Question

At Pizza Palace, the price of a pizza is a quadratic function of the diameter. The prices of three pizzas at Pizza Palace are shown in the following table: Diameter of Pizza (inches) Price of Pizza (dollars) 12" $15.80 16" $22.20 20" $35.00

Answer 1

To model the price of the pizza as a quadratic function of its diameter, we can denote the diameter as \( x \) and the price as \( P(x) \).

We have three data points from the table:

1. \( P(12) = 15.80 \)
2. \( P(16) = 22.20 \)
3. \( P(20) = 35.00 \)

Since the price is a quadratic function, we can express it in the form:

\[
P(x) = ax^2 + bx + c
\]

We need to set up a system of equations using the three data points to solve for the coefficients \( a \), \( b \), and \( c \).

Starting with the data points, we can substitute each \( x \) and \( P(x) \) into the quadratic equation:

1. For \( x = 12 \):
\[ a(12^2) + b(12) + c = 15.80 \]
\[ 144a + 12b + c = 15.80 \quad \text{(1)} \]

2. For \( x = 16 \):
\[ a(16^2) + b(16) + c = 22.20 \]
\[ 256a + 16b + c = 22.20 \quad \text{(2)} \]

3. For \( x = 20 \):
\[ a(20^2) + b(20) + c = 35.00 \]
\[ 400a + 20b + c = 35.00 \quad \text{(3)} \]

Now we have the following system of equations:

\[
\begin{align*}
144a + 12b + c &= 15.80 \quad (1) \\
256a + 16b + c &= 22.20 \quad (2) \\
400a + 20b + c &= 35.00 \quad (3)
\end{align*}
\]

Next, we can eliminate \( c \) by subtracting equations.

Subtract equation (1) from (2):
\[
(256a + 16b + c) - (144a + 12b + c) = 22.20 - 15.80
\]
\[
112a + 4b = 6.40 \quad \text{(4)}
\]

Subtract equation (2) from (3):
\[
(400a + 20b + c) - (256a + 16b + c) = 35.00 - 22.20
\]
\[
144a + 4b = 12.80 \quad \text{(5)}
\]

Now we have a new system with equations (4) and (5):

\[
\begin{align*}
112a + 4b &= 6.40 \quad (4) \\
144a + 4b &= 12.80 \quad (5)
\end{align*}
\]

Next, we can subtract equation (4) from (5) to eliminate \( b \):

\[
(144a + 4b) - (112a + 4b) = 12.80 - 6.40
\]
\[
32a = 6.40 \implies a = \frac{6.40}{32} = 0.20
\]

Now, substitute \( a = 0.20 \) back into equation (4) to find \( b \):
\[
112(0.20) + 4b = 6.40
\]
\[
22.4 + 4b = 6.40 \implies 4b = 6.40 - 22.4 \implies 4b = -16.00 \implies b = -4.00
\]

Now we have \( a = 0.20 \) and \( b = -4.00 \). Substitute these values back into equation (1) to solve for \( c \):
\[
144(0.20) + 12(-4.00) + c = 15.80
\]
\[
28.8 - 48 + c = 15.80 \implies c = 15.80 - 28.8 + 48 = 35.00 - 28.8 = 6.20
\]

Now, we have all coefficients:

\[
a = 0.20, \quad b = -4.00, \quad c = 6.20
\]

Therefore, the quadratic function modeling the price of the pizza is:

\[
P(x) = 0.2x^2 - 4x + 6.2
\]

You can use this equation to estimate or predict prices for pizzas of various diameters.