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A home improvement store is analyzing the pricing of a line of lawnmowers. These are the cost and revenue functions for a month of business, where [tex]$x$[/tex] represents the selling price of a lawnmower:

[tex]\[
\begin{array}{l}
R(x) = -2.075x^2 + 768x \\
C(x) = -134.625x + 84,037.5
\end{array}
\][/tex]

Which selling prices result in a profit that is greater than or equal to zero? Select all the correct answers.

- 120
- 135
- 285
- 300
- 335

Sagot :

GinnyAnsweridn
To determine which selling prices of lawnmowers result in a profit that is greater than or equal to zero, we need to evaluate the profit, defined as the difference between the revenue and cost functions, [tex]\( \pi(x) = R(x) - C(x) \)[/tex], for each given price.

The given revenue and cost functions are:
[tex]\[ R(x) = -2.075x^2 + 768x \][/tex]
[tex]\[ C(x) = -134.625x + 84037.5 \][/tex]

We'll first evaluate the profit at each given price: 120, 135, 285, 300, and 335.

### Evaluation of Profit at Given Prices

1. Price [tex]\( x = 120 \)[/tex]
[tex]\[ R(120) = -2.075(120)^2 + 768(120) \][/tex]
[tex]\[ C(120) = -134.625(120) + 84037.5 \][/tex]
[tex]\[ \pi(120) = R(120) - C(120) = -5602.5 \][/tex]

The profit is [tex]\(-5602.5\)[/tex], so this price does not yield a non-negative profit.

2. Price [tex]\( x = 135 \)[/tex]
[tex]\[ R(135) = -2.075(135)^2 + 768(135) \][/tex]
[tex]\[ C(135) = -134.625(135) + 84037.5 \][/tex]
[tex]\[ \pi(135) = R(135) - C(135) = 0.0 \][/tex]

The profit is [tex]\(0.0\)[/tex], which is a non-negative profit.

3. Price [tex]\( x = 285 \)[/tex]
[tex]\[ R(285) = -2.075(285)^2 + 768(285) \][/tex]
[tex]\[ C(285) = -134.625(285) + 84037.5 \][/tex]
[tex]\[ \pi(285) = R(285) - C(285) = 4668.75 \][/tex]

The profit is [tex]\(4668.75\)[/tex], which is a non-negative profit.

4. Price [tex]\( x = 300 \)[/tex]
[tex]\[ R(300) = -2.075(300)^2 + 768(300) \][/tex]
[tex]\[ C(300) = -134.625(300) + 84037.5 \][/tex]
[tex]\[ \pi(300) = R(300) - C(300) = -2.9103830456733704 \times 10^{-11} \][/tex]

The profit is [tex]\(-2.9103830456733704 \times 10^{-11}\)[/tex], which is effectively zero but slightly negative due to numerical precision.

5. Price [tex]\( x = 335 \)[/tex]
[tex]\[ R(335) = -2.075(335)^2 + 768(335) \][/tex]
[tex]\[ C(335) = -134.625(335) + 84037.5 \][/tex]
[tex]\[ \pi(335) = R(335) - C(335) = -14525.00000000003 \][/tex]

The profit is [tex]\(-14525.00000000003\)[/tex], which is negative.

### Conclusion
The prices that result in a profit that is greater than or equal to zero are:
135, 285.

So, the correct answers are:
- 135
- 285
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