The demand function for Sportsman 5 X 7 tents is given by the following function where p is measured in dollars and x is measured in units of a thousand. (Round your answers to three decimal places.)
p = f(x) = -0.1x^2 - x + 40
(a) Find the average rate of change in the unit price of a tent if the quantity demanded is between the following intervals.
between 5100 and 5150 tents $ per 1000 tents
between 5100 and 5110 tents $ per 1000 tents
(b) What is the rate of change of the unit price if the quantity demanded is 5100?
$ per 1000 tents
To find the average rate of change, we need to calculate the slope of the line connecting two points on the demand curve. In this case, we have the demand function p = f(x) = -0.1x^2 - x + 40.
(a) To find the average rate of change in the unit price between 5100 and 5150 tents, we need to find the values of p at x = 5100 and x = 5150.
1. Average Rate of Change = (p2 - p1) / (x2 - x1)
Let's find the values of p at x = 5100 and x = 5150.
For x = 5100:
p = -0.1(5.1^2) - 5.1 + 40 = -0.1(26.01) - 5.1 + 40 = -2.601 - 5.1 + 40 = 32.299
For x = 5150:
p = -0.1(5.15^2) - 5.15 + 40 = -0.1(26.5225) - 5.15 + 40 = -2.65225 - 5.15 + 40 = 32.197
Now, let's calculate the average rate of change:
Average Rate of Change = (p2 - p1) / (x2 - x1)
Average Rate of Change = (32.197 - 32.299) / (5150 - 5100)
Average Rate of Change = (-0.102) / (50)
Average Rate of Change = -0.00204
Therefore, the average rate of change in the unit price of a tent between 5100 and 5150 tents is approximately $0.002 per 1000 tents.
Now, to find the average rate of change in the unit price between 5100 and 5110 tents:
For x = 5100:
p = -0.1(5.1^2) - 5.1 + 40 = -0.1(26.01) - 5.1 + 40 = -2.601 - 5.1 + 40 = 32.299
For x = 5110:
p = -0.1(5.11^2) - 5.11 + 40 = -0.1(26.1121) - 5.11 + 40 = -2.61121 - 5.11 + 40 = 32.189
Average Rate of Change = (p2 - p1) / (x2 - x1)
Average Rate of Change = (32.189 - 32.299) / (5110 - 5100)
Average Rate of Change = (-0.11) / (10)
Average Rate of Change = -0.011
Therefore, the average rate of change in the unit price of a tent between 5100 and 5110 tents is approximately $0.011 per 1000 tents.
(b) To find the rate of change of the unit price if the quantity demanded is 5100, we need to find the derivative of the demand function with respect to x.
p = f(x) = -0.1x^2 - x + 40
To find the derivative of f(x), we differentiate each term with respect to x:
dp/dx = -0.1 * 2x - 1 = -0.2x - 1
Now, substitute x = 5100 into the derivative:
dp/dx = -0.2(5100) - 1 = -1020 - 1 = -1021
Therefore, the rate of change of the unit price when the quantity demanded is 5100 is approximately $1021 per 1000 tents.