Wilma's Car Repair can repair cars using kryptonite bolts,K,or lithium bolts,L,as long as it uses 10 bolts in toto.The cost of repairing a car is TC = K² + L² - KL.The cost-minimizing combination of kryptonite and lithium bolts is:

Question

Quizplus · Accepted Answer

To find the cost-minimizing combination of kryptonite and lithium bolts, we need to use the formula for total cost and take the partial derivative with respect to each variable and set it equal to 0. Then we can solve for the optimal values of K and L. TC = K² + L² - KL Taking partial derivatives with respect to K and L, we get: dTC/dK = ² - L dTC/dL = ² - K Setting these equal to 0 and solving for K and L, we get: ² - L = 0 ² - K = 0 K = L Plugging this back into the formula for total cost, we get: TC = 10K² - K2 TC = 10L² - L2 We want to minimize TC, so we can add these two equations together and simplify: 2TC = 10(K+L)² - (K2 + L2) 2TC = 10(10)² - 10(K-L)2 TC = 500 - 5(K-L)2 Since K and L have to add up to 10, we can substitute K = 10 - L into the equation: TC = 500 - 5(10 - L - L)2 TC = 500 - 5(2L - 10)2 TC = 500 - 20(L - 5)2 The minimum value of TC occurs when L = 5 and K = 10 - L = 5, so the best combination of kryptonite and lithium bolts is: K = 5 and L = 5.

Wilma's Car Repair Can Repair Cars Using Kryptonite Bolts,K,or Lithium