Question

the height above the ground, measured in feet, of a ball thrown off a 65 ft cliff with an initial velocity of 5 feet per second is given by h(t)=65 + 5t-16t^{2}. what is the error in measuring the height of the ball after 1.4 seconds of motion with an error in time measurement of 0.01 seconds? error in height = feet

Explanation:

Step1: First, find the height function value at $t = 1.4$

Substitute $t = 1.4$ into $h(t)=65 + 5t-16t^{2}$.
$h(1.4)=65+5\times1.4 - 16\times(1.4)^{2}$
$=65 + 7-16\times1.96$
$=72-31.36$
$=40.64$.

Step2: Then, find the derivative of the height - function

$h(t)=65 + 5t-16t^{2}$, so $h^\prime(t)=5 - 32t$.
Evaluate the derivative at $t = 1.4$, $h^\prime(1.4)=5-32\times1.4=5 - 44.8=-39.8$.

Step3: Use the differential formula for error

The differential $dh=h^\prime(t)dt$. Here, $dt = 0.01$.
Substitute $h^\prime(1.4)=-39.8$ and $dt = 0.01$ into the formula, $dh=-39.8\times0.01=- 0.398$.

Answer:

$-0.398$