A building 24 m tall is viewed from the top and bottom of a vertical ladder.The building forms an angle of 45 degrees with the top of the ladder and an angle of 60 degrees with the bottom of the ladder.Find

(i) the height of the vertical ladder
(ii) the distance between the building and the ladder

The top of the building forms an angle of 45 degrees with the top of the ladder...

I believe the question means the top of the building makes 45 and 60 degrees with the horizontal, and not the building itself (i.e. from bottom to top).

In this case, the distance from the building is
d=24 cot(60)=8sqrt(3).
The height of the ladder is
24m - d.tan(45)
= 24 - 8sqrt(3)
= 10.144 m.

If the angles are subtented by the top and bottom of the building, the feasible solution of the height of the ladder is higher than the building, as if it is a ladder from a fire-truck.

In this case, we calculate the centre of a circle which passes through the top and bottom of the building, and subtends an angle of 90 deg. It is a point at 12 m from the face of the building and 12 m. high above ground. The radius of the circle is r=12sqrt(2) m.

From here we find the intersection with the ladder at d=8sqrt(3) from the building. There is one intersection above ground at a height of
h = 12m + sqrt(r2+(d-r)2)
=12 + sqrt(144*2 + (13.856-12)2)
=12 + sqrt(288+ 3.446)
= 12+17.07
= 29.07 m.

A sketch is available at the following link:

http://i263.photobucket.com/albums/ii157/mathmate/Trigo.jpg

To solve this problem, we can use trigonometric ratios such as sine, cosine, and tangent. Let's break down the problem into two parts:

(i) The height of the vertical ladder:
To find the height of the vertical ladder, let's consider the triangle formed by the building, the ladder, and the ground. We can use the tangent ratio, which is defined as the opposite side divided by the adjacent side.

In this triangle, the opposite side is the height of the building (24 m), and the adjacent side is the height of the ladder. Since the angle formed between the building and the top of the ladder is 45 degrees, we can use the tangent of 45 degrees:

tan(45°) = opposite/adjacent

tan(45°) = 24 m/height of the ladder

Using the fact that tan(45°) = 1, we can solve for the height of the ladder:

1 = 24 m/height of the ladder

Rearranging the equation, we get:

height of the ladder = 24 m/1 = 24 m

Therefore, the height of the vertical ladder is 24 m.

(ii) The distance between the building and the ladder:
To find the distance between the building and the ladder, let's consider the triangle formed by the building, the ladder, and the ground. We can use the sine ratio, which is defined as the opposite side divided by the hypotenuse.

In this triangle, the opposite side is the height of the building (24 m), and the hypotenuse is the distance between the building and the ladder. Since the angle formed between the building and the bottom of the ladder is 60 degrees, we can use the sine of 60 degrees:

sin(60°) = opposite/hypotenuse

sin(60°) = 24 m/distance between the building and the ladder

Using the fact that sin(60°) = √3/2, we can solve for the distance between the building and the ladder:

√3/2 = 24 m/distance between the building and the ladder

Rearranging the equation, we get:

distance between the building and the ladder = (24 m * 2)/√3

Simplifying further, we get:

distance between the building and the ladder = 48√3/3

Therefore, the distance between the building and the ladder is approximately 27.86 m (rounded to two decimal places).