Question

1. gardening mr. bigelows garden is a rectangle with dimensions x feet long by y feet wide. its perimeter is 70 feet.

a. write a function to represent the perimeter of his garden.
b. what are the x - and y - intercepts of the function?
c. does either intercept make sense as a solution for this situation? explain.

Explanation:

Step1: Write the perimeter function

The formula for the perimeter \(P\) of a rectangle is \(P = 2(x + y)\). Given \(P=70\), the function is \(2(x + y)=70\), which simplifies to \(y=-x + 35\).

Step2: Find the \(x\) - intercept

Set \(y = 0\) in \(y=-x + 35\). Then \(0=-x + 35\), so \(x = 35\).

Step3: Find the \(y\) - intercept

Set \(x = 0\) in \(y=-x + 35\). Then \(y=35\).

Step4: Analyze the intercepts as solutions

The \(x\) - intercept \((35,0)\) means if \(y = 0\) (width is \(0\)), it's not a valid garden (no width). Similarly, the \(y\) - intercept \((0,35)\) means if \(x = 0\) (length is \(0\)), it's not a valid garden (no length).

Answer:

a. \(y=-x + 35\)
b. \(x\) - intercept: \((35,0)\), \(y\) - intercept: \((0,35)\)
c. No. A garden with length \(0\) ( \(y\) - intercept) or width \(0\) ( \(x\) - intercept) is not a real - world garden (needs both length and width greater than \(0\)).