Question

The following shows the distance-time graph of an athlete in a race who moves at a constant speed. If after 30 minutes of running the speed is reduced to half, then how much time will the athlete take to run 45 miles?

• A. 4 hours
• B. 3.5 hours
• C. 3.25 hours
• D. 3 hours

Answer 1

The correct option is B 3.5 hours


The graph passes through $(0.5,11.25)$ and $(0.75,16.875)$.
$x_1=0.5,y_1=11.25;x_2=0.75,y_2=16.875$
Unit rate $=\frac{\text{Rise}}{\text{Run}}=\frac{y_2-y_1}{x_2-x_1}=\frac{16.875-11.25}{0.75-0.5}=\frac{5.625}{0.25}=\frac{5625}{250}=22.5$ miles/hr
Time required to run $45$ miles with original speed $\frac{45}{22.5}=2$ hours

$60$ minutes $=1$ hour
$30$ minutes $=\frac{1}{60}\times 30=\frac{30}{60}=\frac{1}{2}=0.5$ hour
Distance traveled in $30$ minutes, i.e., $12$ hour $=\frac{22.5}{2}=\frac{225}{40}=11.25$ miles

Remaining distance $=45-11.25=33.75$ miles
After 30 minutes of running, the speed is reduced to half.
New speed $=\frac{1}{2}\times 22.5=11.25$ miles/hr
Time required to run 33.75 miles $\frac{33.75}{11.25}=3$ hours
Total time taken to run 45 miles $=3+0.5=3.5$ hours

Note: After we calculated the total time needed for $45$ miles at original speed as $2$ hours, we could solve the rest in fewer steps as shown.

After $30$ minutes, the speed is reduced to half. Due to this, the time needed will be doubled (speed and time are inversely proportional) so the runner will take $3$ hours in place of $1.5$ hours in normal condition for the remaining distance. Therefore, in total, time taken will be $0.5$ (original speed) $+3$ hours (with half of the speed) $=3.5$ hours.