The profit, in thousands of dollars, of a new product over the first 6 months, 0 ≤ t ≤ 6, is approximated by the model P=5(Sqrt(t)+30). Write the integral used to find the average value of the profit function and solve using your calculator
To find the average value of a function, we need to calculate its definite integral over a given interval and then divide the result by the width of the interval. In this case, we want to find the average value of the profit function P(t).
Given that the time interval is from t = 0 to t = 6, the average value of the profit function can be found using the following integral:
1/(6-0) * ∫[0 to 6] 5(sqrt(t) + 30) dt
Simplifying, we have:
1/6 * ∫[0 to 6] 5sqrt(t) + 150 dt
Now we can use a calculator or software that supports symbolic integration to evaluate this integral. Alternatively, we can break it down into two separate integrals:
1/6 * (∫[0 to 6] 5sqrt(t) dt + ∫[0 to 6] 150 dt)
The first integral corresponds to the profit function, while the second integral is a constant term. Let's calculate them individually:
1/6 * (∫[0 to 6] 5sqrt(t) dt) = 1/6 * [2/3 * t^(3/2)] from 0 to 6
= 1/6 * (2/3 * 6^(3/2) - 2/3 * 0^(3/2))
≈ 2.039
1/6 * (∫[0 to 6] 150 dt) = 1/6 * [150t] from 0 to 6
= 1/6 * (150 * 6 - 150 * 0)
= 150/6
= 25
Adding the two results together, we get:
2.039 + 25 ≈ 27.039
Therefore, the average value of the profit function over the interval [0, 6] is approximately $27,039.
Note: The calculator can vary depending on the software or device you are using, but most scientific calculators or math software should be capable of performing these calculations.
the average value of f(t) on [a,b] is
∫[a,b] f(t) dt
----------------
b-a
Shouldn't need a calculator for this one.