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Exercises

Exercise 1

The capacity of a wind turbine is described by the formula:
P = 0.00013 · v ³ · D ².
For any specified value of $D$ (the rotor diameter) you will find a relation between $P$ and $v$ .

On squared paper, draw a family of graphs for $D = 5$ , $D = 10$ , $D = 15$ en $D = 20$ metres. Take $0 \leq v \leq 40$ .

Read off from your family of graphs the value of $P$ when $v = 8$ and $D = 10$ . Describe how you do that. Check your answer by substitution into the formula.

In the family of graphs you can seen how the capacity, given a specified diameter, depends on the wind speed. Shade the area when the following is given: the diameter is between $10$ and $20$ metres and the maximum wind speed is $90$ km/h.

What is the maximum capacity that can be generated?

Exercise 2

The community council decides to maintain a temperature of 20°C in the outside swimming pool. They think they can achieve this by means of a heating system. Because the heat of the sun will aid the temperature during the summer months, a heating specialist predicts that the cost of heating $k$ will conform to the formula: $k = 800 - 60 u - 50 t$ , where $u$ is the average number of sun-hours per day and $t$ the number of degrees Celsius the outside temperature differs from 20°C. $k$ is given in € per day.

What does the number $800$ represent in this formula?

On a particular day the average temperature is 16°C. So $t = - 4$ . If there are $3.5$ sun-hours on that day, what are the heating costs?

Under which conditions does the temperature of the pool stay at 20°C, without any costs? Describe a few different possibilities.

On square paper draw a family of graphs for $k$ dependent on $u$ given that $t = - 2$ , $t = - 1$ , $t = 0$ , $t = 1$ and $t = 2$ .

Show in your family of graphs how much the heating costs are, given that on a particular day there are $6$ hours of sunshine and the outside temperature is 22°C. Calculate this answer with the formula as well.

During one particular week the outside temperature varies between 18°C and 22°C. The number of hours of sunshine is measured from $4$ hours up to a maximum of $10$ hours. Between which two amounts of money can the heating costs vary for that week?

Exercise 3

The ANWB advises car drivers to keep a distance $d$ (in m) that is half of the speed $v$ in km/h.

Write down the formula of $d$ as a function of $v$ .

On average a car measures $4$ m in length. The distance between the front bumpers of two cars is therefore $s = 4 + d$ m. Assume that all car drivers stick to the advise given by the ANWB, assume all cars are $4$ m long and drive at the same speed $v$ .

The time $t$ in secondes between two cars can now be calculated with the formula: $t = \frac{(3.6 s)}{v}$ .
Explain this formula.

Write down a formula for $t$ as a function of $v$ by combining formulas.

The number of cars $N$ passing a certain check point each minute is: $N = \frac{60}{t}$ .
Write down a formula for $N$ as a function of $v$ .

Some $29.9$ cars pass the checkpoint per minute. What is the speed $v$ of this stream of cars?

Exercise 4

The amount of petrol used by a car depends, amongst others, on the distance travelled, the style of driving and the waiting time at traffic lights. We shall investigate this further using mathematical modelling. In our model the petrol consumption $B$ (in mL) of a car is calculated using the following formula:
$B = a \cdot L + b \cdot S + c \cdot D$
where:

$L$ = distance travelled in km;
$S$ = number of stops en route;
$D$ = total time waiting at traffic lights in secondes.

$a$ and $b$ depend on the driving speed $V$ (in km/h) and $c$ is a constant. For $a$ , $b$ and $c$ the following holds:

$a = 170 - 4, 55 V + 0, 049 V^{2}$
$b = 0, 0077 V^{2}$
$c = 0, 39$

We do not consider acceleration or deceleration, so assume $V$ is constant in the above expressions for $a$ en $b$ .

Consider a journey of $1$ km with an average speed of $50$ km/h, $2$ stops en route and a total waiting time of $40$ secondes. Calculate what percentage of the total petrol consumption is used on the stops and the waiting time.

Two cars are waiting for a traffic light $P$ . Exactly $600$ m further along is another traffic light $Q$ . If the cars drive at a speed of $50$ km/h between $P$ and $Q$ , the traffic light at $Q$ will change to green in time for them to carry on driving. Do not concern yourself with breaking and accelerating in this problem.
Car $1$ drives at a constant speed of $50$ km/h between $P$ and $Q$ and so can carry on driving when approaching $Q$ . Car $2$ drives at an average speed of $70$ km/h, so it will have to break and wait at traffic light Q.

Make a calculation to show that car $2$ wil have to wait for a little over $12$ secondes at traffic light $Q$ .

We consider the first $900$ m along from traffic light $P$ . After $Q$ there are no more traffic lights and car $1$ continues to drive at $50$ km/h whilst car $2$ again drives at $70$ km/h. Investigate whether car $2$ uses more than twice as much petrol as car 1.

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