Brief calculus.

Question

The manager of a company determines that when x units of a particular commodity are produced, the total cost of production C thousand dollars, where C² -3x³ = 4,275.

a) If 15 units are being produced, find the total cost.

b) When 15 units are being produced the level of production is increasing at a rate of 0.2 units per week. ( so, dx/dt = 0.2) Using related rates, find how fast the total cost is changing at the same time. Write your answer as a complete sentence.

Answer 1

C² -3x³ = 4,275

a) This is the simplest one! Just just plugin x=15 then find C.

C² -3(15)³ = 4,275

C² = 4,275 + 3(15)³

C² = 4,275 + 3(15)³

C² = 14400

C = 120 thousand dollars or $120,000

b) We are looking for dC/dt. First, get the derivative of the equation with respect to time (t):

2C(dC/dt) -9x² (dx/dt) = 0

Since dx/dt= 0.2, plugin all the values given and acquired to solve for dC/dt

2 (120) (dC/dt) - 9(15)² (0.2) = 0

240 (dC/dt) = 405

dC/dt = 1.6875

Therefore the total cost is changing at a rate of $1,687.50 per week.