Question

you are manufacturing ceramic lawn ornaments. after several months, your accountant tells you that your profit, p(n), can be modeled by the function p(n) = -0.002n² + 5.2n - 1203, where n is the number of ornaments sold each month. complete parts a through g.
tblstart = 0 and δtbl = 500 to help you set your window. include the x-intercepts and the vertex. choose the correct graphs below.
all graphs have the viewing window -100, 3000, 500 by -200, 3000, 500.
b) determine the x-intercepts of the graph of the profit function. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the x-intercepts are (0,257) (type an ordered pair. use a comma to separate answers as needed. round to the nearest whole number as needed.)
b. there are no x-intercepts.

Explanation:

1. Finding \( x \)-intercepts:

• We start with the profit function \( P(n)=-0.002n^2 + 5.2n - 1203 \). To find the \( x \)-intercepts, we set \( P(n) = 0 \). This gives us a quadratic equation.
• We use the quadratic formula \( n=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \) where \( a=-0.002 \), \( b = 5.2 \) and \( c=-1203 \). After simplifying the equation (by multiplying through by - 500) and applying the quadratic formula, we find the roots of the quadratic equation. Rounding the roots to the nearest whole number, we get \( n\approx257 \) and \( n\approx2343 \). So the \( x \)-intercepts (points where \( P(n) = 0 \)) are \( (257,0) \) and \( (2343,0) \).

1. Identifying the correct graph:

• The profit function is a quadratic function with \( a=-0.002<0 \), so its graph is a downward - opening parabola.
• The vertex of the parabola is at \( n =-\frac{b}{2a}=1300 \) (as calculated above). The graph should have a vertex around \( n = 1300 \) and pass through the \( x \)-intercept \( (257,0) \). Among the given graphs, the one that has the shape of a downward - opening parabola and the correct \( x \)-intercept is Option A.

Answer:

To find the \( x \)-intercepts of the profit function \( P(n) = -0.002n^2 + 5.2n - 1203 \), we set \( P(n) = 0 \) and solve for \( n \).

Step 1: Set up the equation

We have the quadratic equation:

$$ -0.002n^2 + 5.2n - 1203 = 0 $$

Multiply through by \( -500 \) to simplify (to make the coefficients easier to handle):

$$ n^2 - 2600n + 601500 = 0 $$

Step 2: Use the quadratic formula

For a quadratic equation \( ax^2 + bx + c = 0 \), the solutions are given by:

$$ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

Here, \( a = 1 \), \( b = -2600 \), and \( c = 601500 \).

First, calculate the discriminant \( D = b^2 - 4ac \):

$$ D = (-2600)^2 - 4(1)(601500) $$

$$ D = 6760000 - 2406000 $$

$$ D = 4354000 $$

Now, find the square root of the discriminant:

$$ \sqrt{D} = \sqrt{4354000} \approx 2086.62 $$

Now, substitute into the quadratic formula:

$$ n = \frac{2600 \pm 2086.62}{2} $$

Step 3: Calculate the two solutions

First solution:

$$ n_1 = \frac{2600 + 2086.62}{2} = \frac{4686.62}{2} \approx 2343.31 $$

Second solution:

$$ n_2 = \frac{2600 - 2086.62}{2} = \frac{513.38}{2} \approx 256.69 $$

Rounding to the nearest whole number, the \( x \)-intercepts (values of \( n \)) are approximately \( 257 \) and \( 2343 \). So the ordered pairs are \( (257, 0) \) and \( (2343, 0) \). But looking at the options, the first \( x \)-intercept is \( (257, 0) \) (since the option A mentions \( (0,257) \) which is likely a typo and should be \( (257, 0) \)). So the \( x \)-intercepts are \( (257, 0) \) and \( (2343, 0) \), but from the given option, the \( x \)-intercept is \( (257, 0) \) (considering the option A's format might have a typo in the ordered pair, swapping \( x \) and \( y \) coordinates).

For the graph, the profit function is a downward - opening parabola (since the coefficient of \( n^2 \) is negative). The vertex of a quadratic function \( y = ax^2+bx + c \) is at \( n=-\frac{b}{2a} \). For \( P(n)=-0.002n^2 + 5.2n-1203 \), \( a=-0.002 \), \( b = 5.2 \). So \( n=-\frac{5.2}{2\times(-0.002)}=\frac{5.2}{0.004} = 1300 \). Then \( P(1300)=-0.002\times(1300)^2+5.2\times1300 - 1203=-0.002\times1690000+6760 - 1203=-3380 + 6760-1203=2177 \). The graph with vertex around \( n = 1300 \) and passing through \( n = 257 \) (x - intercept) is the correct one. From the given graphs, the graph that matches the shape (downward opening parabola) and the x - intercept around 257 is the one corresponding to option A (assuming the visual of option A has the correct shape and intercept).

So the \( x \)-intercepts are \( (257, 0) \) (and \( (2343, 0) \)) and the correct graph is Option A.