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

Question

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

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 2

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

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!