The table gives the relationship between the... | WAEC SSCE - Further Mathematics - 2013 (Essay)

Question 1 Report

The table gives the relationship between the height, in metres, of a plant and the number of days it is left to grow.

Number of days (x)	10	20	30	40	50	60	70	80
Height (y)	1.0	1.1	1.2	1.4	1.6	1.8	2.0	2.3

(a) Using a scale of 2 cm to represent 0.5 units on the y- axis and 2cm to 10 units on the x- axis, draw a scatter diagram for the information.

(b) Find \(\bar{x}\), the mean of x, and \(\bar{y}\), the mean of y, and plot \((\bar{x}, \bar{y})\) on the diagram.

(d) From graph, find the :

(i) equation of the line of best fit ; (ii) height of plant in 75 days.

Answer

(a) & (c) Scatter diagram, mean point and line of best fit

Plotting the eight pairs of readings using a scale of 2 cm to 10 units on the x-axis and 2 cm to 0.5 units on the y-axis gives the scatter diagram below. The mean point \((\bar{x},\bar{y})=(45,\,1.55)\) is marked, and the line of best fit is drawn straight through \((10,\,1.0)\) and \((45,\,1.55)\).

graph — Scatter diagram with the mean point (45, 1.55) marked and the line of best fit drawn through (10, 1.0) and the mean point.

(b) Mean of x and mean of y

\[\bar{x}=\frac{10+20+30+40+50+60+70+80}{8}=\frac{360}{8}=45\] \[\bar{y}=\frac{1.0+1.1+1.2+1.4+1.6+1.8+2.0+2.3}{8}=\frac{12.4}{8}=1.55\]

Therefore the mean point is \((\bar{x},\bar{y})=(45,\,1.55)\), which is plotted on the diagram above.

(d)(i) Equation of the line of best fit

The line passes through \((\bar{x},\bar{y})=(45,\,1.55)\) and \((10,\,1.0)\). Its gradient is

\[m=\frac{1.55-1.0}{45-10}=\frac{0.55}{35}=0.0157\]

Using the point \((10,\,1.0)\):

\[y-1.0=0.0157\,(x-10)\] \[y-1.0=0.0157x-0.157\] \[\boxed{\,y=0.0157x+0.843\,}\]

Check: at \(x=45\), \(y=0.0157(45)+0.843=0.707+0.843=1.55=\bar{y}\), so the line indeed passes through the mean point.

(d)(ii) Height of plant after 75 days

Substituting \(x=75\) into the equation of the line of best fit:

\[y=0.0157(75)+0.843=1.179+0.843=2.02\]

Reading off the graph at \(x=75\) confirms this value. The height of the plant after 75 days is approximately 2.0 m (2.02 m).

Read lesson note on Matrices And Linear Transformation (WAEC)

Answer Details

(a) & (c) Scatter diagram, mean point and line of best fit

(b) Mean of x and mean of y

\[\bar{x}=\frac{10+20+30+40+50+60+70+80}{8}=\frac{360}{8}=45\] \[\bar{y}=\frac{1.0+1.1+1.2+1.4+1.6+1.8+2.0+2.3}{8}=\frac{12.4}{8}=1.55\]

Therefore the mean point is \((\bar{x},\bar{y})=(45,\,1.55)\), which is plotted on the diagram above.

(d)(i) Equation of the line of best fit

The line passes through \((\bar{x},\bar{y})=(45,\,1.55)\) and \((10,\,1.0)\). Its gradient is

\[m=\frac{1.55-1.0}{45-10}=\frac{0.55}{35}=0.0157\]

Using the point \((10,\,1.0)\):

\[y-1.0=0.0157\,(x-10)\] \[y-1.0=0.0157x-0.157\] \[\boxed{\,y=0.0157x+0.843\,}\]

Check: at \(x=45\), \(y=0.0157(45)+0.843=0.707+0.843=1.55=\bar{y}\), so the line indeed passes through the mean point.

(d)(ii) Height of plant after 75 days

Substituting \(x=75\) into the equation of the line of best fit:

\[y=0.0157(75)+0.843=1.179+0.843=2.02\]

Reading off the graph at \(x=75\) confirms this value. The height of the plant after 75 days is approximately 2.0 m (2.02 m).

Read lesson note on Matrices And Linear Transformation (WAEC)

Matrices And Linear Transformation

View Answer

The table gives the relationship between the height, in metres, of a plant and the number of days it is left to grow. Number of days (x) 10 20 30 40 50 60 7...

(a) & (c) Scatter diagram, mean point and line of best fit

(b) Mean of x and mean of y

(d)(i) Equation of the line of best fit

(d)(ii) Height of plant after 75 days

(a) & (c) Scatter diagram, mean point and line of best fit

(b) Mean of x and mean of y

(d)(i) Equation of the line of best fit

(d)(ii) Height of plant after 75 days

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The table gives the relationship between the height, in metres, of a plant and the number of days it is left to grow. Number of days (x) 10 20 30 40 50 60 7...

(a) & (c) Scatter diagram, mean point and line of best fit

(b) Mean of x and mean of y

(d)(i) Equation of the line of best fit

(d)(ii) Height of plant after 75 days

(a) & (c) Scatter diagram, mean point and line of best fit

(b) Mean of x and mean of y

(d)(i) Equation of the line of best fit

(d)(ii) Height of plant after 75 days

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