Question

The distance an object falls (when released from rest, under the influence of Earth's gravity, and with no air resistance) is given by d(t)=16t 2

, where d is measured in feet and t is measured in seconds A rock climber sits on a ledge on a vertical wall and carefully observes the time it takes for a small stone to fall from the ledge to the ground. a. Compute d ′
(t). What units are associated with the derivative, and what does it measure? b. If it takes 5.5 s for a stone to fall to the ground, how high is the ledge? How fast is the stone moving when it strikes the ground (in miles per hour)?

Answer 1

The derivative of d(t) = 16t^2 is d'(t) = 32t, representing the rate of change of distance with respect to time. The height of the ledge is 484 feet and the stone's velocity when it strikes the ground is 176 ft/s (or 120 mph).

Step-by-step explanation:

a. To find the derivative of d(t), we can apply the power rule for derivatives. The derivative of d(t) = 16t^2 is given by d'(t) = 32t. The units associated with the derivative are feet per second (ft/s), which represent the rate of change of distance with respect to time.

b. To find the height of the ledge, we can substitute the time (t = 5.5s) into the equation d(t) = 16t^2. The height of the ledge is: d(5.5) = 16(5.5)^2 = 484 feet. To find the stone's velocity when it strikes the ground, we can use the derivative d'(t) = 32t. Substituting t = 5.5s, the stone's velocity is: d'(5.5) = 32(5.5) = 176 ft/s. To convert the velocity to miles per hour (mph), we can use the conversion factor: 1 mile = 5280 feet and 1 hour = 3600 seconds. So, the stone's velocity in mph is: 176 ft/s * (1 mile/5280 ft) * (3600 s/1 hour) = 120 mph.