To use an extension ladder safely, the base must be 1 m out from the wall for every 2 m of vertical height.
a) What is the maximum angle of elevation, to the nearest degree, to
the top of the ladder?
b) If the ladder is extended to 4.72 m in length, how high can it safely
reach? Round your answer to the nearest hundredth of a metre.
c) How far out from the wall does a 5.9 m ladder need to be? Round
your answer to the nearest tenth of a metre.
you want an angle of x, where
tan(x) = 2/1
The rest just use the Pythagorean Theorem.
To use an extension ladder safely, the base must be 1 m out from the wall for every 2 m of vertical height.
a) What is the maximum angle of elevation, to the nearest degree, to
the top of the ladder?
b) If the ladder is extended to 4.72 m in length, how high can it safely
reach? Round your answer to the nearest hundredth of a metre.
c) How far out from the wall does a 5.9 m ladder need to be? Round
your answer to the nearest tenth of a metre.
a) To find the maximum angle of elevation to the top of the ladder, we can use the trigonometric relationship between the height, length, and angle of elevation. The tangent function relates the opposite side (height) to the adjacent side (length).
We can set up the equation as follows:
tan(angle) = height / length
Using the given relationship, the base must be 1 m out from the wall for every 2 m of vertical height. So, for a given height (h), the corresponding base distance (b) can be calculated as (1/2) * h.
Substituting this into the equation, we have:
tan(angle) = h / (1/2 * h)
Now, simplify the equation:
tan(angle) = 2
To find the angle, we can take the inverse tangent (arctan) of both sides:
angle = arctan(2)
Using a calculator, we find:
angle ≈ 63.43 degrees
Therefore, the maximum angle of elevation to the top of the ladder is approximately 63 degrees.
b) To determine how high the ladder can safely reach when extended to a length of 4.72 m, we need to calculate the corresponding safe height.
Using the given relationship, we know that the base must be 1 m out from the wall for every 2 m of vertical height. This can be expressed as:
b = (1/2) * h
Rearranging the equation to solve for height, we have:
h = 2 * b
Substituting the value for b (which is half the ladder length), we get:
h = 2 * (4.72 m / 2)
Simplifying:
h = 2.36 m
Therefore, when the ladder is extended to 4.72 m, it can safely reach a height of 2.36 m.
c) To determine how far out from the wall a 5.9 m ladder needs to be, we can use the same relationship as before.
Given that the base must be 1 m out from the wall for every 2 m of vertical height, we can set up the equation:
b = (1/2) * h
Substituting the given ladder height (h = 5.9 m) into the equation, we have:
b = (1/2) * 5.9 m
Calculating:
b ≈ 2.95 m
Therefore, a 5.9 m ladder needs to be approximately 2.95 m out from the wall.