The following function gives the cost, in dollars, of producing x gallons of wood stain: C(x) = 0.0004x^3 -...

Question

The following function gives the cost, in dollars, of producing x gallons of wood stain:
C(x) = 0.0004x^3 - 0.0012x^2 + 0.1x + 3200
What is the instantaneous rate of change of the cost when 100 gallons are produced?
Choose 1 answer:
A) 11.9 dollars
B) 11.9 dollars per gallon
C) 3600 dollars
D) 3600 dollars per gallon

SnapAndSolve AI · Accepted Answer

To find the instantaneous rate of change of the cost when 100 gallons are produced, we need to find the derivative of the cost function C(x) and evaluate it at x = 100.

The derivative C'(x) is calculated as follows:

C'(x) = d/dx [0.0004x^3 - 0.0012x^2 + 0.1x + 3200]
C'(x) = 3 * 0.0004x^2 - 2 * 0.0012x + 0.1
C'(x) = 0.0012x^2 - 0.0024x + 0.1

Now, evaluate C'(x) at x = 100:

C'(100) = 0.0012(100)^2 - 0.0024(100) + 0.1
C'(100) = 0.0012(10000) - 0.24 + 0.1
C'(100) = 12 - 0.24 + 0.1
C'(100) = 11.86

The instantaneous rate of change of the cost when 100 gallons are produced is approximately 11.86 dollars per gallon.

The closest answer choice is:

B) 11.9 dollars per gallon

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The following function gives the cost, in dollars, of producing x gallons of wood stain: C(x) = 0.0004x^3 -...

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