Question

The cost of a flight is related to the distance traveled. Thus the further you fly the more you pay for the flight. Use the data to answer the questions keep in mind that x is the distance in miles and y is the cost in dollars
Table
Miles Cost
225 52.90
1375 95
1675 155
1950 204
2250 210
Find the line of best fit y = ___x + ___ round to the nearest hundredths using the above equation the cost of a flight that travels 525 miles is $__

Answer 1

Where is the line of best fit and the cost of the flight

Answer 2

To find the line of best fit for the given data, we can use linear regression. The data points provided are:

- (225, 52.90)
- (1375, 95)
- (1675, 155)
- (1950, 204)
- (2250, 210)

We can find the values of \(m\) (slope) and \(b\) (y-intercept) using the following formulas:

### Steps to Calculate the Line of Best Fit

1. **Calculate the means of x and y:**
\[
\bar{x} = \frac{\sum x}{n}, \quad \bar{y} = \frac{\sum y}{n}
\]

where \(n\) is the number of points.

- \(\sum x = 225 + 1375 + 1675 + 1950 + 2250 = 10275\)
- \(\sum y = 52.90 + 95 + 155 + 204 + 210 = 716.90\)
- \(n = 5\)

Now we can compute:
\[
\bar{x} = \frac{10275}{5} = 2055 \quad \text{and} \quad \bar{y} = \frac{716.90}{5} = 143.38
\]

2. **Calculate the slope \(m\):**
\[
m = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sum (x - \bar{x})^2}
\]

- Calculate \((x - \bar{x})(y - \bar{y})\) and \((x - \bar{x})^2\):

\[
\begin{align*}
(225 - 2055)(52.90 - 143.38) &= (-1830)(-90.48) = 165100.4 \\
(1375 - 2055)(95 - 143.38) &= (-680)(-48.38) = 32858.4 \\
(1675 - 2055)(155 - 143.38) &= (-380)(11.62) = -4415.6 \\
(1950 - 2055)(204 - 143.38) &= (-105)(60.62) = -6365.1 \\
(2250 - 2055)(210 - 143.38) &= (195)(66.62) = 12909.9 \\
\end{align*}
\]

\[
\sum (x - \bar{x})(y - \bar{y}) = 165100.4 + 32858.4 - 4415.6 - 6365.1 + 12909.9 = 191087
\]

For \(\sum (x - \bar{x})^2\):

\[
(225 - 2055)^2 = 3340250, \quad (1375 - 2055)^2 = 462400, \quad (1675 - 2055)^2 = 14400
\]
\[
(1950 - 2055)^2 = 11025, \quad (2250 - 2055)^2 = 38025
\]

\[
\sum (x - \bar{x})^2 = 3340250 + 462400 + 14400 + 11025 + 38025 = 3828100
\]

Now calculate \(m\):
\[
m = \frac{191087}{3828100} \approx 0.0499
\]

3. **Calculate the y-intercept \(b\):**
\[
b = \bar{y} - m \bar{x} = 143.38 - (0.0499)(2055) \approx 143.38 - 102.59 \approx 40.79
\]

Now, the equation for the line of best fit is:
\[
y = 0.0499x + 40.79
\]

### Cost for a Flight Traveling 525 Miles

To find the cost for a flight of 525 miles, substitute \(x = 525\):
\[
y = 0.0499(525) + 40.79 \approx 26.28 + 40.79 \approx 67.07
\]

### Final Results
- The line of best fit is:
\[ y \approx 0.05x + 40.79 \]
- The cost for a flight that travels 525 miles is approximately:
\[ $67.07 \]

Answer 3

The line of best fit, calculated from the provided data, is:

\[
y = 0.05x + 40.79
\]

### Explanation:
- **Where**:
- \(y\) represents the cost of the flight in dollars.
- \(x\) represents the distance of the flight in miles.

### Cost of the Flight for 525 Miles:
To find the cost of a flight that travels 525 miles, substitute \(x = 525\) into the equation:

\[
y = 0.05(525) + 40.79
\]
\[
y = 26.25 + 40.79
\]
\[
y \approx 67.04
\]

### Final Results:
- The line of best fit is:
\[
y = 0.05x + 40.79
\]

- The estimated cost for a flight traveling 525 miles is approximately:
\[
\text{Cost} \approx \$67.04
\]

If you require further details or clarifications on any aspect, feel free to ask!