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: \[ \boxed{70} \text{ feet} \]