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An experiment measures the fuel consumption at various speeds for a particular model of car - Leaving Cert Mathematics - Question 2 - 2017

Question 2

An experiment measures the fuel consumption at various speeds for a particular model of car. The data collected are shown in Table 1 below. Table 1 Speed (km/hour)... show full transcript

Worked Solution & Example Answer:An experiment measures the fuel consumption at various speeds for a particular model of car - Leaving Cert Mathematics - Question 2 - 2017

Step 1

Find the correlation coefficient of the data in Table 1, correct to 3 decimal places.

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Answer

To find the correlation coefficient, we will use the formula for Pearson's correlation coefficient, given by:

$r = rac{n( ext{Σ}xy) - ( ext{Σ}x)( ext{Σ}y)}{ ext{√}[n( ext{Σ}x^2) - ( ext{Σ}x)^2][n( ext{Σ}y^2) - ( ext{Σ}y)^2]}$

Where:

$n$ is the number of data points
$x$ is the speed
$y$ is the fuel consumption

Calculating each component from the data:

$n = 7$
$ext{Σ}x = 40 + 48 + 56 + 64 + 88 + 96 + 112 = 504$
$ext{Σ}y = 21 + 16 + 18 + 16 + 13 + 11 + 9 = 105$
$ext{Σ}xy = (40*21) + (48*16) + (56*18) + (64*16) + (88*13) + (96*11) + (112*9) = 1716$
$ext{Σ}x^2 = 40^2 + 48^2 + 56^2 + 64^2 + 88^2 + 96^2 + 112^2 = 27936$
$ext{Σ}y^2 = 21^2 + 16^2 + 18^2 + 16^2 + 13^2 + 11^2 + 9^2 = 2673$

Plugging these into the equation gives: $r = rac{7(1716) - (504)(105)}{ ext{√}[7(27936) - (504)^2][7(2673) - (105)^2]}$

After calculating, we find that: $r ext{ is approximately } -0.957$

Step 2

Plot the points from the table on the grid below and draw the line of best fit (by eye).

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Answer

To plot the points:

On a graph, label the x-axis as 'Speed (km/h)' and the y-axis as 'Fuel consumption (km/litre)'.
Plot each point from Table 1 based on the speed and fuel consumption values.
Draw a line of best fit that represents the trend of the data; this should show a downward slope indicating that as speed increases, fuel consumption decreases.

Step 3

The slope of the line of best fit is found to be -0.15. What does this value represent in the context of the data?

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Answer

The slope of the line of best fit being -0.15 indicates that for each increase of 1 km/h in speed, the average distance traveled on 1 litre of fuel decreases by 0.15 km. This suggests an inverse relationship between speed and fuel efficiency; as speed increases, fuel consumption becomes less efficient.

Step 4

Find how much longer it took Mary to complete the journey. Give your answer correct to the nearest minute.

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Answer

To calculate the time taken for both journeys:

For Mary: $ext{Time} = rac{Distance}{Speed} = rac{260 ext{ km}}{96 ext{ km/h}} = 2.7083 ext{ hours}$
For Jane: $ext{Time} = rac{Distance}{Speed} = rac{260 ext{ km}}{112 ext{ km/h}} = 2.3214 ext{ hours}$

Finding the difference in time: $ext{Difference} = 2.7083 - 2.3214 = 0.3869 ext{ hours}$

Convert to minutes: $0.3869 ext{ hours} imes 60 ext{ minutes/hour} ext{ is approximately } 23.21 ext{ minutes}$

Thus, it took Mary about 23 minutes longer.

Step 5

Based on the data in Table 1 and their average speeds, find how much more Jane spent on fuel during the course of this journey.

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Answer

To calculate fuel consumption for each:

For Mary (at 96 km/h): From Table 1, fuel consumption is 11 km/litre. Therefore for 260 km: $ext{Fuel used} = rac{260}{11} ext{ litres} ext{ and at } 132.9 ext{ cents/litre }$ Total cost for Mary: $ext{Cost} = rac{260}{11} imes 132.9 = ext{approximately } 313.645 ext{ cents}$
For Jane (at 112 km/h): Fuel consumption is 9 km/litre. Therefore for 260 km: $ext{Fuel used} = rac{260}{9} ext{ litres}$ Total cost for Jane: $ext{Cost} = rac{260}{9} imes 132.9 = ext{approximately } 419.24 ext{ cents}$

Difference in spending: $ext{Difference} = 419.24 - 313.645 = ext{approximately } 105.595 ext{ cents}$

Thus, Jane spent about 105.60 cents more than Mary.

An experiment measures the fuel consumption at various speeds for a particular model of car - Leaving Cert Mathematics - Question 2 - 2017

Worked Solution & Example Answer:An experiment measures the fuel consumption at various speeds for a particular model of car - Leaving Cert Mathematics - Question 2 - 2017

Find the correlation coefficient of the data in Table 1, correct to 3 decimal places.

Plot the points from the table on the grid below and draw the line of best fit (by eye).

The slope of the line of best fit is found to be -0.15. What does this value represent in the context of the data?

Find how much longer it took Mary to complete the journey. Give your answer correct to the nearest minute.

Based on the data in Table 1 and their average speeds, find how much more Jane spent on fuel during the course of this journey.

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