Question

a building is 2 ft from a 10-ft fence that surrounds the property. a worker wants to wash a window in the building 15 ft from the ground. he plans to place a ladder over the fence so it rests against the building. (see the figure.) he decides he should place the ladder 8 ft from the fence for stability. to the nearest tenth of a foot, how long a ladder will he need?
he will need a \\(\square\\) ft ladder.
(type an integer or decimal rounded to the nearest tenth as needed.)

Explanation:

Step1: Determine horizontal distance

The horizontal distance from the ladder's base to the building is \(8 + 2=10\) ft.

Step2: Determine vertical distance

The vertical distance from the ground to the window is \(15\) ft (since the fence is 10 ft and the window is 5 ft above the fence, \(10 + 5 = 15\) ft).

Step3: Apply Pythagorean theorem

The ladder length \(L\) is the hypotenuse of a right triangle with legs \(10\) ft (horizontal) and \(15\) ft (vertical). Using \(L=\sqrt{a^{2}+b^{2}}\), where \(a = 10\) and \(b = 15\), we get \(L=\sqrt{10^{2}+15^{2}}=\sqrt{100 + 225}=\sqrt{325}\approx18.0\) (rounded to the nearest tenth). Wait, no, wait. Wait, the horizontal distance: the ladder is 8 ft from the fence, and the fence is 2 ft from the building, so total horizontal distance from ladder base to building is \(8 + 2=10\) ft. Vertical distance: the window is 15 ft from the ground (10 ft fence + 5 ft above fence). Wait, but let's check the figure again. Wait, maybe I misread. Wait, the fence is 10 ft tall, and the window is 5 ft above the fence? Wait, the figure shows 10 ft (fence height) and 5 ft (above fence). So total vertical height is \(10 + 5=15\) ft. Horizontal distance: 8 ft from fence, 2 ft from fence to building, so total horizontal distance from ladder base to building is \(8 + 2 = 10\) ft. Then ladder length is hypotenuse of right triangle with legs 10 and 15? Wait, no, wait, maybe the horizontal distance is 8 + 2 = 10, vertical distance is 15? Wait, but let's recalculate. \(a = 10\), \(b = 15\), so \(L=\sqrt{10^{2}+15^{2}}=\sqrt{100 + 225}=\sqrt{325}\approx18.0\)? Wait, no, \(\sqrt{325}\) is approximately 18.027..., so to the nearest tenth, 18.0? Wait, but maybe I made a mistake in vertical distance. Wait, maybe the window is 15 ft from the ground, but let's check again. Wait, the problem says "a window in the building 15 ft from the ground". Oh, right, the window is 15 ft from the ground, so vertical leg is 15 ft, horizontal leg is 8 + 2 = 10 ft. So ladder length is \(\sqrt{10^{2}+15^{2}}=\sqrt{325}\approx18.0\) ft? Wait, but let's confirm. Wait, 10 squared is 100, 15 squared is 225, sum is 325, square root of 325 is about 18.03, so to the nearest tenth, 18.0. Wait, but maybe the horizontal distance is 8 ft from fence, and fence is 2 ft from building, so total horizontal distance from ladder base to building is 8 + 2 = 10 ft. Vertical distance is 15 ft. So yes, Pythagorean theorem: \(L=\sqrt{(8 + 2)^{2}+(10 + 5)^{2}}=\sqrt{10^{2}+15^{2}}=\sqrt{325}\approx18.0\) ft. Wait, but maybe I misread the vertical distance. Wait, the problem says "a window in the building 15 ft from the ground". So that's correct. So the ladder length is approximately 18.0 ft? Wait, no, wait, maybe the vertical distance is 15 ft, horizontal is 10 ft, so ladder length is \(\sqrt{10^2 + 15^2}=\sqrt{325}\approx18.0\) ft.

Wait, but let's check again. Wait, 8 ft from fence, 2 ft from fence to building: horizontal distance from ladder base to building is 8 + 2 = 10 ft. Vertical distance from ground to window is 15 ft. So right triangle with legs 10 and 15. Then hypotenuse is \(\sqrt{10^2 + 15^2}=\sqrt{100 + 225}=\sqrt{325}\approx18.0\) ft (to the nearest tenth).

Answer:

\(18.0\)