Suppose the annual revenue for a company between 2000 and 2005 is approximated by
R(t) = 18t^2 + 4t + 3, where t = 0 represents the year 2000.
(a) Find the Average Value of R(t) between t = 1 and t = 6.
(b) Find the Average Rate of Change of R(t) between t = 1 and t = 6.
6-1 = 5
integral Rdt from 1 to 6 = 6 t^3 + 2 t^2 + 3 t at 6 minus at 1
= 1386 - 11 = 1375
1375/5 = 275
R(6) = 675
R(1) = 25
change = 670
change/5 = 134
To find the average value of R(t) between t = 1 and t = 6, we need to calculate the definite integral of R(t) over the interval [1, 6] and divide it by the length of the interval (6 - 1 = 5).
(a) Average Value of R(t) between t = 1 and t = 6:
Step 1: Find the integral of R(t) over the interval [1, 6].
∫[1, 6] R(t) dt = ∫[1, 6] (18t^2 + 4t + 3) dt.
Integrating each term separately:
∫[1, 6] 18t^2 dt + ∫[1, 6] 4t dt + ∫[1, 6] 3 dt.
Step 2: Evaluate the definite integrals.
Using the power rule of integration:
(18/3) * t^3 + (4/2) * t^2 + 3t | from 1 to 6.
Simplifying and plugging in the limits:
(6 * 6^3 + 4 * 6^2 + 3 * 6) - (6 * 1^3 + 4 * 1^2 + 3 * 1).
Calculating:
(6 * 216 + 4 * 36 + 18) - (6 * 1 + 4 * 1 + 3).
Simplifying further:
(1296 + 144 + 18) - (6 + 4 + 3).
Final calculation:
1458 - 13 = 1445.
Step 3: Divide the sum by the length of the interval.
Average Value = Sum / Length of Interval = 1445 / 5 = 289.
Therefore, the average value of R(t) between t = 1 and t = 6 is 289.
(b) Average Rate of Change of R(t) between t = 1 and t = 6:
The average rate of change of R(t) is the slope of the secant line connecting the points (1, R(1)) and (6, R(6)).
Step 1: Calculate R(1):
Substitute t = 1 into the equation R(t) = 18t^2 + 4t + 3.
R(1) = 18(1)^2 + 4(1) + 3 = 18 + 4 + 3 = 25.
Step 2: Calculate R(6):
Substitute t = 6 into the equation R(t) = 18t^2 + 4t + 3.
R(6) = 18(6)^2 + 4(6) + 3 = 648 + 24 + 3 = 675.
Step 3: Calculate the average rate of change using the formula:
Average Rate of Change = (R(6) - R(1)) / (6 - 1).
Plugging in the values:
(675 - 25) / (6 - 1) = 650 / 5 = 130.
Therefore, the average rate of change of R(t) between t = 1 and t = 6 is 130.