Question

1. katrina buys a 64-ft roll of fencing to make a rectangular play area for her dogs. use 2(l + w) = 64 to write a function for the length, given the width. graph the function. what is a reasonable domain for the situation? explain. see example 3

Explanation:

Step1: Solve the perimeter formula for length

Given \(2(l + w)=64\). First, divide both sides by 2:
\(\frac{2(l + w)}{2}=\frac{64}{2}\)
\(l + w = 32\). Then, solve for \(l\): \(l=32 - w\).

Step2: Determine the domain

Since \(l>0\) and \(w>0\) (length and width of a rectangle must be positive).
From \(l = 32 - w>0\), we get \(w<32\). And from \(w>0\).

Answer:

The function for the length is \(l(w)=32 - w\). The reasonable domain is \(0 < w<32\) because the width of a rectangle must be a positive number and less than half of the perimeter (since \(l = 32 - w\) and \(l>0\)).