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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


Question
Use the three points from the table to write the equation of the parabola that passes through the points. If using a calculator, you MUST upload picture of the calculator, or send it to your teacher in a webmail.



Note: Enter your answer and SHOW ALL THE STEPS NECESSARY to solve this problem in the space provided. You will not receive credit without all work shown.

1 answer

To find a quadratic function \( P(d) = ad^2 + bd + c \) that represents the price of pizza \( P \) as a function of the diameter \( d \), we can use the three given points from the table.

The points we need are:

  • \( (12, 15.80) \)
  • \( (16, 22.20) \)
  • \( (20, 35.00) \)

We can represent these points as equations by substituting \( d \) into the quadratic function:

  1. For the first point \( (12, 15.80) \): \[ 15.80 = a(12)^2 + b(12) + c \implies 15.80 = 144a + 12b + c \quad \text{(Equation 1)} \]

  2. For the second point \( (16, 22.20) \): \[ 22.20 = a(16)^2 + b(16) + c \implies 22.20 = 256a + 16b + c \quad \text{(Equation 2)} \]

  3. For the third point \( (20, 35.00) \): \[ 35.00 = a(20)^2 + b(20) + c \implies 35.00 = 400a + 20b + c \quad \text{(Equation 3)} \]

This gives us the following system of equations:

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

Next, we will eliminate \( c \) by subtracting the equations in pairs:

Step 1: Eliminate \( c \)

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

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

Step 2: Solve for \( a \) and \( b \)

Now, we have a system of two equations in two variables:

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

Next, we can eliminate \( b \) by subtracting Equation 4 from Equation 5: \[ (144a + 4b) - (112a + 4b) = 12.80 - 6.40 \] This gives: \[ 32a = 6.40 \implies a = \frac{6.40}{32} = 0.20 \]

Step 3: Substitute \( a \) back to find \( b \)

Now substitute \( a = 0.20 \) into Equation 4: \[ 112(0.20) + 4b = 6.40 \] \[ 22.4 + 4b = 6.40 \] \[ 4b = 6.40 - 22.4 \] \[ 4b = -16.00 \implies b = -4.00 \]

Step 4: Substitute \( a \) and \( b \) to find \( c \)

Now substitute \( a \) and \( b \) back into Equation 1 to find \( c \): \[ 144(0.20) + 12(-4.00) + c = 15.80 \] \[ 28.8 - 48 + c = 15.80 \] \[ c = 15.80 + 48 - 28.8 \] \[ c = 35.00 - 28.8 = 6.20 \]

Conclusion

Now we have \( a \), \( b \), and \( c \):

  • \( a = 0.20 \)
  • \( b = -4.00 \)
  • \( c = 6.20 \)

Thus, the equation for the price of the pizza as a function of its diameter is: \[ P(d) = 0.20d^2 - 4.00d + 6.20 \]