Question
Question 7 . Biscuit Co. has determined that their total monthly costs \( \$ C \) depend on how many biscuits \( b \) they bake by the quadratic \( C=(b-500)^{2}+150 \). Using this quadratic, by baking the optimal number of biscuits, their minimum monthly costs should be
Ask by Fowler Ayala. in Australia
Feb 19,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
By baking 500 biscuits, the minimum monthly cost is $150.
Solution
The quadratic cost function is given in vertex form as C = (b – 500)² + 150. In this form, the minimum value occurs at b = 500 biscuits, and substituting this into the equation gives:
C = (500 – 500)² + 150
= 0 + 150
= $150
So, by baking 500 biscuits, the minimum monthly cost is $150.
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Simplify this solution Bonus Knowledge
To minimize the costs given by the quadratic equation \( C=(b-500)^{2}+150 \), observe that the term \( (b-500)^{2} \) reaches its minimum value of 0 when \( b=500 \). Substituting \( b=500 \) back into the equation gives \( C=0+150=150 \). Thus, by baking 500 biscuits, Biscuit Co. achieves their minimum monthly costs of \$150. This formula demonstrates the classic U-shaped curve of quadratic functions, where the vertex represents the minimum point. The concept applies in various businesses, from muffins to macaroons. So whether you’re scaling your cookie production or optimizing other production lines, always look for that sweet spot where costs dip!
