Question

buddys magical journey

1. buddys journey from the north pole to new york city can be modeled by a line. he starts at the point (0, 0) representing the north pole and moves at a rate of 4 miles per day. write the equation of his epic travel route.

1. the height of an elf changes dramatically when entering the human world. buddy grows 0.5 inches taller each month after arriving in new york, starting at 30 inches tall. write an equation relating buddys height to the number of months since he arrived.

buddys elevator button adventure

1. buddy creates a unique pattern while pushing elevator buttons in the chrysler building. he starts at the point (1, 3) representing the lobby floor and ends at the point (5, 11) representing a higher floor. write the equation of the line that represents his elevator button - pushing journey!

Explanation:

Question 3 Solution:

Step1: Identify slope and intercept

The journey is a line. Start point \((0,0)\) means \(y\)-intercept \(b = 0\). Rate (slope \(m\)) is 4 miles per day.

Step2: Use slope - intercept form

Slope - intercept form is \(y=mx + b\). Substitute \(m = 4\) and \(b = 0\).

Step1: Identify slope and intercept

Growth rate (slope \(m\)) is \(0.5\) inches per month. Starting height ( \(y\)-intercept \(b\)) is 30 inches.

Step2: Use slope - intercept form

Let \(x\) be months, \(y\) be height. Formula \(y=mx + b\). Substitute \(m = 0.5\), \(b = 30\).

Step1: Calculate slope

Given points \((1,3)\) and \((5,11)\). Slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{11 - 3}{5 - 1}=\frac{8}{4}=2\).

Step2: Find \(y\)-intercept

Use point \((1,3)\) and \(m = 2\) in \(y=mx + b\). \(3=2(1)+b\Rightarrow b = 1\).

Step3: Write equation

Substitute \(m = 2\), \(b = 1\) into \(y=mx + b\).

Answer:

\(y = 4x\) (where \(x\) is days, \(y\) is miles)

Question 4 Solution: