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A 6.0 m ladder leans against a wall at an angle of 80 degrees to the horizontal. How far is the foot of the ladder from the wall, and how high up the wall does the ladder reach?

Mathematics

ScanSolve AI Answer

Answer

b

1.0 m\text{1.0 m}

5.9 m\text{5.9 m}

Step-by-step solution

4 steps

Here is the solution to the problem:

Step 1: Identify the given information and what needs to be found.
The ladder, the ground, and the wall form a right-angled triangle.
Length of the ladder (hypotenuse), L=6.0L = 6.0 m.
Angle of inclination to the horizontal, θ=80∘\theta = 80^\circ.
We need to find:

Part a

The distance of the foot of the ladder from the wall (adjacent side), let's call it xx.

Part b

1.0 m\text{1.0 m}

The height above the ground where the ladder touches the wall (opposite side), let's call it hh.

  1. Calculate the distance of the foot of the ladder from the wall (xx).

    We use the cosine function, which relates the adjacent side, hypotenuse, and angle:

    cos⁡(θ)=adjacenthypotenuse\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} cos⁡(80∘)=xL\cos(80^\circ) = \frac{x}{L} x=L⋅cos⁡(80∘)x = L \cdot \cos(80^\circ)

  2. Substitute the values and calculate xx.

    x=6.0 m×cos⁡(80∘)x = 6.0 \text{ m} \times \cos(80^\circ) x=6.0 m×0.173648...x = 6.0 \text{ m} \times 0.173648... x≈1.04188 mx \approx 1.04188 \text{ m}

    Rounding to one decimal place:

    x≈1.0 mx \approx \boxed{\text{1.0 m}}

  3. Calculate the height above the ground (hh).

    We use the sine function, which relates the opposite side, hypotenuse, and angle:

    sin⁡(θ)=oppositehypotenuse\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} sin⁡(80∘)=hL\sin(80^\circ) = \frac{h}{L} h=L⋅sin⁡(80∘)h = L \cdot \sin(80^\circ)

  4. Substitute the values and calculate hh.

    h=6.0 m×sin⁡(80∘)h = 6.0 \text{ m} \times \sin(80^\circ) h=6.0 m×0.984807...h = 6.0 \text{ m} \times 0.984807... h≈5.90884 mh \approx 5.90884 \text{ m}

    Rounding to one decimal place:

    h≈5.9 mh \approx \boxed{\text{5.9 m}}
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