Could someone answer this question 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]
only the 0.02x is ^ on the e
give the answer as an integer or decimal rounded to two decimal places.
To find the average price over the supply interval [31, 36], we need to calculate the definite integral of the supply function, S(x), over that interval and then divide the result by the length of the interval.
The integral of S(x) with respect to x is given by:
∫(S(x)) dx = ∫[5(e^(0.02x) - 1)] dx
Integrating the function, we get:
∫[5(e^(0.02x) - 1)] dx = [5(50e^(0.02x) - x)] / 0.02 + C
Next, we evaluate the integral over the supply interval [31, 36]:
Average Price = (1 / (b - a)) * ∫[a to b] S(x) dx
where a is the lower limit (31) and b is the upper limit (36).
Average Price = (1 / (36 - 31)) * ∫[31 to 36] S(x) dx
Plugging in the values, we have:
Average Price = (1/5) * [5(50e^(0.02x) - x)] / 0.02 | [31 to 36]
Now, substitute x = 36 into the expression and find the value:
Average Price = (1/5) * [5(50e^(0.02*36) - 36)] / 0.02 - [(1/5) * [5(50e^(0.02*31) - 31)] / 0.02
Evaluating this expression will give you the average price over the supply interval [31, 36]. Round the result to two decimal places.