An organization hosting a 5K race is looking to sell t-shirts at the event.Demand for t-shirts is believed to be normally distributed with a mean of 100 and standard deviation of 15.With a marginal loss of $4 and marginal profit of $5, how many t-shirts should the organization stock for the 5K race?
A)98
B)85
C)103
D)115

Question

Quizplus · Accepted Answer

To determine the optimal number of t-shirts to stock, we need to calculate the expected profit for each possible quantity of t-shirts. Let x represent the number of t-shirts.

The revenue from selling x t-shirts is: 
Revenue = price x quantity = $5x

The cost from producing x t-shirts is: 
Cost = (mean demand - x) x marginal loss = (100 - x) x $4

The expected profit from selling x t-shirts is therefore: 
Profit = Revenue - Cost = $5x - (100 - x) x $4 = $x - $4x + $400

Simplifying: 
Profit = -$3x + $400

To find the optimal quantity of t-shirts, we need to maximize the expected profit. We can do this by finding the vertex of the parabola that represents the profit function.

The x-coordinate of the vertex is: 
x = -b / 2a = -400 / (-6) = 66.67

We should round up to the nearest integer, since we can't stock a fraction of a t-shirt.

Therefore, the optimal quantity of t-shirts to stock is 67.

An Organization Hosting a 5K Race Is Looking to Sell