Video Transcript
A rock is dropped from a height of
81 feet. Its position 𝑡 seconds after it is
dropped until it hits the ground is given by the function 𝑠 of 𝑡 equals negative
16𝑡 squared plus 81. Determine how long it will take for
the rock to hit the ground. Find the average velocity of the
rock from the point of release until it hits the ground. And find the time 𝑡 according to
the mean value theorem when the instantaneous velocity of the rock is equal to the
average velocity.
The rock will reach the ground when
its position 𝑠 of 𝑡 is equal to zero. We can therefore set this
expression negative 16𝑡 squared plus 81 equal to zero and solve for 𝑡. We add 16𝑡 squared to both sides
and then divide through by 16. And we obtain 𝑡 squared to be
equal to 81 over 16. We then take the square root of
both sides, remembering to take by the positive and negative square root of 81 over
16. And we see that 𝑡 is equal to plus
or minus nine-quarters. Now, we can actually disregard
negative nine-quarters since we’re working in time. And we find that the rock hits the
ground after nine-quarters of a second.
Our next job is to find the average
velocity of the rock over this period of time. The definition for average velocity
is total displacement divided by time taken. The displacement of our rock is its
change in position. That’s negative 81 feet. And it takes nine-quarters of a
second to travel this far. So the velocity is negative 81
divided by nine over four. Remember, to divide by a fraction,
we can multiply by the reciprocal of that fraction. So we have negative 81 times four
over nine. And then we cancel this factor of
nine. And so we obtain that the average
velocity of our rock is negative 36 feet per second.
For the final part of this
question, we’ll need to quote the mean value theorem. Remember, this says that if 𝑓 is a
continuous function over some closed interval 𝑎 to 𝑏 and differentiable at every
point of that open interval 𝑎 to 𝑏. Then there’s a point 𝑐 in this
interval, such that 𝑓 prime of 𝑐 is equal to 𝑓 of 𝑏 minus 𝑓 of 𝑎 over 𝑏 minus
𝑎. We know that the average velocity
is negative 36 feet per second. That’s the equivalent of this
quotient. The instantaneous velocity can be
found by differentiating our function for position. That’s 𝑠 prime of 𝑡 equals
negative 32𝑡. In this case then, we can say that
𝑠 prime of 𝑐 is equal to negative 32𝑐. And we obtain the equation negative
32𝑐 equals negative 36. We solve for 𝑐 by dividing both
sides by negative 32. And we find that the time at which
the instantaneous velocity of the rock is equal to the average velocity is equal to
nine-eighths of a second.
