Question

suppose your dog weighed 2.2 pounds at birth and weighed 15.7 pounds one year later. based on these two data points, find a linear function that describes how weight varies with age. use this function to predict your dogs weight at 6 and 11 years of age. comment on the validity of this model. the equation used to model this situation is \\(\square\\), where w is the weight of the dog in pounds and t is time in years.

Explanation:

Step1: Recall linear function form

A linear function has the form \( w(t) = mt + b \), where \( m \) is the slope and \( b \) is the y - intercept.
We have two points: when \( t = 0 \) (birth), \( w=2.2 \) pounds, and when \( t = 1 \) (one year later), \( w = 15.7 \) pounds.

Step2: Find the slope \( m \)

The formula for the slope \( m \) between two points \((t_1,w_1)\) and \((t_2,w_2)\) is \( m=\frac{w_2 - w_1}{t_2 - t_1} \).
Substituting \( t_1 = 0,w_1 = 2.2,t_2=1,w_2 = 15.7 \) into the slope formula:
\( m=\frac{15.7 - 2.2}{1 - 0}=\frac{13.5}{1}=13.5 \)

Step3: Find the y - intercept \( b \)

We know that when \( t = 0 \), \( w = 2.2 \). Substituting \( t = 0 \), \( w = 2.2 \) and \( m = 13.5 \) into the linear function \( w(t)=mt + b \):
\( 2.2=13.5\times0 + b \), so \( b = 2.2 \)

Step4: Write the linear function

Substituting \( m = 13.5 \) and \( b = 2.2 \) into the linear function \( w(t)=mt + b \), we get \( w(t)=13.5t+2.2 \)

Answer:

\( w(t) = 13.5t + 2.2 \)