The problem asks to find the length of a rectangular garden given that its perimeter is 32 cm and its area is 39 cm$^2$.

1. Problem Description

The problem asks to find the length of a rectangular garden given that its perimeter is 32 cm and its area is 39 cm$^2$.

2. Solution Steps

Let $l$ be the length and $w$ be the width of the rectangular garden.

We are given the perimeter $P = 32$ cm and the area $A = 39$ cm$^2$.

The formulas for the perimeter and area of a rectangle are:

$P = 2(l + w)$

$A = l \times w$

We have the following equations:

$2(l + w) = 32$

$l \times w = 39$

From the first equation, we can write:

$l + w = \frac{32}{2} = 16$

$w = 16 - l$

Substitute this into the second equation:

$l(16 - l) = 39$

$16l - l^2 = 39$

$l^2 - 16l + 39 = 0$

This is a quadratic equation. We can solve it by factoring:

$(l - 3)(l - 13) = 0$

Thus, $l = 3$ or $l = 13$.

If $l = 3$, then $w = 16 - 3 = 13$.

If $l = 13$, then $w = 16 - 13 = 3$.

Since length is usually greater than width, we can say that the length is 13 cm and the width is 3 cm. Therefore, the length of the rectangular garden is 13 cm.

3. Final Answer

C. 13cm