The monthly profit, in hundreds of dollars, made by selling  
x
  custom built electric scooters is given by  
P
(
x
)
=
−
2
x
2
+
172
x
−
2448
 ,  
0
≤
x
≤
83
 .

a) What are the zeros of  
P
(
x
)
 ?
Smaller: 
18
Correct 
 
Larger: 
68
Correct 
 

b) What does  
P
(
x
)
=
0
  mean in the context of the problem? 
Number of scooters to break even*
Correct

c) How many electric scooter should be sold to maximize profit? 
43
Correct 
 

d) What is the maximum profit? 

To find the maximum profit, we first need to recognize that the function $ P(x) = -2x^2 + 172x - 2448 $ is a quadratic function. Since the coefficient of $ x^2 $ is negative, this parabola opens downward, and thus its maximum value occurs at the vertex.

The formula to find the vertex of a quadratic function $ ax^2 + bx + c $ is:

\[
x = \frac{-b}{2a}
\]

For the given profit function:

\[
P(x) = -2x^2 + 172x - 2448
\]

Here, $ a = -2 $ and $ b = 172 $.

Substitute these values into the formula for the vertex:

\[
x = \frac{-172}{2(-2)} = \frac{-172}{-4} = 43
\]

So, the maximum profit occurs when 43 scooters are sold, which we already know.

To find the **maximum profit**, substitute $ x = 43 $ back into the profit function:

\[
P(43) = -2(43)^2 + 172(43) - 2448
\]

Let's calculate that:

\[
P(43) = -2(1849) + 172(43) - 2448
\]
\[
P(43) = -3698 + 7396 - 2448
\]
\[
P(43) = 7396 - 6146
\]
\[
P(43) = 1250
\]

Therefore, the **maximum profit** is $ \mathbf{1250} $ hundreds of dollars, or **$125,000**.

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. How does the coefficient of $ x^2 $ affect the shape of the profit function?
2. What would happen if we increased the price per scooter?
3. Can we calculate the profit if 50 scooters are sold?
4. What is the break-even point and how is it derived from the profit function?
5. How would the profit change if the cost to produce each scooter decreased?

**Tip:** Always remember that the vertex formula for quadratics is your go-to tool for finding maximums or minimums when dealing with profit and cost functions.

The monthly profit, in hundreds of dollars, made by selling x custom built electric scooters is given by P(x) = -2x^2 + 172x - 2448, 0 ≤ x ≤ 83.
a) What are the zeros of P(x)?
b) What does P(x) = 0 mean in the context of the problem?
c) How many electric scooters should be sold to maximize profit?
d) What is the maximum profit?

Solve the electric scooter sales problem by analyzing the quadratic profit function P(x) = -2x^2 + 172x - 2448. Discover how to find the zeros, interpret the break-even point, and calculate the number of scooters needed to maximize profit. Learn key concepts like the vertex formula and profit maximization. Suitable for Grades 10-12.

Math Problem Statement

The monthly profit, in hundreds of dollars, made by selling
x custom built electric scooters is given by
P ( x )

b) What does
P ( x )

Solution

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