2. An automobile is braking to a stop and its velocity is recorded at various times and listed in the following table. Time t is measured in seconds and velocity v in feet per second. t (seconds) | 0 | 2 | 4 | 6 | 8 | 10 --- | --- | --- | --- | --- | --- | --- v(t) (feet per second) | 22 | 14 | 8 | 5 | 2 | 1 (a) Use a trapezoidal approximation with ?t = 2 seconds to estimate the distance required for the driver to bring the car to a stop from the initial speed. Show the work that lead to your conclusion. (b) The function V(t) = 0.18t² - 4t + 22 is proposed as a model for the velocity of the braking car at time t, where t is measured in seconds and V(t) is measured in feet per second. Use V(t) to approximate the average velocity of the braking car during the 10-second time period. (c) Assume that the driver applies a braking force in such a way that the deceleration at time t is proportional to the velocity of the car at that time. Write and solve a differential equation that models the braking of the car.
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The formula for the trapezoidal rule is: \[ \text{Distance} = \frac{\Delta t}{2} \sum_{i=1}^{n} (V_{i-1} + V_i) \] where \(V_i\) is the velocity at time \(i\), and \(\Delta t\) is the time interval between measurements. Given \(\Delta t = 2\) seconds, we can Show more…
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