Could someone work this out so I understand it. Thanks
Given the supply function
p=S(x)=5(e^0.02x-1)
Find the average price (in dollars)over the supply interval [31,36]
The 0.02x in the only thing ^ on the e
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To find the average price over the supply interval [31, 36], we need to calculate the average value of the supply function over this interval. Here's how you can work it out:
Step 1: Substitute the upper and lower limits of the interval into the supply function. In this case, we substitute x = 31 and x = 36 into the function S(x):
S(31) = 5(e^(0.02*31) - 1)
S(36) = 5(e^(0.02*36) - 1)
Step 2: Evaluate the supply function at these values to find the corresponding prices:
S(31) ≈ 5(e^0.62 - 1)
S(36) ≈ 5(e^0.72 - 1)
Use a calculator to compute the values inside the parentheses and simplify:
S(31) ≈ 5(1.85917 - 1)
S(36) ≈ 5(2.05585 - 1)
Step 3: Calculate the average value by taking the average of the prices at the lower and upper limits:
Average price = (S(31) + S(36))/2
Substitute the previously calculated values:
Average price ≈ (5(1.85917 - 1) + 5(2.05585 - 1))/2
Simplify the expression:
Average price ≈ (5*0.85917 + 5*1.05585)/2
Step 4: Calculate the final average price by performing the arithmetic:
Average price ≈ (4.29585 + 5.27925)/2
Average price ≈ 4.78755
Therefore, the average price over the supply interval [31, 36] is approximately $4.79.