The rate of change R, in kilometers per hour, of the altitude of a hot air balloon is given by R(t)=t^3-4t^2+6 for time on the interval [0,4], where t is measured in hours. Assume the balloon is on ground level.
What is the maximum altitude of the balloon during the interval [0,4]?
9.727
To find the maximum altitude of the balloon during the interval [0,4], we need to determine the highest point on the graph of the function R(t) = t^3 - 4t^2 + 6.
1. Calculate the derivative of R(t) with respect to t. This will give us the rate of change of altitude with respect to time.
R'(t) = 3t^2 - 8t
2. Set the derivative equal to zero and solve for t to find the critical points.
3t^2 - 8t = 0
t(3t - 8) = 0
t = 0 or t = 8/3
3. Evaluate the altitude at the critical points and the endpoints of the interval [0,4].
R(0) = 0^3 - 4(0)^2 + 6 = 6
R(4) = 4^3 - 4(4)^2 + 6 = 22
R(8/3) = (8/3)^3 - 4(8/3)^2 + 6 ≈ 13.037
4. Compare the altitudes to find the maximum.
The maximum altitude is the highest of R(0) = 6, R(4) = 22, and R(8/3) ≈ 13.037.
Therefore, the maximum altitude of the balloon during the interval [0,4] is 22 kilometers.
To find the maximum altitude of the balloon during the interval [0,4], we need to find the maximum value of the function R(t) = t^3 - 4t^2 + 6.
One way to find the maximum value of a function is by finding its critical points. Critical points occur where the derivative of the function is either zero or undefined.
To find the critical points of R(t), we need to take the derivative of the function with respect to t. Let's do that:
R'(t) = d/dt (t^3 - 4t^2 + 6)
= 3t^2 - 8t
Now, to find the critical points, we set the derivative R'(t) equal to zero and solve for t:
3t^2 - 8t = 0
Factoring out t, we get:
t(3t - 8) = 0
Setting each factor equal to zero, we have:
t = 0 or (3t - 8) = 0
The first solution, t = 0, is not within the interval [0,4], so it can be ignored.
Solving the second equation, we have:
3t - 8 = 0
3t = 8
t = 8/3
Since the interval is [0,4], our critical point t = 8/3 is within the interval.
Now, we need to evaluate the function R(t) at the critical point(s) and at the endpoints of the interval to find the maximum value. Let's do that:
R(0) = (0)^3 - 4(0)^2 + 6 = 6
R(4) = (4)^3 - 4(4)^2 + 6 = -22
R(8/3) = (8/3)^3 - 4(8/3)^2 + 6 = 8
Comparing these values, we see that R(8/3) = 8 is the maximum altitude of the balloon during the interval [0,4].
Find dR/dt, set equal to zero, find time. PUt that in the original equation for altitude.
R'=3t^2-8t=0
t(3t-8)=0
t=0, t= 8/3 sec
Rmax=R(8/3)= you do it.