To determine the car's average speed over each interval of time, we need to calculate the change in distance divided by the change in time within that interval. Let's calculate the average speed for each interval:
a. t = 2 to t = 5 seconds:
For this interval, we need to find the change in distance from t = 2 to t = 5 seconds.
d1 = t1^2 + 2t1 = 2^2 + 2(2) = 8 feet (distance at t = 2 seconds)
d2 = t2^2 + 2t2 = 5^2 + 2(5) = 35 feet (distance at t = 5 seconds)
Change in distance = d2 - d1 = 35 - 8 = 27 feet
Change in time = t2 - t1 = 5 - 2 = 3 seconds
Average speed = Change in distance / Change in time = 27 feet / 3 seconds = 9 feet per second
b. t = 5 to t = 5.5 seconds:
For this interval, we need to find the change in distance from t = 5 to t = 5.5 seconds.
d1 = t1^2 + 2t1 = 5^2 + 2(5) = 35 feet (distance at t = 5 seconds)
d2 = t2^2 + 2t2 = 5.5^2 + 2(5.5) = 45.75 feet (distance at t = 5.5 seconds)
Change in distance = d2 - d1 = 45.75 - 35 = 10.75 feet
Change in time = t2 - t1 = 5.5 - 5 = 0.5 seconds
Average speed = Change in distance / Change in time = 10.75 feet / 0.5 seconds = 21.5 feet per second
c. t = 5.5 to t = 6 seconds:
For this interval, we need to find the change in distance from t = 5.5 to t = 6 seconds.
d1 = t1^2 + 2t1 = 5.5^2 + 2(5.5) = 45.75 feet (distance at t = 5.5 seconds)
d2 = t2^2 + 2t2 = 6^2 + 2(6) = 54 feet (distance at t = 6 seconds)
Change in distance = d2 - d1 = 54 - 45.75 = 8.25 feet
Change in time = t2 - t1 = 6 - 5.5 = 0.5 seconds
Average speed = Change in distance / Change in time = 8.25 feet / 0.5 seconds = 16.5 feet per second
Therefore, the car's average speed over each interval is:
a. t = 2 to t = 5 seconds: 9 feet per second
b. t = 5 to t = 5.5 seconds: 21.5 feet per second
c. t = 5.5 to t = 6 seconds: 16.5 feet per second