We have three points corresponding to the given diameters and prices of pizzas at Pizza Palace:
- \( (12, 15.80) \)
- \( (16, 22.20) \)
- \( (20, 35.00) \)
We can express the price \( P \) as a quadratic function of the diameter \( d \) in the form:
\[ P(d) = ad^2 + bd + c \]
where \( a \), \( b \), and \( c \) are coefficients we need to determine.
Using the three points, we can set up a system of equations:
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From \( (12, 15.80) \): \[ a(12^2) + b(12) + c = 15.80 \] \[ 144a + 12b + c = 15.80 \quad \text{(Equation 1)} \]
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From \( (16, 22.20) \): \[ a(16^2) + b(16) + c = 22.20 \] \[ 256a + 16b + c = 22.20 \quad \text{(Equation 2)} \]
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From \( (20, 35.00) \): \[ a(20^2) + b(20) + c = 35.00 \] \[ 400a + 20b + c = 35.00 \quad \text{(Equation 3)} \]
Now we will organize our equations:
- \( 144a + 12b + c = 15.80 \)
- \( 256a + 16b + c = 22.20 \)
- \( 400a + 20b + c = 35.00 \)
Next, we will eliminate \( c \) from these equations. We can do this by subtracting Equation 1 from Equation 2, and Equation 2 from Equation 3.
Subtract Equation 1 from Equation 2:
\[ (256a + 16b + c) - (144a + 12b + c) = 22.20 - 15.80 \] \[ (256 - 144)a + (16 - 12)b = 6.40 \] \[ 112a + 4b = 6.40 \quad \text{(Equation 4)} \]
Subtract Equation 2 from Equation 3:
\[ (400a + 20b + c) - (256a + 16b + c) = 35.00 - 22.20 \] \[ (400 - 256)a + (20 - 16)b = 12.80 \] \[ 144a + 4b = 12.80 \quad \text{(Equation 5)} \]
Now we have a new system of two equations with two unknowns:
- \( 112a + 4b = 6.40 \)
- \( 144a + 4b = 12.80 \)
We can now eliminate \( b \) by subtracting Equation 4 from Equation 5:
\[ (144a + 4b) - (112a + 4b) = 12.80 - 6.40 \] \[ 32a = 6.40 \] \[ a = \frac{6.40}{32} = 0.20 \]
Now we can substitute \( a = 0.20 \) back into Equation 4 to find \( b \):
\[ 112(0.20) + 4b = 6.40 \] \[ 22.40 + 4b = 6.40 \] \[ 4b = 6.40 - 22.40 \] \[ 4b = -16 \] \[ b = -4 \]
Now that we have \( a \) and \( b \), we can find \( c \) by substituting \( a \) and \( b \) back into one of the original equations; we can use Equation 1:
\[ 144(0.20) + 12(-4) + c = 15.80 \] \[ 28.8 - 48 + c = 15.80 \] \[ c = 15.80 - 28.8 + 48 \] \[ c = 15.80 - 28.8 + 48 = 35.00 \]
Thus, we have found:
- \( a = 0.20 \)
- \( b = -4 \)
- \( c = -35.00 \)
Putting it all together, the equation of the parabola that models the price of pizza as a function of its diameter is:
\[ P(d) = 0.20d^2 - 4d + 35.00 \]