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1234Linear models

Solutions to the exercises

Exercise 1

f : y = 1 3 x + 3 1 3 ; g : y = 2 3 x + 1 ; h : y = - 2 x + 4 ; k : y = - 5 x - 2

Exercise 2
a

Own answer

b

The plotted points do seem to fit a straight line, so yes: linear.

c

Best fit, in this case, is a line where an equal number of plotted points are above the line, as there are below the line.

d

The line goes through the points ( 16 , 57 ) and ( 44 , 120 ) . The formula is given by H = 21 + 2 . 25 B .

e

The difference is small.

f

P = 21 + 2 . 25 32 = 93

g

Linear interpolation: ( 91 + 94 ) 2 = 93 .

Exercise 3

After 3 hours a decrease of 6 cm, so the gradient is - 2 . 2 hours after the start, the candle is 12 cm, so 2 hours earlier the length of the candle was 12 + 4 = 16 cm. The y-intercept is 16.
Formula: L = 16 - 2 t .

Exercise 4
a

V ( t ) = V ( 0 ) 273 ( t + 273 ) = V ( 0 ) ( t 273 + 1 )

b

V ( 0 ) is a constant coefficient, therefore the formula can be seen as V ( t ) = a t + b .
The pressure must remain constant though. The domain is D = [ - 273 ,

c

Enter: Y1=1+1/273X. Window settings: - 300 x 300 and - 1 Y 3 .

d

V ( 20 ) = 1 + 20 273 = 1 . 073 m3

e

1 . 5 = 1 + 1 273 t leads to t = 136 . 5 . So at 136.5°C.

Exercise 5
a

s ( 0 ) is the distance travelled at t = 0 and v is the speed in m/s.

b

s ( t ) = 20 t . Enter: Y1=20X. Window settings: 0 x 50 and 0 y 1000 .

c

s ( t ) = 400 + 15 t , so enter Y2=400+15X.

d

You may use the graphic calculator. 20 t = 400 + 15 t leads to t = 80

Exercise 6
a

In a graph you can see that it is quite reasonable to assume a linear relation: N = 300 L - 10000 .

b

N = 300 85 - 10000 = 15500 .

c

You may use the graphic calculator: 300 L - 10000 = 4500 leads to L = 48 .

d

Yes, there is a linear relation between L and N , so linear extrapolation is possible. The question remains whether a salmon can grow to 120 cm though.

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