Question

A boutique handmade umbrella factory currently sells 37500 umbrellas per year at a cost of 7 dollars each. In the previous year when they raised the price to 15 dollars, they only sold 17500 umbrellas that year. Assuming the amount of umbrellas sold is in a linear relationship with the cost, what is the maximum revenue?

Answer 1

$302,500

Explanation:

If cost (C) = $7, then Sales (S) = 37,500 units

If cost (C) = $15, then Sales (S) = 17,500 units

The slope of the linear relationship between units sold and cost is:

`m=(37,500-17,500)/(7-15)\\m= -2,500`

The linear equation that describes this relationship is:

`s-s_0 =m(c-c_0)\\s-17500 =-2500(c-15)\\s(c)=-2500c + 55,000`

The revenue function is given by:

`R(c) = c*s(c)\\R(c)=-2500c^2 + 55,000c`

The cost at which the derivative of the revenue equals zero is the cost that yields the maximum revenue.

`(d(R(c)))/(dc)=0 =-5000c + 55,000\\c=\$11`

The optimal cost is $11, therefore, the maximum revenue is:

`R(11)=-2500*11^2 + 55,000*11\\R(11)=\$ 302,500`