Question

1. the family designed their vegetable garden to have a perimeter of 32 feet with the length measuring two feet more than twice the width.

a. using ( l ) to represent the length of the garden and ( w ) to represent its width, write and solve a system of equations that describes this situation.
b. what are the dimensions of the garden?

Explanation:

Step1: Set up perimeter equation

The perimeter of a rectangle is $2(l + w) = 32$, which simplifies to $l + w = 16$.

Step2: Set up length-width relation

Length is 2 more than twice width: $l = 2w + 2$.

Step3: Substitute $l$ into perimeter equation

Substitute $l = 2w + 2$ into $l + w = 16$:
$(2w + 2) + w = 16$

Step4: Solve for $w$

Combine like terms: $3w + 2 = 16$
Subtract 2: $3w = 14$
Calculate width: $w = \frac{14}{3} \approx 4.67$

Step5: Solve for $l$

Substitute $w = \frac{14}{3}$ into $l = 2w + 2$:
$l = 2\times\frac{14}{3} + 2 = \frac{28}{3} + \frac{6}{3} = \frac{34}{3} \approx 11.33$

Answer:

a. The system of equations is:

$$\begin{cases} 2(l + w) = 32 \\ l = 2w + 2 \end{cases}$$

b. The width of the garden is $\frac{14}{3}$ feet (or approximately 4.67 feet) and the length is $\frac{34}{3}$ feet (or approximately 11.33 feet).