To solve this problem, we will model the relationship between speed and gas mileage using a quadratic equation. A quadratic model takes the form:
[tex]\[ y = ax^2 + bx + c \][/tex]
where [tex]\( y \)[/tex] is the gas mileage, and [tex]\( x \)[/tex] is the speed in miles per hour (mph). We need to determine the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] that best fit the given data. After finding the quadratic model, we can use it to predict the gas mileage at 80 mph.
### Step-by-Step Solution:
1. Fit a Quadratic Model to the Data:
We have the data points for speed ([tex]\( x \)[/tex]) and gas mileage ([tex]\( y \)[/tex]). By fitting a quadratic model to these data points, we determine the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]. The coefficients that best fit the given data are:
[tex]\[ a = -0.008895104895104906 \][/tex]
[tex]\[ b = 0.8047352647352655 \][/tex]
[tex]\[ c = 12.612787212787183 \][/tex]
2. Form the Quadratic Equation:
Substituting the obtained coefficients into the quadratic model, we get:
[tex]\[ y = -0.008895104895104906x^2 + 0.8047352647352655x + 12.612787212787183 \][/tex]
3. Predict the Gas Mileage at 80 mph:
To find the predicted gas mileage at [tex]\( x = 80 \)[/tex] mph, we substitute [tex]\( x = 80 \)[/tex] into the quadratic equation:
[tex]\[ y = -0.008895104895104906(80)^2 + 0.8047352647352655(80) + 12.612787212787183 \][/tex]
4. Calculate the Mileage:
Evaluating the equation at [tex]\( x = 80 \)[/tex]:
[tex]\[ y = -0.008895104895104906 \cdot 6400 + 0.8047352647352655 \cdot 80 + 12.612787212787183 \][/tex]
[tex]\[ y = -56.92867117283964 + 64.37882117882124 + 12.612787212787183 \][/tex]
[tex]\[ y \approx 20.062937062937017 \][/tex]
5. Round the Result:
Rounding the result to the nearest tenth, we get:
[tex]\[ y \approx 20.1 \][/tex]
Thus, the predicted gas mileage of the boat at 80 miles per hour is:
[tex]\[ \boxed{20.1} \text{ miles per gallon} \][/tex]
Among the provided options (16.9, 19.6, 22, 25 miles per gallon), the closest value is not present. Therefore, the correct predicted mileage, according to our calculation, is approximately 20.1 miles per gallon.
