Question

Suppose customers in a hardware store are willing to buy​ N(p) boxes of nails at p dollars per​ box, as given by the following function.
N(p)= 100- 3p² ​; 1 <= p <= 4

a. Find the average rate of change of demand for a change in price from ​$2 to ​$3.
b. Find the instantaneous rate of change of demand when the price is ​$2.
c. Find the instantaneous rate of change of demand when the price is ​$3.

Answer 1

Answer:

a) -15 unites per dollar

b) -12 units per dollar

c) -18 units per dollar

Step-by-step explanation:

We are given the following in the question:

$N(p) = 100-3p^2, 1\leq p \leq 4$

where N(p) is the number of boxes of nails at p dollars per​ box.

a) average rate of change of demand for a change in price from ​$2 to ​$3.

Average rate of change =

$\displaystyle\frac{N(3) -N(2)}{3-2}\\\\=\frac{100-3(3)^2-(100 - 3(2)^2)}{1}\\\\= -15\text{ units per dollar}$

b) instantaneous rate of change of demand when the price is ​$2

Instantaneous rate =

$N'(p) = \displaystyle\frac{d(N(p))}{dp} = -6p$

$N'(2) = -6(2) = -12$

The instantaneous rate of change of demand when the price is ​$2 is decreasing at 12 units per dollar.

c) instantaneous rate of change of demand when the price is ​$3

Instantaneous rate =

$N'(3) = -6(8) = -18$

The instantaneous rate of change of demand when the price is ​$3 is decreasing at 18 units per dollar.