Question

Suppose thatthe monthly cost, in dollars, of producing x chairs is C(x) =0.001x3 + 0.07x2 + 19x +700, and currently 25chairs are produced monthly.

a.)What isthe current monthly cost?
b.)Whatwould be the additional cost of the increasing production to 26chairs monthly?
c.)What isthe marginal cost when x=25?
d.)usemarginal cost to estimate the difference in cost between producing25 and 27 chairs per month?
e.)Use theanswer from part (d) to predict C(27).

Answer 1

a.) The current monthly cost is $1240.625. b.) The additional cost of increasing production to 26 chairs monthly is $23.071. c.) The marginal cost when x = 25 is $24.375. d.) The estimated difference in cost between producing 25 and 27 chairs per month is $48.75. e.) C(27) is predicted to be $1289.375.

Step-by-step explanation:

a.) To find the current monthly cost, you need to substitute x = 25 into the cost function: C(25) = 0.001(25)^3 + 0.07(25)^2 + 19(25) + 700. Simplifying this expression gives C(25) = 21.875 + 43.75 + 475 + 700 = $1240.625.

b.) The additional cost of increasing production to 26 chairs monthly can be calculated by finding the difference between the cost of producing 26 chairs and the cost of producing 25 chairs. Substitute x = 26 into the cost function: C(26) = 0.001(26)^3 + 0.07(26)^2 + 19(26) + 700. Simplifying this expression gives C(26) = 23.176 + 46.52 + 494 + 700 = $1263.696. The additional cost is C(26) - C(25) = $1263.696 - $1240.625 = $23.071.

c.) Marginal cost represents the rate of change of cost with respect to the number of chairs produced. The marginal cost when x = 25 can be found by taking the derivative of the cost function with respect to x and evaluating it at x = 25. Differentiating the cost function, we get C'(x) = 0.003x^2 + 0.14x + 19. When x = 25, C'(25) = 0.003(25)^2 + 0.14(25) + 19 = 1.875 + 3.5 + 19 = $24.375.

d.) Using marginal cost, we can estimate the difference in cost between producing 25 and 27 chairs per month. The marginal cost at x = 25 is $24.375. To estimate the difference, multiply the marginal cost by the change in quantity, which is 27 - 25 = 2 chairs. The estimated difference in cost is $24.375 * 2 = $48.75.

e.) To predict C(27), add the estimated difference in cost to the current monthly cost. C(27) = C(25) + $48.75 = $1240.625 + $48.75 = $1289.375.

Answer 2

Given the cost function of chair produced monthly as

C(x) =0.001x³ + 0.07x² + 19x +700

where x =no of chairs produced

a) Current monthly cost?

For x=25, C = 0.001 (25)³ + 0.07(25)² + 19(25) +700

C = 15.625+43.75+475+700

C = $1234.38

b) additional cost of the increasing production to 26 chairs monthly?

that is C(26) - C(25) = ?

For C(26) = 0.001 (26)³ + 0.07(26)² + 19(26) +700

C(26) =$1258.89

C(26) - C(25) = $24.52

c) the marginal cost when x=25?

This is dC/dx = 0.003x² + 0.14x +19

Marginal Cost at x =25, = 0.003(25)² + 0.14(25) +19

= $24.375/chair

d) To estimate the difference in cost between producing 25 and 27 chairs per month?

we will find dC/dx(27)?

Marginal Cost at x =27, = 0.003(27)² + 0.14(27) +19

C(27) = $81.83/chair

For increase in chairs per month = 2,

The difference in cost between C(25) and C(27), = ($81.83 - $24.375) x 2 =$114.91

e) To predict C(27) = ?

From (d) to predict this, it means for a single chair, we have the cost as $57.454/chair

so for 27 chairs, the cost will be $57.454 x 27 = $1551.26/month