A ladder leans against a building. The angle of elevation of the ladder is 70°. The top of the ladder is 25 ft from the ground.

To the nearest tenth of a foot, how long is the ladder?

24.7

sin 70° = 25/x

As usual for trig problems, draw a diagram. In this case, label the angle at the ground 70°. You want the hypotenuse (x), and have the side opposite the angle (25).

To find the length of the ladder, we can use trigonometry. The opposite side of the right triangle (from the angle of elevation) is the height of the building, and the hypotenuse is the length of the ladder.

Let's define the length of the ladder as 'x' feet.

Using the trigonometric function tangent, which is defined as the opposite side divided by the adjacent side:

tan(70°) = height of the building / distance from the building to the ladder base

Rearranging the equation to solve for the height of the building:

height of the building = tan(70°) * distance from the building to the ladder base

height of the building = tan(70°) * 25 ft (since we know the distance from the building to the ladder base is 25 ft)

Using a calculator, we can evaluate tan(70°) ≈ 2.7474

height of the building ≈ 2.7474 * 25 ft ≈ 68.685 ft (rounded to the nearest thousandth)

Now we can use the Pythagorean theorem to find the length of the ladder:

x^2 = height of the building^2 + (distance from the building to the ladder base)^2

x^2 = (68.685 ft)^2 + (25 ft)^2

x^2 ≈ 4708.537225 ft^2 + 625 ft^2

x^2 ≈ 5333.537225 ft^2

Taking the square root of both sides:

x ≈ √5333.537225 ft ≈ 73.0 ft (rounded to the nearest tenth)

Therefore, the length of the ladder is approximately 73.0 feet to the nearest tenth of a foot.

To find the length of the ladder, we can use the trigonometric function called sine. In this case, we can use the sine function because we have the angle of elevation and the opposite side length, which is the height of the building where the ladder leans against.

The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse. In this case, the opposite side is 25 ft and we need to find the hypotenuse, which is the length of the ladder.

So we can set up the equation:

sin(70°) = opposite / hypotenuse

Rearranging the equation, we have:

hypotenuse = opposite / sin(70°)

Plugging in the values, we get:

hypotenuse = 25 ft / sin(70°)

Calculating this using a calculator, we find that the length of the ladder to the nearest tenth of a foot is approximately 26.6 ft.