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

Answer:

$302,500

Step-by-step 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:

$m=\frac{37,500-17,500}{7-15}\\m= -2,500$

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.

$\frac{d(R(c))}{dc}=0 =-5000c + 55,000\\c=\ 11$

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

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