Q. 55 - Calculus

Question

If Katie walked at $3$ miles per hour for $20$ minutes and then sprinted at $10$ miles an hour for $8$ minutes, how fast would Dave have to walk or run to go the same distance as Katie did at the same time while moving at a constant speed? Sketch a graph of Katie’s position over time and a graph of Dave’s position over time on the same set of axes.

Step-by-Step Solution

Answer

Ans:   The speed of Dave is: $5$ miles per hrs.

   Graph for the position of Katie and Dave over time.

Step 1. Given information.

given,  

        Katie walked at $3$ miles per hour for $20$ minutes and then sprinted at $10$ miles an hour for $8$ minutes.

Step 2. The main objective is to determine the speed of a Dave to cover the same distance as Katie did at the same time with constant speed.

The distance d covered by Katie is calculated as $d=v.t$, where v is the speed of a person and t is the time taken by a Katie.

There are $60$ minutes in one hour.

Divide $20$ minutes by $60$ minutes to calculate the time in hours.

       $\frac{20}{60}=\frac{1}{3}$

Therefore, Katie covers a certain distance in $\frac{1}{3}$ hours with $3$ miles per hour.

Divide $8$ minutes by $60$ minutes to calculate the time in hours.

           $\frac{8}{60}=\frac{2}{15}$

Therefore, Katie covers a certain distance in $\frac{2}{15}$ hours with $10$ miles per hour.

Use the formula $d=v.t$ to calculate the total distance covered by Katie.

         $\begin{array}{r}d=3\cdot \frac{1}{3}+10\cdot \frac{2}{15}\\ =1+\frac{4}{3}\\ =\frac{7}{3}\end{array}$

Therefore the total distance covered by Katie is $\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ miles.

Add $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ hours and $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$15$}\right.$ hours to calculate the total time taken by Katie.

          $\begin{array}{r}\frac{1}{3}+\frac{2}{15}=\frac{5+2}{15}\\ =\frac{7}{15}\end{array}$

Therefore, the total time taken is $\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$15$}\right.$ hours.

It is given that Dave takes the same distance and same time as Katie will take. The speed of any object can be calculated as $v=\raisebox{1ex}{$d$}\!\left/ \!\raisebox{-1ex}{$t$}\right.$, where d is the covered distance and t is the time.

Use the formula $v=\raisebox{1ex}{$d$}\!\left/ \!\raisebox{-1ex}{$t$}\right.$ to calculate the speed of Dave.

               $\begin{array}{r}v=\frac{\left(\frac{7}{3}\right)}{\left(\frac{7}{15}\right)} =\frac{7}{3}\times \frac{15}{7}=5\end{array}$

Thus, the speed of Dave is 5 miles per hour.

For any time t, the distance covered by Dave is $5t$.

In 20 minutes Katie, runs 1 mile and in the next $8$ minutes, Katie sprinted $\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ miles which is equal to $1.33$ miles.

Step 3. Graph the distance for Dave which is 5 t and for Katie on the same coordinate plane.

Figure $1$ shows the position of Katie and Dave over time.