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

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.