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Question

A study was done to determine how fuel economy of a particular car was related to the speed at which it was driven.The result are shown in the table below. Use this data to create an accurate scatterplot and a line of best fit using quadratic regression. then answer the questions below. I also need a graph because there is a grid here. thanks

FUEL ECONOMY RESEARCH

speed (km/h)	fuel economy (L/100km)
24	10.5
32	9.2
40	8.5
48	8.1
56	8.2
64	8.0
72	7.9
80	7.8
88	7.7
96	8.2
104	8.6
112	9.3

 

a) write your regression using F for fuel economy and K for km/h. Round these values to the nearest thousandth.

b) Show how to use your regression equation to predict the fuel economy if you were to drive this car at 130km/h.

c) what is the best fuel economy you could expect ( round to 2 decimals) from this car and at what speed ( round to 1 decimal) would this occur? Explain how you found these values on your calculator.

d) If you were travelling on a highway with the cruise control set at a speed of 108 km/h and had only 3.0 litres of fuel remaining in the gas tank, how far could you expect to travel before running out of fuel?

Thank you so much :) help me with those explainations pleaseeee ! :)

Answer 1

(a)

The Speed and Fuel Economy Data table is

Speed of the Car(km/hr)	Fuel Economy (L/100km)
24	10.5
32	9.2
40	8.5
48	8.1
56	8.2
64	8.0
72	7.9
80	7.8
88	7.7
96	8.2
104	8.6
112	9.3

We make use of Qudratic Expression.

The Quadratic Equation is in the form of F = aK² + bK + c

Where F is the Fuel Economy in L/100km and K is the Speed of the Car.

Now consider the three points from the table, (24, 10.5) , (64, 8.0) and (112, 9.8).

Consider the Point (24, 10.5) where K = 24 and F = 10.5.

The Quadratic Equation is

F = aK² + bK + c

10.5 = a(24)² + 24b + c

10.5 = 576a + 24b + c

576a + 24b + c = 10.5   .............................Equation (1)

Consider the Point (64, 8.0) where K = 64 and F = 8.0.

The Quadratic Equation is

F = aK² + bK + c

8.0 = a(64)² + 64b + c

8.0 = 4096a + 64b + c

4096a + 64b + c = 8.0   .............................Equation (2)

Consider the Point (112, 9.8) where K = 112 and F = 9.8.

The Quadratic Equation is

F = aK² + bK + c

9.8 = a(112)² + 112b + c

9.8 = 12544a + 112b + c

12544a + 112b + c = 9.8   .............................Equation (3)

Solving Equations (1), (2) and (3), we get

a = 0.001

b = -0.152

c = 13.566

The Quadratic Regression F = 0.001K² - 0.152K + 13.566

Graphing the data table and the Regression

Therefore the Qudratic Regression is F = 0.001K² - 0.152K + 13.566.

Answer 2

(b)

The Speed and Fuel Economy Data table is

Speed of the Car(km/hr)	Fuel Economy (L/100km)
24	10.5
32	9.2
40	8.5
48	8.1
56	8.2
64	8.0
72	7.9
80	7.8
88	7.7
96	8.2
104	8.6
112	9.3

We make use of Qudratic Expression.

The Quadratic Equation is in the form of F = aK² + bK + c

Where F is the Fuel Economy in L/100km and K is the Speed of the Car.

Now consider the three points from the table, (24, 10.5) , (64, 8.0) and (112, 9.8).

Solving the Qudratic Equation for these points we get,

F = 0.001K² - 0.152K + 13.566.

If the Speed of the car is 130 km/hr then

F = 0.001(130)² - 0.152(130) + 13.566

F = 0.001*16900 - 0.152(130) + 13.566

F = 16.9 - 19.76 + 13.566

F = 10.7

Therefore the Fuel Economy for the Speed of 130 km/hr is 10.7 L/100km.

Answer 3

(c)

The Best Fuel ecomomy is the instant after which the consumption of the fuel starts increasing as we increase the speed.

The Speed and Fuel Economy Data table is

Speed of the Car(km/hr)	Fuel Economy (L/100km)
24	10.5
32	9.2
40	8.5
48	8.1
56	8.2
64	8.0
72	7.9
80	7.8
88	7.7
96	8.2
104	8.6
112	9.3

Observe the Table.

The Fuel Economy at the speed of 88 km/hr is 7.7 L/100km.

If we increase the speed, the Fuel Economy also increases.

Therefore the best Fuel Economy at the speed 88 km/hr is 7.7.

Answer 4

(d)

The Equation of the Fuel and Speed is F = 0.001K² - 0.152K + 13.566.

Speed of the Car is 108 km/hr.

Fuel Economy F is

F = 0.001K² - 0.152K + 13.566

F =0.001(108)² - 0.152(108) + 13.566

F = 8.8 L/100km

Now we can see that

8.8 L of fuel is consumed to travel a distance of 100 km

1 L of Fuel is consumed to travel a distance of (100 / 8.8) km.

Then if we have 3 L of fuel in the car,

Then 3 L of fuel is consumed to travel a distance of (100 / 8.8) * 3 km.

3 L of fuel is consumed by 34.09 km.

Therefore the car travels 34.09 km if tank has 3 L of Fuel filled in it.