Geometry and Angles (My weakness)

sanchu wrote:13th Edition Question Number 92.

A ladder of a fire truck is elevated to an angle of 60 degrees and extended to a length of 70 feet. If the base of the ladder is 7 feet above ground, how many feet above the ground does the ladder reach?

The answer set mentions that this is a 30-60-90 triangle. How would I have deduced this part? I visualized correctly, but I didn't know how to solve.

When ladder BC is elevated to an angle of 60 degrees, the result is ∆ABC.
Since AB is perpendicular to the ground, ∠BAC is a right angle.
Thus, ∆ABC is a 30-60-90 triangle.

In a 30-60-90 triangle, the sides are in the following ratio:
s : s√3 : 2s.
In ∆ABC, 2s = 70, implying that s=35.
Thus, the length of AB -- the middle side -- is 35√3.

Since the base of the ladder -- point C -- is 7 feet off the ground, the total distance between point B and the ground = [spoiler]35√3 + 7[/spoiler].

Your scratch paper should look like the attachment below.

sanchu wrote:13th Edition Question Number 92.

A ladder of a fire truck is elevated to an angle of 60 degrees and extended to a length of 70 feet. If the base of the ladder is 7 feet above ground, how many feet above the ground does the ladder reach?

The answer set mentions that this is a 30-60-90 triangle. How would I have deduced this part? I visualized correctly, but I didn't know how to solve.

A ladder of a fire truck is elevated to an angle of 60 degrees and extended to a length of 70 feet. If the base of the ladder is 7 feet above the ground, how many feet above the ground does the ladder reach?

A/ 35
B/ 42
C/ 35√3
D/ 7 + 35√3
E/ 7 + 42√3

Solution:

We are given that the ladder of a fire truck is elevated to an angle 60 degrees above the ground and that the ladder has a length of 70 feet. We are also given that the ladder is 7 feet above the ground. The best thing to do in this situation is to draw a diagram.



Notice that the resulting triangle in the sketch is a 30-60-90 right triangle. The ratio of the sides of a 30-60-90 right triangle is x : x√3 : 2x. We see that the hypotenuse length of 70 feet is equal to the "2x" from the 30-60-90 ratio. Thus, we can set up an equation and solve for x.

70 = 2x

x = 35

Because x = 35, we know that the side opposite the 60-degree angle or, in this case, the height of the ladder, is 35√3. The height of the ladder is 35√3 feet and the base of the ladder is 7 feet above the ground; thus, we know that the ladder reaches a total height above the ground of 35√3 + 7 feet.

The Answer is D

I worked through this problem yesterday and was conceptually fine on it -- i.e. recognized the 30-60-90 triangle, the fixed relationships for side lengths, and the need to add the 7 feet to the final answer -- but I ended up choosing the wrong answer because I didn't assign the 30 and 60 degree angles correctly, and obviously that gave me the wrong answer when I calculated the lengths via the fixed ratios for 30-60-90 triangles.

Reading back over this, the prompt seems to be really vague about where 60 degree angle is formed (i.e. between the ladder and the truck, or between the ladder and the building). Am I just misreading it when I think of it that way?

mcdesty wrote:Your scratch paper should look like the attachment below.

Great visual. Thanks for providing the scratchwork, and Rich, thank you for the insight as well.