A cyclist rides 42 kilometers from Town A to Town B. - For the first of the distance, the cyclist rides at a speed of 28 kilometers per hour. - The cyclist then rests for 15 minutes. - The cyclist rides the remaining distance at a speed of 16 kilometers per hour. What is the cyclist's average speed for the entire trip, in meters per second? (1 kilometer = 1,000 meters and 1 hour = 3,600 seconds)

Solution available with step-by-step explanation

Super-Hard SAT Math Question 170 | Free SAT Prep | Aniko

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Question 170·200 Super-Hard SAT Math Questions·Problem Solving and Data Analysis

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A cyclist rides 42 kilometers from Town A to Town B.

For the first $\frac{2}{3}$ of the distance, the cyclist rides at a speed of 28 kilometers per hour.
The cyclist then rests for 15 minutes.
The cyclist rides the remaining distance at a speed of 16 kilometers per hour.

What is the cyclist's average speed for the entire trip, in meters per second?

(1 kilometer = 1,000 meters and 1 hour = 3,600 seconds) Sourсе: аn⁠ікo.aі

When average speed includes breaks, always use $\text{average speed}=\frac{\text{total distance}}{\text{total elapsed time}}$ . Split the trip using the given fraction of distance (not time), compute each travel time with $t=\frac{d}{r}$ , add the rest time, and only then convert units. A quick unit shortcut is $1\text{ km/h}=\frac{5}{18}\text{ m/s}$ . © аnik‍ο.aі

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Split the trip by distance

Find $\frac{2}{3}$ of 42 km to get the distance of the first riding segment, and subtract from 42 km to get the remaining distance. Anіко‌.ai - ‍ЅAT Prе‌р

Add all the time

Compute the riding time for each segment using $\text{time}=\frac{\text{distance}}{\text{speed}}$ , and don’t forget to include the 15-minute rest converted to hours.

Average speed and unit conversion

Average speed is $\frac{\text{total distance}}{\text{total time}}$ . Then convert km/h to m/s using $\frac{1000}{3600}=\frac{5}{18}$ .

Desmos Guide

Enter the segment times

In Desmos, enter:

t1 = 28/28
t2 = 14/16
t = t1 + 1/4 + t2

Compute average speed in km/h

Enter v_kmh = 42/t and note the value shown for v_kmh.

Convert km/h to m/s

Enter v_ms = v_kmh*(5/18). аnі⁠ko‍.аi ЅАТ Quеѕtі‍on В‍аnk

Match to an answer choice

Compare the value of v_ms to the four choices and select the one that matches exactly.

Step-by-step Explanation

Find each distance segment

Total distance is 42 km. F‍rom anіko.aі

First segment distance:

\frac{2}{3}\cdot 42 = 28\text{ km}

Remaining distance:

42-28=14\text{ km}

Compute total time (including the rest)

Time for first segment:

\frac{28\text{ km}}{28\text{ km/h}}=1\text{ h}

Rest time: $15$ minutes $=\frac{15}{60}=\frac{1}{4}$ hour.

Time for second segment:

\frac{14\text{ km}}{16\text{ km/h}}=\frac{7}{8}\text{ h}

Total time:

1+\frac{1}{4}+\frac{7}{8}=\frac{17}{8}\text{ h}

Find the average speed in kilometers per hour

Average speed $=$ total distance $\div$ total time:

\frac{42}{\frac{17}{8}}=42\cdot \frac{8}{17}=\frac{336}{17}\text{ km/h}

Convert to meters per second and choose the answer

To convert km/h to m/s, multiply by $\frac{1000}{3600}=\frac{5}{18}$ :

\frac{336}{17}\cdot \frac{5}{18}=\frac{1680}{306}=\frac{280}{51}

So the cyclist's average speed is $\frac{280}{51}$ meters per second.

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