Linear relations

Exercise 1

A student group has organised a movie night for all students of the school. They spent € 400 organising this event. In order to recoup these costs, they are going to charge an entrance fee of € 2,50. Take $P$ to be the total profit made and $s$ the number of students going to the movie night.

a

Write down a formula of $P$ dependent on $s$ .

b

How many tickets does the student group need to sell in order to make a profit and not a loss?

c

How many tickets were sold if their profit was more than € 1000?

Exercise 2

Look at the figure and solve: $p \leq q$ .

Exercise 3

Two persons want to take a taxi from the station to their house. They have a choice between a 'rail-taxi' and a regular taxi. The rail-taxi costs € 3, the regular taxi charges € 2,25 for the ride plus € 0,75 per minute journey time.

a

Write down a formula for the cost of the regular taxi depending on the journey time (in minutes) $m$

b

Determine the minimum journey time after which it would be cheaper for the two people to take a rail-taxi.

c

The average speed of the taxis in town is $60$ kilometers per hour. The two people live six kilometers away from the station. Which type of taxi would you recommend to them? Show your calculation.

Exercise 4

Solve the following equations and inequalities:

a

$55 - 6 k = 4 k - 25$

b

$12 - 4 x \geq 36 + 2 x$

c

$25 - 1 \frac{2}{3} t > 30 - 3 t$

d

$1200 + 0, 08 a \geq 1045 + 0, 11 a$

e

$\frac{(6 - 2 x)}{5} = \frac{(4 - x)}{4}$

f

$200 - (80 - x) = 4 (x + 15)$

Exercise 5

A factory produces an article that is sold at a unit price of € 10. The actual production of this article costs € 6,50 per unit and there are fixed costs (maintaining the factory and machines, salaries etc) of € 83000. You may assume that every unit that is produced is sold.

a

Write down formulas for the total revenue $T R$ and the total costs $T C$ depending on the number of units produced $q$ .

b

The value of $q$ where revenue and costs are equal is called the "break-even-point". Use your graphing calculator to determine this "break-even-point."

c

Calculate this point.

d

At what values of $q$ will there be a profit?

Exercise 6

A pottery sells vases at € 5 a piece. For making the vases they calculate € 3 per vase and fixed costs of € 1600 per day. You may assume that all vases that are produced are also sold.

a

Write down a formula of the costs $C$ (in euro) dependent on the number of vases produced $v$ .

b

Write down a formula of the revenue $R$ (in euro) dependent on the number of vases sold $v$ .

c

At least how many vases does the pottery need to sell to make a profit? (Show clearly which method you use to calculate this!)

d

What daily production volume results in a daily profit of € 2000?

Exercises