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 two points given to find the rate of change and then apply it to the $\$$ $4$ . $60$ fare. $\newline$ Calculation: First, find the slope (m) using the two points $(0, 2.8)$ and $(12, 6.8)$ . $\newline$ $m = \frac{6.8 - 2.8}{12 - 0} = \frac{4}{12} = \frac{1}{3}$ . $\newline$ Now, use the point-slope form to find the distance for a $\$$ $4$ . $60$ fare. $\newline$ $4.60 = 2.8 + \left(\frac{1}{3}\right)d$ $\newline$ $d = \frac{4.60 - 2.8}{\frac{1}{3}}$ $\newline$ $d = \frac{1.8}{\frac{1}{3}}$ $\newline$ $\$$ $0$ $\newline$ $\$$ $1$ km
Calculate Gradient: (c) The gradient of the graph is the slope calculated in step (b). Calculation: The gradient is $\frac{1}{3}$ .
Find Equation of the Graph: (d) The equation of the graph is in the form $y = mx + b$ , where $m$ is the gradient and $b$ is the y-intercept. $\newline$ Calculation: Using the gradient $\frac{1}{3}$ and the y-intercept $2.8$ , the equation is $y = \left(\frac{1}{3}\right)x + 2.8$ .