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) = 80-5p2; Find the average rate of change of demand for a change in price from $2 to $3. The average rate of change of demand for a change in price from $2 to $3 is -25 boxes per dollar. (Type an integer or a decimal.) Find the instantaneous rate of change of demand when the price is $2. The instantaneous rate of change of demand when the price is $2 is - 20 boxes per dollar. (Type an integer or a decimal.) Find the instantaneous rate of change of demand when the price is $3. The instantaneous rate of change of demand when the price is $3 is boxes per dollar. (Type an integer or a decimal.)

Answer 1

Answer:

(a) - 25 boxes per dollar

(b) - 20 boxes per dollar

Step-by-step explanation:

Given that,

Consumer's willing to buy boxes of nails at p dollars per box:

N(p) = 80 - 5p^{2}

(a) Change in price from $2 to $3.

N(2) = 80 - 5(2)^{2}

       = 80 - 20

       = 60

N(3) = 80 - 5(3)^{2}

       = 80 - 45

       = 35

Therefore, the average rate of change of demand is

= [N(3) - N(2)] ÷ (3 - 2)

= 35 - 60

= - 25 boxes per dollar.

(b) N(p) = 80 - 5p^{2}

Now, differentiating the above function with respect to p,

N'(p) = -10p

Therefore, the instantaneous rate of change of demand when the price is $2 is calculated as follows:

N'(p) = -10p

N'(2) = -10 × 2

        = -20 boxes per dollar