Question

e. the graph shows a linear relation.
a) what is the approximate value of the y - coordinate when x = 1.5?
b) what is the approximate value of the x - coordinate when y = 0.5?

1. a) the table of values represents the distance that sophie cycles in relation to time.

time, t (h)	1	2	3	4	5	6

a) plot the linear relation on a graph.
b) from the graph, determine the approximate distance sophie has cycled after 2.5 h.
c) from the graph, approximate how long it takes sophie to cycle 44 km.
for help with #8 to #11, refer to example 2 on pages 223 - 224.

1. the graph shows a linear relation between distance and time.

a) what is the approximate value of the d - coordinate when t = 10?
b) what is the approximate value of the t - coordinate when d = 33?

Explanation:

Part 5 (Linear Relation Graph)

a) When \( x = 1.5 \), find \( y \)-coordinate

Assume the linear equation is \( y = mx + b \). From the graph, when \( x = 0 \), \( y = -2 \) (y-intercept \( b = -2 \)). When \( x = 2 \), \( y = -4 \). Slope \( m=\frac{-4 - (-2)}{2 - 0}=\frac{-2}{2}=-1 \). So equation: \( y=-x - 2 \).
Substitute \( x = 1.5 \): \( y=-1.5 - 2=-3.5 \).

b) When \( y = 0.5 \), find \( x \)-coordinate

Use \( y=-x - 2 \). Substitute \( y = 0.5 \):
\( 0.5=-x - 2 \)
\( x=-2 - 0.5=-2.5 \).

Part 7 (Sophie’s Cycling)

a) Plotting the linear relation

• Time (\( t \)) on x-axis: 1, 2, 3, 4, 5, 6.
• Distance (\( d \)) on y-axis: 12.5, 25, 37.5, 50, 62.5, 75.
• Plot points (1,12.5), (2,25), (3,37.5), etc., and draw a straight line.

b) Distance after \( 2.5 \) h

The relation is linear: \( d = 12.5t \) (slope \( 12.5 \), y-intercept 0).
Substitute \( t = 2.5 \): \( d = 12.5 \times 2.5 = 31.25 \) km.

c) Time to cycle 44 km

Use \( d = 12.5t \). Solve for \( t \):
\( t=\frac{44}{12.5}=3.52 \) hours (≈3.5 h).

Part 8 (Distance-Time Graph)

a) When \( t = 10 \), find \( d \)-coordinate

From the graph, the slope \( m=\frac{\Delta d}{\Delta t} \). At \( t = 2 \), \( d = 5 \); \( t = 4 \), \( d = 10 \). Slope \( m=\frac{10 - 5}{4 - 2}=2.5 \). Equation: \( d = 2.5t \).
Substitute \( t = 10 \): \( d = 2.5 \times 10 = 25 \).

b) When \( d = 33 \), find \( t \)-coordinate

Use \( d = 2.5t \). Solve for \( t \):
\( t=\frac{33}{2.5}=13.2 \) minutes.

Final Answers

5a) \( \boldsymbol{-3.5} \)
5b) \( \boldsymbol{-2.5} \)
7b) \( \boldsymbol{31.25} \) km
7c) \( \boldsymbol{3.52} \) h (≈3.5 h)
8a) \( \boldsymbol{25} \)
8b) \( \boldsymbol{13.2} \) min

Answer:

Part 5 (Linear Relation Graph)

a) When \( x = 1.5 \), find \( y \)-coordinate

Assume the linear equation is \( y = mx + b \). From the graph, when \( x = 0 \), \( y = -2 \) (y-intercept \( b = -2 \)). When \( x = 2 \), \( y = -4 \). Slope \( m=\frac{-4 - (-2)}{2 - 0}=\frac{-2}{2}=-1 \). So equation: \( y=-x - 2 \).
Substitute \( x = 1.5 \): \( y=-1.5 - 2=-3.5 \).

b) When \( y = 0.5 \), find \( x \)-coordinate

Use \( y=-x - 2 \). Substitute \( y = 0.5 \):
\( 0.5=-x - 2 \)
\( x=-2 - 0.5=-2.5 \).

Part 7 (Sophie’s Cycling)

a) Plotting the linear relation

• Time (\( t \)) on x-axis: 1, 2, 3, 4, 5, 6.
• Distance (\( d \)) on y-axis: 12.5, 25, 37.5, 50, 62.5, 75.
• Plot points (1,12.5), (2,25), (3,37.5), etc., and draw a straight line.

b) Distance after \( 2.5 \) h

The relation is linear: \( d = 12.5t \) (slope \( 12.5 \), y-intercept 0).
Substitute \( t = 2.5 \): \( d = 12.5 \times 2.5 = 31.25 \) km.

c) Time to cycle 44 km

Use \( d = 12.5t \). Solve for \( t \):
\( t=\frac{44}{12.5}=3.52 \) hours (≈3.5 h).

Part 8 (Distance-Time Graph)

a) When \( t = 10 \), find \( d \)-coordinate

From the graph, the slope \( m=\frac{\Delta d}{\Delta t} \). At \( t = 2 \), \( d = 5 \); \( t = 4 \), \( d = 10 \). Slope \( m=\frac{10 - 5}{4 - 2}=2.5 \). Equation: \( d = 2.5t \).
Substitute \( t = 10 \): \( d = 2.5 \times 10 = 25 \).

b) When \( d = 33 \), find \( t \)-coordinate

Use \( d = 2.5t \). Solve for \( t \):
\( t=\frac{33}{2.5}=13.2 \) minutes.

Final Answers

5a) \( \boldsymbol{-3.5} \)
5b) \( \boldsymbol{-2.5} \)
7b) \( \boldsymbol{31.25} \) km
7c) \( \boldsymbol{3.52} \) h (≈3.5 h)
8a) \( \boldsymbol{25} \)
8b) \( \boldsymbol{13.2} \) min