Question

About how many feet of fencing are needed to enclose a rectangular garden with a 12-foot-long side and a 26-foot-long diagonal? Round your answer to the nearest whole foot.

Answer 1

To solve this problem, we need to find the dimensions of the rectangular garden given one side and one diagonal.

We are given:
- One side of the rectangle (\(a\)) = 12 feet
- Diagonal of the rectangle (\(d\)) = 26 feet

According to the Pythagorean theorem, in a right triangle formed by the length, width, and diagonal of a rectangle:
\[
d^2 = a^2 + b^2
\]
where \(b\) is the other side of the rectangle.

Substituting the known values into this equation:
\[
26^2 = 12^2 + b^2
\]
Calculating \(26^2\) and \(12^2\):
\[
676 = 144 + b^2
\]
Now, we will isolate \(b^2\):
\[
b^2 = 676 - 144 = 532
\]
Taking the square root of both sides to find \(b\):
\[
b = \sqrt{532} \approx 23.065
\]

Now we can find the perimeter \(P\) of the rectangle, which is given by the formula:
\[
P = 2(a + b)
\]
Substituting the known values:
\[
P = 2(12 + \sqrt{532}) = 2(12 + 23.065) \approx 2(35.065) \approx 70.13
\]

Rounding to the nearest whole foot, the total amount of fencing needed is approximately:
\[
\boxed{70} \text{ feet}
\]