Question

1. An engineer collected the data below showing the speed, s, of a Toyota Camry and its average miles per gallon, M

Speed, s	30	35	40	40	45	50	55	60	65	65	70
Mpg, M	18	20	23	25	25	28	30	29	26	25	25

a. Plot the data above, treating speed as the independent variabl (x). What type of relation, linear or quadratic, appears to exist between speed and miles per gallon?

b. Based on your response above, find either a linear or a quadratic model that describes the relationship between the two variables.

c. Use your model to predict the mpg of a Toyota Camry that is travelling 63 mph

d. Does this model have a maximum value? if so, what is it? if the model does not have a maximum value, find the y-intercept. Does this intercept make sense in the context of the problem

Answer 1

a. A scatterplot of the given data is attached. It appears that the relation between speed (s) and miles per gallon(M) is quadratic as the scatterplot resembles a downwards opening parabola.

b. Let the quadratic equation be M = as²+bs +c, where a, b, c are arbitrary real numbers. When s =55, M = 18 so that a(30)²+ 30b +c = 18 or, 900a+30b+c = 18…(1). Also, when s = 55, M = 30 so that 3025a+55b+c = 30…(2) and when s = 70, M = 25 so that 4900a+70b+c = 25…(3). The augmented matrix of this linear system is A =

900	30	1	18
3025	55	1	30
4900	70	1	25

The RREF of A is

1	0	0	-61/3000
0	1	0	53/24
0	0	1	-599/20

It implies that a = -61/3000, b = 53/24 and c = -599/20 so that M = -(61/3000)s²+(53/24)s -599/20

c. When s = 63, we have M =-(61/3000)*63²+(53/24)*63 -599/20= -80.703 +139.125 – 29.95 = 28.472 say, 28( on rounding off to the nearest whole number).

d. A graph of the function M =-(61/3000)s²+(53/24)s -599/20 is attached. It may be observed that M has a maximum value of 30.01 (say 30) when s = 54.303 (say 53) as the vertex of the downwards opening parabola represented by this equation is at (54.303,30.01).