Question

Over the summer, Bryce offers landscaping services throughout his community. His most popular request is to install small flower beds in his neighbors' yards. After conducting a survey of his neighbors, Bryce determines that if he charges x dollars per flower bed, he will install
–
3x+270 flower beds this summer.
The materials to make each flower bed cost Bryce $10. So, he will earn x–10 dollars in profit for each flower bed he installs.
Which equation can Bryce use to predict the price he can charge per flower bed to earn $4,500 in profit this summer?
What two prices can Bryce charge per flower bed to earn exactly $4,500 in profit this summer?
$
or $

Answer 1

To determine the equation Bryce can use to predict the price he can charge per flower bed to earn $4,500 in profit, we start by setting up the profit formula.

1. **Flower beds installed**:
Bryce installs \( -3x + 270 \) flower beds when he charges \( x \) dollars per flower bed.

2. **Profit per flower bed**:
Each flower bed generates a profit of \( x - 10 \) dollars.

3. **Total profit**:
The total profit \( P \) can be expressed as:
\[
P = (\text{Number of flower beds}) \times (\text{Profit per flower bed})
\]
Substituting in the expressions for number of flower beds and profit per flower bed:
\[
P = (-3x + 270)(x - 10)
\]

4. **Setting the total profit equal to $4,500**:
To find the equation that Bryce can use, we set \( P \) to 4500:
\[
(-3x + 270)(x - 10) = 4500
\]

Next, we can expand this equation:
\[
(-3x + 270)(x - 10) = -3x^2 + 30x + 270x - 2700
\]
Combining like terms gives:
\[
-3x^2 + 300x - 2700 = 4500
\]

Now, rearranging this to set the equation to zero:
\[
-3x^2 + 300x - 7200 = 0
\]
Multiplying through by -1 to simplify:
\[
3x^2 - 300x + 7200 = 0
\]

Now, we can use the quadratic formula to find the values of \( x \). The quadratic formula is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 3 \), \( b = -300 \), and \( c = 7200 \).

Calculating the discriminant:
\[
b^2 - 4ac = (-300)^2 - 4(3)(7200) = 90000 - 86400 = 3600
\]

Now substituting into the quadratic formula:
\[
x = \frac{-(-300) \pm \sqrt{3600}}{2(3)} = \frac{300 \pm 60}{6}
\]

Calculating the two possible values for \( x \):
1.
\[
x_1 = \frac{300 + 60}{6} = \frac{360}{6} = 60
\]
2.
\[
x_2 = \frac{300 - 60}{6} = \frac{240}{6} = 40
\]

Thus, the two prices that Bryce can charge per flower bed to earn exactly $4,500 in profit this summer are \(\$60\) or \(\$40\).

## Summary:
- **Equation**: \((-3x + 270)(x - 10) = 4500\)
- **Prices**: $40 or $60