Bob owns a watch repair shop. He has found that the cost of operating his shop is given by $c(x)=3 x^{2}-174 x+87$, where $c$ is cost and $x$ is the number of watches repaired. How many watches must he repair to have the lowest cost?
A. 44 watches
B. 87 watches
C. 30 watches
D. 29 watches
Answer
Final Answer: \(\boxed{29 \text{ watches}}\). The correct answer is D. 29 watches.
Step 1 ：The cost function is a quadratic function. The minimum value of a quadratic function \(f(x) = ax^2 + bx + c\) is achieved at \(x = -\frac{b}{2a}\). In this case, \(a = 3\) and \(b = -174\). So, we need to calculate \(x = -\frac{-174}{2*3}\).
Step 2 ：Substitute the values of a and b into the formula, we get \(x = -\frac{-174}{2*3} = 29.0\)
Step 3 ：Final Answer: \(\boxed{29 \text{ watches}}\). The correct answer is D. 29 watches.
