a 35-foot long ladder is leaninig against the wall of a building. how far from the ground is the top of the ladder and how far from the base of the buildings is its foot if it makes a 20° angle with the wall ?
well, you know that
y/35 = cos 20°
x/35 = sin 20°
This problem is slightly sly, since most of the time they give you the angle with the ground!
To determine how far from the ground the top of the ladder is and how far from the base of the building its foot is, we can use trigonometry and the given information about the ladder's length and angle.
Let's label the distance from the top of the ladder to the ground as "h" and the distance from the base of the building to the foot of the ladder as "d".
First, let's focus on the triangle formed by the ladder, the wall, and the ground. In this triangle:
- The length of the ladder is the hypotenuse, which is 35 feet.
- The angle between the ladder and the wall is given as 20 degrees.
Using trigonometry, we can apply the sine function:
sin(angle) = opposite/hypotenuse
sin(20°) = h/35
Now, to determine the value of h, we can rearrange the equation:
h = sin(20°) * 35
Using a calculator, we find:
h ≈ 11.98 feet (rounded to two decimal places)
So, the top of the ladder is approximately 11.98 feet from the ground.
Next, let's focus on the triangle formed by the ladder, the wall, and the base of the building. In this triangle:
- The length of the ladder is still the hypotenuse, which is 35 feet.
- The same 20-degree angle is between the ladder and the wall.
- We are looking for the distance from the base of the building to the foot of the ladder, which is labeled as "d".
Using trigonometry again, but this time applying the cosine function:
cos(angle) = adjacent/hypotenuse
cos(20°) = d/35
Now, let's solve for d:
d = cos(20°) * 35
Using a calculator, we find:
d ≈ 32.92 feet (rounded to two decimal places)
Therefore, the foot of the ladder is approximately 32.92 feet from the base of the building.