Question

Estimate the cost of producing an additional 400 lb of paper once 10 tons have been produced.

Suppose C(t) is the cost, in thousands of dollars, of producing t tons of white paper. If C’(10)=370, estimate the cost of producing an additional 400 lb of paper once 10 tons have been produced.

Answer 1

We can begin by converting the additional 400 lb of paper to tons:

400 lb * (1 ton / 2000 lb) = 0.2 tons

Now, we want to estimate the cost of producing an additional 0.2 tons of paper once 10 tons have been produced. To do this, we can use the linear approximation:

ΔC ≈ C’(10) Δt

where ΔC is the change in cost, C’(10) is the derivative of the cost function at t=10, and Δt is the change in tons of paper.

Since we are given that C’(10) = 370, we have:

ΔC ≈ 370 * 0.2 = 74

Therefore, the estimated cost of producing an additional 400 lb of paper once 10 tons have been produced is $74,000.