Questions: A businessman dropped a coin from the top floor of his office building and it fell according to the formula S(t)=-16 t^2+12 t^0.5, where t is the time in seconds and S(t) is the distance in feet from the top of the building. What was the average speed of the fall? Use the fact that the coin hit the ground in exactly 2.2 seconds. Round your answer to 2 decimal places.

Solution Steps

To find the average speed, we first need to determine the total distance the coin fell. This is given by the function \( S(t) = -16t^2 + 12t^{0.5} \). We need to evaluate this function at \( t = 2.2 \) seconds, which is when the coin hits the ground.

\[ S(2.2) = -16(2.2)^2 + 12(2.2)^{0.5} \]

Calculating each term:

\[ -16(2.2)^2 = -16 \times 4.84 = -77.44 \]

\[ 12(2.2)^{0.5} = 12 \times 1.4832 \approx 17.7984 \]

Thus, the total distance fallen is:

\[ S(2.2) = -77.44 + 17.7984 = -59.6416 \]

Since the distance is negative, it indicates the direction of the fall. The magnitude of the distance is \( 59.6416 \) feet.

The average speed is calculated by dividing the total distance by the total time taken. The total distance is \( 59.6416 \) feet, and the time taken is \( 2.2 \) seconds.

\[ \text{Average Speed} = \frac{59.6416}{2.2} \approx 27.1098 \text{ feet per second} \]

Rounding to two decimal places, the average speed is \( 27.11 \) feet per second.

Final Answer