Question

Kelly has 16 m of fencing to enclose her flower garden. The function: f(x) = -(x - 4)^2 + 16 can be used to model the area of the garden, where x is the length of fencing in metres. What is the domain for the measure of fencing Kelly can use for the length?

Answer 1

To determine the domain for the measure of fencing Kelly can use for the length, we need to consider the conditions of the problem.

Kelly has a total of 16 meters of fencing to enclose her garden. The function \( f(x) = -(x - 4)^2 + 16 \) represents the area of the garden in terms of the length \( x \) of one dimension (presumably the length of one side of a rectangular garden).

Since the fencing is used to form a perimeter, if we consider \( x \) as one side of a rectangular garden, we can denote the opposite side as \( y \). The perimeter \( P \) of a rectangle is given by:

\[
P = 2x + 2y
\]

Setting the perimeter equal to the available fencing, we have:

\[
2x + 2y = 16
\]

which simplifies to:

\[
x + y = 8 \quad \Rightarrow \quad y = 8 - x
\]

From this relationship, since \( y \) must be non-negative, we must have:

\[
8 - x \geq 0 \quad \Rightarrow \quad x \leq 8
\]

Also, the fencing length \( x \) must be positive, so:

\[
x \geq 0
\]

Thus, combining these two inequalities, we find that \( x \) must satisfy:

\[
0 \leq x \leq 8
\]

Therefore, the domain of \( f(x) \) corresponding to the measure of fencing Kelly can use for the length is:

\[
[0, 8]
\]