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Becca earns money mowing her neighbors' lawns. The revenue for mowing \(x\) lawns is \(r(x)=18x\). Becca's cost for gas and the mower rental is \(c(x)=5x+20\).

Her profit from mowing \(x\) lawns is \(p(x)=(r-c)(x)\). What is \(p(x)\)?

A. \(p(x)=23x-2\)

B. \(p(x)=23x+20\)

C. \(p(x)=13x-20\)

D. [tex]\(p(x)=13x+20\)[/tex]

Sagot :

GinnyAnsweridn
Sure, let's find Becca's profit function step-by-step.

1. Revenue Function \( r(x) \):
The revenue Becca earns from mowing \( x \) lawns is given by:
[tex]\[ r(x) = 18x \][/tex]
This means Becca earns $18 per lawn she mows.

2. Cost Function \( c(x) \):
The cost for mowing \( x \) lawns, which includes gas and mower rental, is:
[tex]\[ c(x) = 5x + 20 \][/tex]
This means there is a fixed cost of [tex]$20 plus $[/tex]5 per lawn she mows.

3. Profit Function \( p(x) \):
Profit is the revenue minus the cost:
[tex]\[ p(x) = r(x) - c(x) \][/tex]
Substituting in the given functions for \( r(x) \) and \( c(x) \):
[tex]\[ p(x) = 18x - (5x + 20) \][/tex]

4. Simplify the Profit Function:
Distribute the negative sign through the cost function:
[tex]\[ p(x) = 18x - 5x - 20 \][/tex]
Combine like terms:
[tex]\[ p(x) = (18x - 5x) - 20 \][/tex]
[tex]\[ p(x) = 13x - 20 \][/tex]

Therefore, Becca's profit function \( p(x) \) is given by:
[tex]\[ p(x) = 13x - 20 \][/tex]

Using the multiple choices provided:
A. \( p(x) = 23x - 2 \)

B. \( p(x) = 23x + 20 \)

C. \( p(x) = 13x - 20 \)

D. \( p(x) = 13x + 20 \)

The correct answer is:
C. [tex]\( p(x) = 13x - 20 \)[/tex]
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