Formulas

Exercise 1

The volume of a cylindrical tin can is given as: $V = π \cdot r^{2} \cdot h$ . Here $V$ is the volume, $r$ the radius in centimetres and $h$ the height in centimetres. Consider a tin can with height $h = 10$ cm. Now $V$ is a function of $r$ .

a

Write down the formula of this function.

b

Plot the corresponding graph. Use the table option to find a value of $r$ corresponding to $V = 1000$ . Give the value of $r$ correct to one decimal place.

For a particular cylindrical tin can the diametre and height are the same, so: $h = 2 r$ .

c

Write down the corresponding formula for the volume $V$ as a function of $r$ .

d

Give the value of $r$ , correct to one decimal place, when it is given that the volume of the tin can is $0.5$ L.

Exercise 2

The lease on a photocopying machine is € 200 per month. Over and above the cost of one copy is $4$ cents.
$K$ respresents the total cost (in €), which consists of the lease costs and the costs per page, and $a$ is the number of copies that are made (on average) monthly.

a

Write down a formula for $K$ as a function of $a$ .

b

Someone using the photocopier pays $10$ cents per copy. Write down a formula for the monthly takings $I$ as a function of $a$ .

c

How many copies should be made per month when $10$ cents per copy covers all costs?

Exercise 3

Plot the graphs of the formulas below. Pay attention to the use of brackets and set the window adequately!

a

$R = 250 p - 0.5 p^{2}$

b

$k = 0.04 + \frac{200}{a}$

c

$N = \frac{60}{(30 + 0.5 d)}$

Exercise 4

The total costs ( $T K$ ) in euros for production of a specific product is given by:
$T K = 100 + 0.1 q^{2}$ where $q$ is the number of products.

a

Calculate the average cost per product item when $120$ items are produced, correct to two decimal places.

b

Give the formula for the average cost per item ( $G T K$ ) as a function of $q$ .

c

Give an equation of the vertical asymptote of the function $G T K$ .

d

What is the reason that there is no horizontal asymptote?

Exercise 5

Your company wants to produce posters. For good visible impact, the area of such a poster needs to measure $1$ square metre. The poster is printed so that on both sides and on the top a white border of $10$ cm remains. The bottom has a border of $15$ cm. The company management is wondering what the dimensions of the poster could be. They reach the following formula: $(l + 25) \cdot (b + 20) = 10000$ .

a

Show how this formula is derived and explain what the variables $l$ en $b$ represent.

b

Re-write the formula so that $l$ is a function of $b$ . Plot a graph of this formula.

c

Check whether all of the plotted dimensions are indeed possible.

d

After consideration the management would like the printed part of the poster to be a square. What is your recommendation for the dimensions of the poster?

Exercises