drive twice as fast as when you can see ahead for $50$ metres."

Exercise 3

This question concerns the volume $V$ of a cube with edges $r$ in centimetres.

a

Calculate the volume of a cube with edge length $r = 2$ . Then calculate the volume if $r = 6$ .

b

The edge of the second cube is three times longer than that of the first cube. What consequence does this have for the volume of the cube?

c

A cube has a volume of $50$ cm. Determine $r$ .

d

Write down the formula that expresses volume $V$ in terms of $r$ .

e

Write down the formula that expresses the length of the edge $r$ in terms of volume $V$ .

Exercise 4

The formula for the total surface area $A$ of a cube with edge length $r$ is: $A = 6 r^{2}$ .

a

Explain how you can derive this formula.

b

Calculate the total surface area of a cube with edges of $3$ cm, and of a cube with edges of $6$ cm.

c

What happens to the surface area if the length of the edges doubles?

d

The total surface area of a cube is $500$ cm². Determine the length of its edges.

e

Write down a formula that expresses the edge length $r$ in terms of surface area $A$ .

Exercise 5

You have a solid iron cube with edge length $r$ in cm. The specific weight of iron is $7.9$ g/cm³.

a

Write down a formula of the weight of the cube $G$ as a function of $r$ .

b

Write down a formula of the surface area $A$ of the cube as a function of $r$ .

c

Derive a formula of the form $A = c \cdot G^{\frac{2}{3}}$ . Hence give the constant of proportionality $c$ .

d

Calculate the weight of such a cube if the total surface area is $150$ cm².

Exercises