Bret's bakery must decide how many loaves of fresh bread to produce in a single day.Daily demand for fresh bread is normally distributed with a mean of 70 loaves and standard deviation of 18.If the marginal loss is $2 and the marginal profit is $1,how much bread should Bret's bakery produce in a single day?
A)88
B)52
C)63
D)78
E)70

Question

Quizplus · Accepted Answer

The goal is to find the quantity of bread that maximizes profit. Profit is calculated by subtracting the cost (marginal loss) from the revenue (marginal profit) per loaf. Therefore, profit per loaf is $1 - $2 = -$1. This means that for each loaf of bread produced, there is a loss of $1.

To maximize profit, we need to find the quantity of bread that has the greatest profit contribution per unit. Since we're dealing with a normal distribution, we can use the z-score formula to calculate the number of standard deviations above or below the mean of the distribution:

z = (x - mu) / sigma

Where:
x = the number of loaves of bread
mu = the mean (70 loaves)
sigma = the standard deviation (18 loaves)

We want to find the z-score that corresponds to the quantity that maximizes profit. This would be the z-score at which the profit contribution per loaf is 0:

0 = $1 - $2
0 = -$1

Solving for z, we get:

z = 1 / 18
z = 0.06

We can use this z-score to find the corresponding number of loaves using the same formula:

z = (x - mu) / sigma

0.06 = (x - 70) / 18

Solving for x:

x = 70 + (0.06 * 18)
x = 71.08

Since Bret's bakery cannot produce 0.08 of a loaf, the closest integer value of bread to produce would be 71 loaves. However, since we're dealing with a continuous distribution, it would be better to take a range of values around 71 loaves. A reasonable range would be from 63 to 78 loaves, which corresponds to a z-score range of -0.45 to 0.67. To maximize profit, Bret's bakery should produce around 63 to 78 loaves of bread in a single day.

Out of the answer choices, the closest to this range is 63, so the best choice is C.

Bret's Bakery Must Decide How Many Loaves of Fresh Bread