Question

1. a taxi driver records the time required to complete various trips and the distance for each trip. the best fit line is given by the equation y = 0.457x + 0.417, where y represents the distance in miles, and x represents the time for the trip in minutes. a. use the best fit line to predict the distance for a trip that takes 20 minutes. show your reasoning. b. use the best fit line to predict the time for a trip that is 6 miles long. show your reasoning.

Explanation:

Part a

Step1: Identify the formula and variables

The best - fit line is given by \(y = 0.457x+0.417\), where \(y\) is the distance (in miles) and \(x\) is the time (in minutes). We need to find \(y\) when \(x = 20\).

Step2: Substitute \(x = 20\) into the equation

Substitute \(x = 20\) into \(y=0.457x + 0.417\).
\(y=0.457\times20+0.417\)
First, calculate \(0.457\times20\): \(0.457\times20 = 9.14\)
Then, add \(0.417\) to the result: \(y=9.14 + 0.417=9.557\)

Step1: Identify the formula and variables

The best - fit line is \(y = 0.457x+0.417\), where \(y\) is the distance (in miles) and \(x\) is the time (in minutes). We need to find \(x\) when \(y = 6\).

Step2: Substitute \(y = 6\) into the equation and solve for \(x\)

Substitute \(y = 6\) into \(y = 0.457x+0.417\):
\(6=0.457x + 0.417\)
Subtract \(0.417\) from both sides:
\(6 - 0.417=0.457x\)
\(5.583=0.457x\)
Then, solve for \(x\) by dividing both sides by \(0.457\):
\(x=\frac{5.583}{0.457}\approx12.22\)

Answer:

The distance for a 20 - minute trip is approximately \(9.56\) miles (or \(9.557\) miles).

Part b