Question

Henry drives 150 miles at 30 miles per hour and then another 200 miles at 50 miles per hour. What was his average speed, in miles per hour, for the entire journey, to the nearest hundredth? \[\]
A.38.89\[\]
B.48.00\[\]
C.42.33\[\]
D.34.58\[\]

Answer 1

Hint: We find the taken by taken by Henry to drive the first 30 miles as ${{t}_{1}}=\dfrac{{{d}_{1}}}{{{v}_{1}}}$ and to drive the latter 50 miles ${{t}_{2}}=\dfrac{{{d}_{2}}}{{{v}_{2}}}$ where ${{d}_{1}},{{d}_{2}}$ are distances and ${{v}_{1}},{{v}_{2}}$ are speeds. We found the total time take for the journey ${{t}_{1}}+{{t}_{2}}$ and total distance travelled ${{d}_{1}}+{{d}_{2}}$. We find the average speed $\dfrac{{{d}_{1}}+{{d}_{2}}}{{{t}_{1}}+{{t}_{2}}}$ and round off to hundredth digit after decimal. \[\]

Complete step by step answer:
We know the distance covered $d$ by an object is the length of shortest possible path from the initial position to the latter position. If we measure the time from point of start as $t$ then the speed $v$ is the distance covered per unit time. It is given by the formula
\[v=\dfrac{s}{t}\]
The time taken in terms of speed and distance can be obtained from above equation as,
\[t=\dfrac{s}{v}\]
We are given in the question that Henry drives 150 miles at 30 miles per hour and then another 200 miles at 50 miles per hour. Let us denote the distance covered by Henry in the initially as ${{d}_{1}}$ and the speed as ${{v}_{1}}$. So we have${{d}_{1}}=150$ and ${{v}_{1}}=30$miles/hour. So the time taken ${{t}_{1}}$ by Henry to cover the first 30 miles in hours is
\[{{t}_{1}}=\dfrac{{{d}_{1}}}{{{v}_{1}}}=\dfrac{150}{30}=50\]
Similarly let us denote the distance covered by Henry in the after the increase in speed as ${{d}_{2}}$ and the speed as ${{v}_{2}}$. So we have${{d}_{2}}=200$ and ${{v}_{2}}=50$miles/hour. So the time taken ${{t}_{1}}$ by Henry to cover the latter 50 miles in hours is
 \[{{t}_{1}}=\dfrac{{{d}_{2}}}{{{v}_{2}}}=40\]
The total time taken $t$ by Henry to travel the entire journey is the sum of time taken for first 30 miles and then latter 50 miles which means
\[t={{t}_{1}}+{{t}_{2}}=50\text{ hours}+40\text{ hours}=90\text{ hours}\]
The total distance travelled by Henry $d$ in miles is
\[d={{d}_{1}}+{{d}_{2}}=150+200=450\text{ miles}\]
So the average speed $v$ of Henry to cover the entire journey is total distance travelled divided by total time taken that is,
\[v=\dfrac{d}{t}=\dfrac{350}{9}=38.8888...\]
We are asked to find the speed to the nearest hundred. So we round off up to two digits after the decimal and conclude that the speed was 38.89 miles per hour.

So, the correct answer is “Option A”.

Note: We note that speed is different from velocity because speed is a scalar and velocity is vector. Other measures of speed are meter per second, kilometre per hour etc. If an object covers a certain distance at speed ${{v}_{1}}$ and an equal distance at speed ${{v}_{2}}$, the average speed of the whole journey is $\dfrac{2{{v}_{1}}{{v}_{2}}}{{{v}_{1}}+{{v}_{2}}}$.