The total weekly cost (in dollars) incurred by Lincoln Records in pressing x compact discs is given by the following function. (Round your answers to two decimal places.) C(x) = 2000 + 2x - 0.0001x2

Question

(a) What is the actual cost incurred in producing the 981st disc?

(b)What is the actual cost incurred in producing the 1871st disc?

(c) What is the marginal cost when x = 980?

(d)What is the marginal cost when x = 1870?

Answer 1

For parts (a) and (b), we are finding the actual cost incurred in producing an additional disc. We can't simply plug the value into the equation [for example C(981) for part (a)] as this will provide the total cost to produce all 981 discs, not the additional cost incurred for only the 981^st disc. Instead, we must subtract C(n) - C(n-1) to find the net cost incurred in producing the next disc (where "n" is the n^th disc).

This makes more sense when we think about it using smaller numbers. If we take, for example, the total cost required to make 3 discs and subtract from it the total cost required to make 2 discs, we are left with the actual cost incurred in producing the third disc. Basically, the jump from 2 to 3.

C(x) = 2000 + 2x - 0.0001x²

(a) Actual cost incurred = C(981) - C(980)

C(981) = 2000 + 2*(981) - 0.0001*(981)² = $3865.76

C(980) = 2000 + 2*(980) - 0.0001*(980)² = $3863.96

C(981) - C(980) = $3865.76 - $3863.96 = $1.80

(b) Actual cost incurred = C(1871) - C(1870)

C(1871) = 2000 + 2*(1871) - 0.0001*(1871)² = $5391.94

C(1870) = 2000 + 2*(1870) - 0.0001*(1870)² = $5390.31

C(1871) - C(1870) = $5391.94 - $5390.31 = $1.63

What does this mean? Well, as we produce a greater number of total discs, our additional cost to produce the next disc decreases. This makes sense intuitively. It's similar to buying items in bulk.

Now for parts (c) and (d) we are asked for the marginal cost. This can be defined as the change in total cost that arises when the quantity produced changes by one unit. This is very similar to what we did in parts (a) and (b), but is a more exact method as we will calculate the derivative of the cost function to determine the marginal cost. Basically parts (a) and (b) is an approximation of the more exact answers we will calculate in parts (c) and (d). From a graphical standpoint, parts (a) and (b) were analogous to drawing a secant line between two points on a curve. Parts (c) and (d) are analogous to drawing a tangent line at the exact point we are interested in.

C'(x) = 2 - 0.0002x

(c) Marginal cost = C'(980), (note that this is the marginal cost to produce the 981st disc, not the 980th)

C'(980) = 2 - 0.0002*(980) = $1.80

(d) Marginal cost = C'(1870)

C'(1870) = 2 - 0.0002*(1870) = $1.63

Note that although these answers appear to be the same to (a) and (b), they are actually slightly different. Rounding to two decimal places makes it look like they are equal but if we used more decimal places we would see they are not quite. This does show us however, that our approximation in parts (a) and (b) is a good one.