The Cost( $\$$ ) $\times$ Distance(km) graph of starting point from $(0, 2.8)$ to end point of $(12, 6.8)$ shows the cost of taxi fares for journeys up to $12$ kilometres. * (a) Find the cost shown on the taxi meter when a passenger first boards a taxi. * (b) The cost of a particular journey is $\$4.60$ . How far was the journey? * (c) Find the gradient of the graph. * (d) State the equation of the graph.

Full solution

Q. The Cost(

\$

)

\times

Distance(km) graph of starting point from

(0, 2.8)

to end point of

(12, 6.8)

shows the cost of taxi fares for journeys up to

12

kilometres. * (a) Find the cost shown on the taxi meter when a passenger first boards a taxi. * (b) The cost of a particular journey is

\$4.60

. How far was the journey? * (c) Find the gradient of the graph. * (d) State the equation of the graph.

Calculate Initial Cost: (a) The cost shown on the taxi meter when a passenger first boards a taxi is the $y$ -intercept of the graph. $\newline$ Calculation: The starting point is $(0, 2.8)$ , so the initial cost is $\$2.80$ .
Find Distance for $\$4.60$ Fare: (b) To find the distance for a $\$4.60$ fare, we need to use the linear equation that will be derived from the graph. $\newline$ Calculation: We will find the equation in step (d) and then use it to calculate the distance.
Calculate Gradient: (c) Find the gradient of the graph by using the two points given. $\newline$ Calculation: Gradient $m$ = (Change in $y$ ) / (Change in $x$ ) = $(6.8 - 2.8) / (12 - 0) = 4 / 12 = 1/3$ .
Derive Equation of the Graph: (d) State the equation of the graph using the gradient and y-intercept. $\newline$ Calculation: The equation of a line is $y = mx + b$ , where $m$ is the gradient and $b$ is the y-intercept. $\newline$ Substitute $m = \frac{1}{3}$ and $b = 2.8$ to get the equation $y = (\frac{1}{3})x + 2.8$ .
Calculate Distance for $\$4.60$ Fare: Now, we use the equation $y = \frac{1}{3}x + 2.8$ to find the distance for a $\$4.60$ fare. $\newline$ Calculation: $4.60 = \frac{1}{3}x + 2.8; 4.60 - 2.8 = \frac{1}{3}x; 1.8 = \frac{1}{3}x; x = 1.8 \times 3; x = 5.4$ km.