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Answer :
To find the profit function [tex]\( P(x) \)[/tex], we need to subtract the cost function [tex]\( C(x) \)[/tex] from the revenue function [tex]\( R(x) \)[/tex]. Let's go through the steps of simplifying this expression.
1. Write Down the Revenue and Cost Functions:
The revenue function [tex]\( R(x) \)[/tex] is given by:
[tex]\[
R(x) = -0.0665x^3 + 1.822x^2 - 12.45x + 116.2
\][/tex]
The cost function [tex]\( C(x) \)[/tex] is given by:
[tex]\[
C(x) = -0.0223x^3 + 0.834x^2 - 6.74x + 99.1
\][/tex]
2. Subtract the Cost Function from the Revenue Function:
The profit function [tex]\( P(x) \)[/tex] is calculated as:
[tex]\[
P(x) = R(x) - C(x)
\][/tex]
Plugging in the expressions for [tex]\( R(x) \)[/tex] and [tex]\( C(x) \)[/tex], we have:
[tex]\[
P(x) = (-0.0665x^3 + 1.822x^2 - 12.45x + 116.2) - (-0.0223x^3 + 0.834x^2 - 6.74x + 99.1)
\][/tex]
3. Simplify the Expression:
When we subtract the two polynomials, we handle each term separately:
- For the [tex]\( x^3 \)[/tex] terms:
[tex]\[
-0.0665x^3 - (-0.0223x^3) = -0.0665x^3 + 0.0223x^3 = -0.0442x^3
\][/tex]
- For the [tex]\( x^2 \)[/tex] terms:
[tex]\[
1.822x^2 - 0.834x^2 = 0.988x^2
\][/tex]
- For the [tex]\( x \)[/tex] terms:
[tex]\[
-12.45x + 6.74x = -5.71x
\][/tex]
- For the constant terms:
[tex]\[
116.2 - 99.1 = 17.1
\][/tex]
4. Combine All the Terms:
Putting all the simplified terms together, the profit function [tex]\( P(x) \)[/tex] is:
[tex]\[
P(x) = -0.0442x^3 + 0.988x^2 - 5.71x + 17.1
\][/tex]
Thus, the simplified profit function for the years 2004-2012 is:
[tex]\[ P(x) = -0.0442x^3 + 0.988x^2 - 5.71x + 17.1 \][/tex]
1. Write Down the Revenue and Cost Functions:
The revenue function [tex]\( R(x) \)[/tex] is given by:
[tex]\[
R(x) = -0.0665x^3 + 1.822x^2 - 12.45x + 116.2
\][/tex]
The cost function [tex]\( C(x) \)[/tex] is given by:
[tex]\[
C(x) = -0.0223x^3 + 0.834x^2 - 6.74x + 99.1
\][/tex]
2. Subtract the Cost Function from the Revenue Function:
The profit function [tex]\( P(x) \)[/tex] is calculated as:
[tex]\[
P(x) = R(x) - C(x)
\][/tex]
Plugging in the expressions for [tex]\( R(x) \)[/tex] and [tex]\( C(x) \)[/tex], we have:
[tex]\[
P(x) = (-0.0665x^3 + 1.822x^2 - 12.45x + 116.2) - (-0.0223x^3 + 0.834x^2 - 6.74x + 99.1)
\][/tex]
3. Simplify the Expression:
When we subtract the two polynomials, we handle each term separately:
- For the [tex]\( x^3 \)[/tex] terms:
[tex]\[
-0.0665x^3 - (-0.0223x^3) = -0.0665x^3 + 0.0223x^3 = -0.0442x^3
\][/tex]
- For the [tex]\( x^2 \)[/tex] terms:
[tex]\[
1.822x^2 - 0.834x^2 = 0.988x^2
\][/tex]
- For the [tex]\( x \)[/tex] terms:
[tex]\[
-12.45x + 6.74x = -5.71x
\][/tex]
- For the constant terms:
[tex]\[
116.2 - 99.1 = 17.1
\][/tex]
4. Combine All the Terms:
Putting all the simplified terms together, the profit function [tex]\( P(x) \)[/tex] is:
[tex]\[
P(x) = -0.0442x^3 + 0.988x^2 - 5.71x + 17.1
\][/tex]
Thus, the simplified profit function for the years 2004-2012 is:
[tex]\[ P(x) = -0.0442x^3 + 0.988x^2 - 5.71x + 17.1 \][/tex]
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