The demand 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 5700 and 5750 tents= $ per 1000 tents
between 5700 and 5710 tents = $ per 1000 tents
(b) What is the rate of change of the unit price if the quantity demanded is 5700?
$ per 1000 tents
a) between 5700 and 5750 tents = -$0.100 per 1000 tents; between 5700 and 5710 tents = -$0.100 per 1000 tents
b) rate of change of the unit price if the quantity demanded is 5700 = -$0.100 per 1000 tents
(a) To find the average rate of change in the unit price of a tent, we need to calculate the difference in unit price divided by the difference in quantity demanded.
First, we calculate the unit price at the two given intervals. Let's substitute the quantities demanded into the demand function and find the corresponding unit prices:
For 5700 tents:
p(5.7) = -0.1(5.7)^2 - 5.7 + 40
= -0.1(32.49) - 5.7 + 40
= -3.249 - 5.7 + 40
= 31.051
For 5750 tents:
p(5.75) = -0.1(5.75)^2 - 5.75 + 40
= -0.1(33.0625) - 5.75 + 40
= -3.30625 - 5.75 + 40
= 31.94375
The unit price for 5700 tents is $31.051 per 1000 tents, and for 5750 tents, it is $31.944 per 1000 tents.
Now, we can calculate the average rate of change in the unit price within the given intervals.
For the interval between 5700 and 5750 tents:
Average rate of change = (Unit price at 5750 - Unit price at 5700) / (Quantity at 5750 - Quantity at 5700)
= (31.944 - 31.051) / (5750 - 5700)
= 0.893 / 50
= 0.01786
Therefore, the average rate of change in the unit price of a tent between 5700 and 5750 tents is $0.01786 per 1000 tents.
For the interval between 5700 and 5710 tents:
Unit price at 5700 tents = $31.051 per 1000 tents
Unit price at 5710 tents:
p(5.71) = -0.1(5.71)^2 - 5.71 + 40
= -0.1(32.5441) - 5.71 + 40
= -3.25441 - 5.71 + 40
= 31.03559
Average rate of change = (Unit price at 5710 - Unit price at 5700) / (Quantity at 5710 - Quantity at 5700)
= (31.03559 - 31.051) / (5710 - 5700)
= -0.01541 / 10
= -0.001541
Therefore, the average rate of change in the unit price of a tent between 5700 and 5710 tents is -$0.001541 per 1000 tents.
(b) To find the rate of change of the unit price when the quantity demanded is 5700 tents, we can take the derivative of the demand function and evaluate it at x = 5.7 (since 5700/1000 = 5.7).
Taking the derivative of the demand function:
f'(x) = -0.2x - 1
Now, substitute x = 5.7 into the derivative:
f'(5.7) = -0.2(5.7) - 1
= -1.14 - 1
= -2.14
Therefore, the rate of change of the unit price when the quantity demanded is 5700 tents is -$2.14 per 1000 tents.
To find the average rate of change in the unit price of a tent, we need to calculate the difference in unit price divided by the difference in quantity demanded for each interval.
(a) Between 5700 and 5750 tents:
Quantity demanded at 5700 tents: x₁ = 5700/1000 = 5.7
Quantity demanded at 5750 tents: x₂ = 5750/1000 = 5.75
Unit price at 5700 tents: p₁ = f(x₁) = -0.1(5.7)^2 - 5.7 + 40
Unit price at 5750 tents: p₂ = f(x₂) = -0.1(5.75)^2 - 5.75 + 40
Average rate of change = (p₂ - p₁) / (x₂ - x₁)
= (f(x₂) - f(x₁)) / (x₂ - x₁)
= (-0.1(5.75)^2 - 5.75 + 40) - (-0.1(5.7)^2 - 5.7 + 40) / (5.75 - 5.7)
= (-0.1(32.9075) - 5.75 + 40) - (-0.1(32.49) - 5.7 + 40) / 0.05
= (-3.29075 - 5.75 + 40) - (-3.249 - 5.7 + 40) / 0.05
= (-3.29075 + 40 - 5.75) - (-3.249 + 40 - 5.7) / 0.05
= (31.95925) - (31.051) / 0.05
= 0.90825 / 0.05
= 18.165 per 1000 tents
Between 5700 and 5710 tents:
Quantity demanded at 5700 tents: x₁ = 5700/1000 = 5.7
Quantity demanded at 5710 tents: x₂ = 5710/1000 = 5.71
Unit price at 5700 tents: p₁ = f(x₁) = -0.1(5.7)^2 - 5.7 + 40
Unit price at 5710 tents: p₂ = f(x₂) = -0.1(5.71)^2 - 5.71 + 40
Average rate of change = (p₂ - p₁) / (x₂ - x₁)
= (f(x₂) - f(x₁)) / (x₂ - x₁)
= (-0.1(5.71)^2 - 5.71 + 40) - (-0.1(5.7)^2 - 5.7 + 40) / (5.71 - 5.7)
= (-0.1(32.6641) - 5.71 + 40) - (-0.1(32.49) - 5.7 + 40) / 0.01
= (-3.26641 - 5.71 + 40) - (-3.249 - 5.7 + 40) / 0.01
= (-3.26641 + 40 - 5.71) - (-3.249 + 40 - 5.7) / 0.01
= (31.72359) - (31.051) / 0.01
= 0.67259 / 0.01
= 67.259 per 1000 tents
(b) If the quantity demanded is 5700 tents, the rate of change of the unit price can be calculated.
Rate of change = f'(x)
= -0.1(2)(5.7) - 1
= -1.14 - 1
= -2.14
Therefore, the rate of change of the unit price if the quantity demanded is 5700 is $2.14 per 1000 tents.