A bakery produces two types of desserts: cakes and muffins. Let c be the number of cakes and m be the number of muffins…

Let c be the number of cakes and m be the number of muffins. The problem gives two constraints based on time, and states that the bakery uses the maximum available time. This allows us to set up a system of two linear equations.

First, model the baking time constraint: The time for cakes (30c) plus the time for muffins (20m) equals the total available time (360).

Equation 1: 30c + 20m = 360

Second, model the frosting time constraint: The time for cakes (10c) plus the time for muffins (20m) equals the total available time (200).

Equation 2: 10c + 20m = 200

To solve this system, we can use the elimination method. Subtracting Equation 2 from Equation 1 is an efficient first step:

(30c + 20m) - (10c + 20m) = 360 - 200

20c = 160

c = 8

Now substitute the value of c back into one of the original equations. Using Equation 2:

10(8) + 20m = 200

80 + 20m = 200

20m = 120

m = 6

The bakery produces 8 cakes and 6 muffins. The question asks for the total number of desserts, which is the sum c + m.

Total desserts = 8 + 6 = 14.

A: This is the number of muffins (m), not the total number of desserts. The student correctly solves the system but answers with only one of the variables.

B: This is the number of cakes (c), not the total number of desserts. The student correctly solves the system but answers with only one of the variables.

D: This is the result of a modeling error, specifically swapping the frosting times for cakes and muffins (using 20c + 10m = 200 instead of 10c + 20m = 200). Solving the system 30c + 20m = 360 and 20c + 10m = 200 yields c=4 and m=12, for an incorrect total of 16.