Email: althauser@math.ucsb.edu
Instructions: You have $\mathbf{1}$ hour to complete this quiz and upload a single PDF of your work to Gradescope. You may either work in the space provided or on a separate piece of paper. Make sure to show all work and justify your solutions clearly. Please also follow the Quiz Guidelines outlined on the course Canvas page.
Problem 1
You have decided to make and sell friendship bracelets as a hobby. When bracelets are sold for $\$ 7$ each, you sell 85 bracelets each month. For every $\$ 0.50$ the price is increased you sell 2 fewer bracelets. It costs you $\$ 4$ to make each bracelet.
(a) Use calculus to determine what price bracelets should be sold at to maximize profit.
(b) How much money will you make each month at this price?

Step 1 ：Given that the quantity of bracelets sold is a function of the price, we can express this as \(Q = 85 - 2\frac{(P - 7)}{0.5}\), where \(Q\) is the quantity sold and \(P\) is the price.

Step 2 ：The cost of making each bracelet is $4, so the total cost is \(C = 4Q\).

Step 3 ：The revenue from selling the bracelets is the price times the quantity sold, or \(R = PQ\).

Step 4 ：The profit is the difference between the revenue and the cost, or \(P = R - C\).

Step 5 ：To find the price that maximizes the profit, we take the derivative of the profit function with respect to the price, set it equal to zero, and solve for the price.

Step 6 ：Doing this, we find that the optimal price to sell the bracelets at to maximize profit is \(\boxed{\$16.125}\).

Step 7 ：To find out how much money will be made each month at this price, we substitute the optimal price into the profit function.

Step 8 ：Doing this, we find that the amount of money made each month at this price is \(\boxed{\$588.06}\).