Question

juan wants to change the shape of his vegetable garden from a square to a rectangle, but keep the same area so he can grow the same amount of vegetables. the rectangular garden will have a length that is 2 times the length of the square garden, and the width of the new garden will be 16 feet shorter than the old garden. the square garden is x feet by x feet. old garden area = new garden area $x^2 = (2x)(x - 16)$ $x^2 = 2x^2 - 32x$ $0 = x^2 - 32x$ what is the value of x that makes sense in this context? what are the dimensions of the new garden?

Explanation:

First Sub - Question: Find the value of \(x\)

Step 1: Factor the equation

We have the equation \(0 = x^{2}-32x\). We can factor out an \(x\) from the right - hand side.
\(0=x(x - 32)\)

Step 2: Solve for \(x\)

Using the zero - product property, which states that if \(ab = 0\), then either \(a = 0\) or \(b = 0\).
If \(x=0\), then the side length of the square garden is \(0\) feet, which does not make sense in the context of a garden.
If \(x - 32=0\), then \(x = 32\). This is a valid solution because a side length of \(32\) feet for a garden is a reasonable length.

Step 1: Find the length of the new garden

The length of the new (rectangular) garden is \(2x\). Since we found that \(x = 32\), we substitute \(x = 32\) into the expression for the length.
Length \(=2x=2\times32 = 64\) feet.

Step 2: Find the width of the new garden

The width of the new garden is \(x - 16\). Substitute \(x = 32\) into the expression for the width.
Width \(=x - 16=32-16 = 16\) feet.

Answer:

The value of \(x\) that makes sense in this context is \(32\).

Second Sub - Question: Find the dimensions of the new garden