Linear relations

Exercise 1

Cyclist $1$ cycles at a constant speed of $20$ km/h from A to B. Cyclist $2$ cycles at a constant speed of $25$ km/h from B to A. The distance between A and B is $150$ km for both cyclists. $a$ represents the distance to A and $t$ represents the time in hours.

a

In an $a, t$ -coordinate system draw the graph for both cyclists.

b

Give separate formulas describing the relation between $a$ and $t$ for both cyclists.

c

Which of the two formulas describes a directly proportional relation between $a$ and $t$ ?

d

Use your graphic calculator to find how much time has passed before the two cyclists meet each other.

Exercise 2

In which of the formulas below is $y$ directly proportional to $x$ ? If so, give the constant of proportionality.

a

$y = 3 x + 1$

b

$y = 3 x$

c

$y = x + 3$

d

$y = \frac{x}{3}$

e

$y = \frac{1}{3} x$

f

$x + 3 y = 0$

Exercise 3

Explain which of the situations below conform to a directly proportional relation.

a

Onions are paid by the kg. Is the price of a quantity of onions directly proportional to the weight?

b

Onions are paid by the kg. Is the price of a quantity of onions directly proportional to the number of onions?

c

Is the distance travelled directly proportional to the time when you drive at a constant speed?

d

Is the distance travelled directly proportional to the time when a car accelerates from rest?

e

Is the length of a rectangle directly proportional to the width when you consider similar rectangles?

f

Is the length of a rectangle directly proportional to the width when you consider rectangles with a fixed perimeter?

Exercise 4

Lead free petrol costs € 1.42 per litre. On average a car does $1$ in $12$ (1 L needed for $12$ km). Diesel costs € 1.10 per litre. On average a car does $1$ in $20$ . The number of km someone travels per year is represented by $a$ .

a

Give a formula for the fuel costs $K$ for a car running on lead free petrol.

b

Give a formula for the fuel costs $K$ for a car running on diesel.

c

A car running on diesel is more expensive to buy and more expensive for road tax and insurance than a similar car running on petrol. If this were to mean that a car running on diesel is $1400$ euros more expensive per year, for which number of km per year is it profitable to buy a diesel?

Exercise 5

A plumber charges € 30 call-out charge and an hourly rate of € 22.50.

a

Explain why there is a direct proportional relationship between his wages and the number of hours he has worked.

b

Explain why the relation between the total costs of the bill is not directly proportional to the number of hours he has worked.

c

What is the formula describing the total costs $T K$ as a function of the number of hours $u$ worked?

Exercise 6

The Elfstedentocht is a tour on ice passing through eleven Frisian towns. The last part of it is a pretty straight journey from Dokkum to Leeuwarden with a length of $26$ km. A contestant arrives in Dokkum after $7$ hours of ice skating. On the last straight part he enjoys wind in the back, and skates almost exactly at a constant speed. Three quarters of an hour later he arrives in Leeuwarden and has completed a total of $200$ km.

a

What was his speed on this last part of the tour?

b

Represent the time in hours by $t$ , and define $t = 0$ as the moment the contestant starts the last part of the Elfstedentocht. Distance skated is represented by $a$ . Give a formule for $a$ , describing the last part of his tour.

Exercises