Q. 2 TF - Calculus

Question

What was your average distance from the stop sign from the time that you first saw it to the time that you came to a stop? 

Step-by-Step Solution

Answer

The solution is $21.67$ feet.

Step 1. Given information

The distance from the stop sign (in feet) seconds after stepping on the breaks is given by the function,

$s\left(t\right)=3t^3-12t^2-9t+54$

Step 2. Calculation

The average value of the distance function is given by,

$$\begin{gathered} =\frac{1}{b-a}{\int }_a^{b}s\left(t\right)dt \\ =\frac{1}{4.6}{\int }_0^{4.6}s\left(t\right)dt \\ =\frac{1}{4.6}{\int }_0^{4.6}(3t^3-12t^2-9t+54)dt \\ =\frac{1}{4.6}{[\frac{3t^4}{4}-12\frac{t^3}{3}-9\frac{t^2}{2}+54t]}_0^{4.6}=\frac{1}{4.6}[335.8-389.3-95.2+248.4] \\ =\frac{1}{4.6}[99.7] \\ =21.67 \end{gathered}$$

Thus, the average distance is $21.67$ feet.