Question

A recipe will contain 250 total units of ingredients. Ingredient A costs $0.50 more per unit than ingredient B. Find the units of each ingredient to minimize the cost.

Answer 1

To minimize the cost of a recipe that uses two ingredients (A and B) under the constraint that the total number of units is 250, we can set up the problem as follows:

Let:
- \( x \) = units of ingredient A
- \( y \) = units of ingredient B

From the problem, we know:
1. \( x + y = 250 \) (the total units of ingredients)
2. The cost of ingredient A is \( c_A = c_B + 0.50 \). Let \( c_B \) be the cost per unit of ingredient B, hence \( c_A = c_B + 0.50 \).

The total cost \( C \) can be expressed as:
\[
C = c_A \cdot x + c_B \cdot y
\]
Substituting \( c_A = c_B + 0.50 \) into the equation gives:
\[
C = (c_B + 0.50) \cdot x + c_B \cdot y
\]
Rearranging the cost equation using the relationship \( y = 250 - x \):
\[
C = (c_B + 0.50) \cdot x + c_B \cdot (250 - x)
\]
Expanding this we obtain:
\[
C = (c_B + 0.50)x + c_B(250 - x)
\]
\[
= c_B \cdot x + 0.50x + 250c_B - c_B \cdot x
\]
The \( c_B \cdot x \) terms cancel out:
\[
C = 0.50x + 250c_B
\]
Now, the cost function \( C \) can be simplified to:
\[
C = 0.50x + 250c_B
\]
To minimize the cost of ingredients, only the term with \( x \) (the amount of ingredient A) will affect the cost, as \( 250c_B \) is constant.

Next, we note that since the cost function is linear in \( x \) (with a positive coefficient of 0.50), to minimize cost:
- If \( c_B \) is the lower cost ingredient, we should use the maximum units of ingredient B, which means setting \( x = 0 \) and therefore \( y = 250 \).

Thus, the cost is minimized by using:
- \( x = 0 \) units of ingredient A
- \( y = 250 \) units of ingredient B

In conclusion, to minimize the cost for the recipe:
- Use 0 units of ingredient A.
- Use 250 units of ingredient B.