Question

the height of a bottle rocket, in meters, is given by the function s(t)=−9t2+103t+403. find the average velocity of the bottle rocket over the intervals [4.499,4.5] and [4.5,4.501]. use this information to approximate the instantaneous velocity of the bottle rocket at time t=4.5.

Answer 1

The average velocity of the bottle rocket are two given time intervals is 30 m/s and 20 m/s respectively, while the instantaneous velocity at 4.5 seconds is 22 m/s.

The given parameters;

• s(t) = -9t² + 103t + 403
• time interval (4.499, 4.5), and (4.5, 4.501)

The average velocity of the bottle rocket over the intervals is calculated as follows;

when the interval = [4.499, 4.5]

$s(4.499)= -9(4.499)^2 + 103(4.499) + 403 = 684.22 \ m\\\\s(4.5) = -9(4.5)^2 + 103(4.5) + 403 = 684.25 \ m$

$v = \frac{\Delta \ s}{\Delta \ t} = \frac{684.25 - 684.22}{4.5 - 4.499} = 30 \ m/s$

when the interval = [4.5, 4.501]

$s(4.5) = -9(4.5)^2 + 103(4.5) + 403 = 684.25 \ m\\\\s(4.501) = -9(4.501)^2 + 103(4.501) + 403 =684.27 \ m\\\\v= \frac{684.27 - 684.25}{4.501-4.5} \\\\v = 20 \ m/s$

The instantaneous velocity of the bottle rocket at time t=4.5;

$v = \frac{d \ s(t)}{dt} = -18t + 103\\\\v(4.5) = -18(4.5) + 103\\\\v(4.5) = 22 \ m/s$

Thus, the average velocity of the bottle rocket are two given time intervals is 30 m/s and 20 m/s respectively, while the instantaneous velocity at 4.5 seconds is 22 m/s.

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