Question

The speedometer reading (v) on a car was observed at 1-minute intervals and recorded in the chart. Use Simpson’s Rule to estimate the distance travelled by the car

Short answer

The distance travelled by the car is approximately 8.58 miles.

Step-by-step solution

Step 1: Applied the interval from table

r (t) on the interval [0,24]

\(\begin{aligned}{l}d &= \int\limits_0^{10} {v(t)dt} \\\Delta t &= \frac{1}{{60}}\end{aligned}\)

Step 2: Solution of approximate integral

Use the Simpson’s Rule

\({S_n} = \frac{{\Delta t}}{3}\left( {v\left( {{t_0}} \right) + 4v\left( {{t_1}} \right) + 2v\left( {{t_2}} \right) + \cdots + v\left( {{t_n}} \right)} \right)\)

\(\begin{aligned}{l}{S_{10}} &= \frac{{\Delta t}}{3}\left( {v\left( {{t_0}} \right) + 4v\left( {{t_1}} \right) + 2f\left( {{t_2}} \right) + \cdots + v\left( {{t_1}0} \right)} \right)\\ &= \frac{{\frac{1}{{60}}}}{3}(v(0) + 4v(1) + 2v(2) + 4v(3) + 2v(4) + 4v(5) + 2v(6) + 4v(7) + 2v(8) + 4v(9) + v(10))\\ &= \frac{1}{{180}}(40 + 4 \cdot 42 + 2 \cdot 45 + 4 \cdot 49 + 2 \cdot 52 + 4 \cdot 54 + 2 \cdot 56 + 4.57 + 2.57 + 4.55 + 56)\\ \approx 8.58\end{aligned}\)

Step 3: Final Solution

\(\begin{aligned}{l}d = \int\limits_0^{10} {v(t)dt} \approx {S_{10}}\\ \approx 8.58miles\end{aligned}\)