Linear relations

Linear relations > Linear models

Overview

1234Linear models

Exercises

Exercise 1

In the figure four straight lines are given. Find the formula corresponding to each of these lines.

Exercise 2

There is a relation between the number of breaths $B$ taken by a human being per minute and the heart rate $H$ in beats per minute. A physician examines a group of fifteen people and finds the following test results:

B	16	16	19	20	20	23	24	26	27	28	30	34	36	41	44
H	57	59	66	68	71	70	72	84	82	80	91	94	105	116	120

Plot a graph of the data given in the table. Plot $B$ on the horizontal axis.

Do you think there might be a linear relation between $B$ and $H$ ? Explain your answer.

Draw the line of best fit through the plotted points. What, in this case, is "best fit"?

Give a formula corresponding to the line you have drawn.

Use the formula to calculate the heart rate for all values of $B$ in the table. Do the calculated values differ greatly from the experimental data?

Use your formula to calculate the heart rate of someone taking $32$ breaths per minute.

The heart rate calculated in subquestion f could have been obtained by averaging $91$ and $94$ . Why? How closely does this average resemble the value calculated in f?

Exercise 3

A cylindrical candle is burning down. The length of the candle $L$ (in cm) is a linear function of the time $t$ (in hours). After $2$ hours, the length of the candle is $12$ cm, after $5$ hours there is still $6$ cm of candle left.
Give a formula for $L$ as a function of $t$ .

Exercise 4

Gasses conform to the law of Gay-Lussac. The volume $V$ (in m³) of a fixed quantity of gas at a fixed pressure depends on the temperature $t$ (in °C). The law states :
$\frac{V (t)}{t + 273} = \frac{V (0)}{273}$
where $- 273$ °C is the absolute minimum temperature and $V (0)$ the volume at $0$ °C.

Re-write this formula to: $V (t) = V (0) \cdot (1 + \frac{1}{273} t)$ .

Explain that this conforms to a linear model. What supposition is made for the model to be true? What is the domain of the function?

Take $V (0) = 1$ m³ and make a plot of the corresponding graph. Write down which window settings are suitable.

What is the volume of this gas at room temperature?

At what temperature is the volume $1.5$ times larger than the volume at temperature 0 °C ?

Exercise 5

Uniform motion is defined as a motion of an object with a constant velocity (same speed, same direction). In physics the following formula is used:
$s (t) = s (0) + v \cdot t$ , where $s (t)$ is the distance travelled (in m) after $t$ seconds.

What does $s (0)$ represent? What does $v$ represent?

Take $s (0) = 0$ and $v = 20$ m/s for a certain object. Plot the corresponding graph of $s (t)$ .

A second object has a headstart of $400$ m , moving along the same track at a speed of $15$ m/s. Give the formula describing the motion of this object and plot the corresponding graph as well.

Calculate at what time the first object passes the second object.

Exercise 6

A biologist wishes to find out whether there is a relation between the length $L$ (in cm) of salmon and the number of eggs $N$ they spawn. He has found the following data:

`L`	52	58	66	68	73	74	78	90
`N`	5620	7410	9805	10390	11890	12200	13380	17010

Investigate whether there may be a linear relation between $L$ and $N$ . If so, write down a formula for $N$ .

Give an estimate of the number of eggs spawn by a salmon, measuring $85$ cm.

A salmon lays $4500$ eggs. What will be the approximate length of this salmon?

Someone uses linear extrapolation to estimate the number of eggs spawned by a salmon with a length of $120$ cm. Do you think this will give a realistic estimate? Explain your answer.

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B	16	16	19	20	20	23	24	26	27	28	30	34	36	41	44
H	57	59	66	68	71	70	72	84	82	80	91	94	105	116	120

B	16	16	19	20	20	23	24	26	27	28	30	34	36	41	44
H	57	59	66	68	71	70	72	84	82	80	91	94	105	116	120

B	16	16	19	20	20	23	24	26	27	28	30	34	36	41	44
H	57	59	66	68	71	70	72	84	82	80	91	94	105	116	120