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The following function gives the cost, in dollars, of producing
x gallons of wood stain:

C(x)=0.0004x^(3)-0.001x^(2)+1
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

The following function gives the cost, in dollars, of producing $x$ gallons of wood stain: $\newline$ $C(x)=0.0004 x^{3}-0.001 x^{2}+1$ $\newline$ What is the instantaneous rate of change of the cost when $100$ gallons are produced? $\newline$ Choose $1$ answer: $\newline$ (A) $11$ . $9$ dollars $\newline$ (B) $11$ . $9$ dollars per gallon $\newline$ (C) $3600$ dollars $\newline$ (D) $3600$ dollars per gallon

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Solve quadratic equations: word problems

Full solution

Q. The following function gives the cost, in dollars, of producing

x

gallons of wood stain:

\newline

C(x)=0.0004 x^{3}-0.001 x^{2}+1

\newline

What is the instantaneous rate of change of the cost when

100

gallons are produced?

\newline

Choose

1

answer:

\newline

(A)

11

.

9

dollars

\newline

(B)

11

.

9

dollars per gallon

\newline

(C)

3600

dollars

\newline

(D)

3600

dollars per gallon

Find Derivative of $C(x)$ : The instantaneous rate of change of the cost function $C(x)$ at a specific value of $x$ is found by taking the derivative of $C(x)$ with respect to $x$ .
Calculate $C'(x)$ : Differentiate $C(x) = 0.0004x^3 - 0.001x^2 + 1$ with respect to $x$ to find $C'(x)$ . $\newline$ $C'(x) = 0.0012x^2 - 0.002x$
Substitute $x = 100$ : Substitute $x = 100$ into $C'(x)$ to find the instantaneous rate of change at $100$ gallons. $\newline$ $C'(100) = 0.0012(100)^2 - 0.002(100)$
Calculate $C'(100)$ : Calculate the value of $C'(100)$ .
$C'(100) = 0.0012(10000) - 0.002(100)$
$C'(100) = 12 - 0.2$
$C'(100) = 11.8$

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\newline

\begin{tabular}{lllll}

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x

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