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=1and t=6.
(b)Find the Average Rate of Change of R(t) between t=1 and t=6.
averageRevenue*time= int (R(t))dt from t=0 to 6
average Revenue* 6)=int(18t^2+4t+3)dt over interval
=6t^3+2t^2 + 3t over interval= 1296+72+18=
average revenue= 1/6 (1386)=231
check my work.
To find the average value of R(t) between t=1 and t=6, you need to calculate the definite integral of R(t) over the interval [1, 6] and then divide the result by the width of the interval (6 - 1).
(a) Average Value of R(t):
To solve this, follow these steps:
Step 1: Calculate the definite integral of R(t) over the interval [1, 6]:
∫[1, 6] (18t^2 + 4t + 3) dt
Step 2: Integrate each term separately:
= 18∫[1, 6] t^2 dt + 4∫[1, 6] t dt + 3∫[1, 6] 1 dt
Step 3: Apply the power rule of integration:
= 18 * (t^3/3) ∣[1, 6] + 4 * (t^2/2) ∣[1, 6] + 3 * (t) ∣[1, 6]
Step 4: Evaluate each term at the upper and lower limits:
= 18 * (6^3/3 - 1^3/3) + 4 * (6^2/2 - 1^2/2) + 3 * (6 - 1)
Step 5: Simplify the expression:
= 18 * (216/3 - 1/3) + 4 * (36/2 - 1/2) + 3 * 5
= 18 * (215/3) + 4 * (35/2) + 15
= 645 + 70 + 15
= 730
Step 6: Divide the result by the width of the interval (6 - 1):
Average Value = 730 / (6 - 1)
Average Value = 730 / 5
Average Value ≈ 146
Therefore, the average value of R(t) between t=1 and t=6 is approximately 146.
(b) Average Rate of Change of R(t):
To find the average rate of change of R(t) between t=1 and t=6, you need to calculate the change in R(t) divided by the change in t.
Step 1: Calculate the change in R(t):
ΔR = R(6) - R(1)
Substitute t = 6 into the equation of R(t):
R(6) = 18(6^2) + 4(6) + 3
R(6) = 648 + 24 + 3
R(6) = 675
Substitute t = 1 into the equation of R(t):
R(1) = 18(1^2) + 4(1) + 3
R(1) = 18 + 4 + 3
R(1) = 25
ΔR = 675 - 25
ΔR = 650
Step 2: Calculate the change in t:
Δt = 6 - 1
Δt = 5
Step 3: Calculate the average rate of change:
Average Rate of Change = ΔR / Δt
Average Rate of Change = 650 / 5
Average Rate of Change = 130
Therefore, the average rate of change of R(t) between t=1 and t=6 is 130.