A scientist recorded the movement of an eagle and modeled it with the function x(t)=2t^2 where x is in
miles and t is in hours. What was the eagle's average velocity between times t = 5 and t = 7?
The average velocity of the eagle between times t=a and t=b is given by the formula:
average velocity = (distance traveled between times a and b) / (time elapsed between times a and b)
To apply this formula, we need to find the distance traveled by the eagle between times t=5 and t=7. We can do this by subtracting the distance traveled at time t=5 from the distance traveled at time t=7:
distance traveled between t=5 and t=7 = x(7) - x(5)
= 2(7^2) - 2(5^2)
= 2(49) - 2(25)
= 48 miles
The time elapsed between t=5 and t=7 is 7 - 5 = 2 hours. Therefore, the average velocity of the eagle between times t=5 and t=7 is:
average velocity = (distance traveled between t=5 and t=7) / (time elapsed between t=5 and t=7)
= 48 / 2
= 24 miles per hour
Therefore, the eagle's average velocity between times t=5 and t=7 was 24 miles per hour.
To find the average velocity between times t = 5 and t = 7, we need to find the change in position divided by the change in time.
The position function is x(t) = 2t^2.
Let's find the position at t = 5:
x(5) = 2(5)^2
x(5) = 2(25)
x(5) = 50 miles
Now, let's find the position at t = 7:
x(7) = 2(7)^2
x(7) = 2(49)
x(7) = 98 miles
The change in position is given by the difference in the two positions:
Δx = x(7) - x(5)
Δx = 98 - 50
Δx = 48 miles
The change in time is given by the difference in the two time values:
Δt = 7 - 5
Δt = 2 hours
Now, we can find the average velocity:
Average Velocity = Δx / Δt
Average Velocity = 48 miles / 2 hours
Average Velocity = 24 miles/hour
Therefore, the eagle's average velocity between times t = 5 and t = 7 is 24 miles per hour.