Question

A company sells 1000 packs of cards per day at a price of $5 per pack. With every $0.02 reduction in price, 10 more packs a day are sold. Under these conditions, what is the maximum possible income per day, and what price per pack of cards will produce this income? Include how much extra money is made with this new price structure.

Answer 1

Step-by-step explanation

From the statement, we know that:

0. the company sells 1000 packs of cards per day for $5 per pack,

,

1. with every $0.02 reduction in price, 10 more packs a day are sold.

To solve this problem, we define the following variables and functions:

• r = # of price reductions,

• n(r) = # of packs sold by the company as a function of r,

,

• p(r) = price per pack (in $) as a function of the # of price reductions,

,

• I(r) = income (in $) as a function of the # of price reductions.

Using points 1 and 2, we write the following functions: function of # of packs sold n(r) as:

• the i

`n(r)=1000+10*r,`

• the function of the price per pack p(r):

`p(r)=5-0.02*r,`

• the function for the income I(r) is given by the product of the # of packs sold n(r) and the price per pack p(r):

`I(r)=n(r)*p(r)=(1000+10*r)*(5-0.02*r)=-0.2r^2+30r+5000.`

(1) To find the maximum income, we must maximize the function I(r) for r. To do that, we compute and make equal to zero its first derivative:

`I^(\prime)(r)=-0.2*2r+30=0.`

Solving for r, we get:

`\begin{gathered} -0.4r+30=0, \\ 0.4r=30, \\ r=(30)/(0.4)=75. \end{gathered}`

We have found that the maximum income is achieved when the # of price reductions is equal to r = 75.

(2) s

`I(75)=-0.2*75^2+30*75+5000=6125.`

We have found that the maximum possible income per day is $6125.

(3) s

`p(r)=5-0.02*75=3.5.`

We have found that the price per pack that maximizes the income is $3.5.

(4) s

`1000*\text{ \$}5=\text{ \$}5000.`

With this new price structure, the company wins $6125 s

Answer

• The maximum possible income per day is $6125.

,

• The price per pack that maximizes the income is $3.5.

,

• The company makes $1125 extra with this new price structure.