a ladder 24 feet long leans against the side of a building, and the angle between the ladder and the building is 25°.
a) approximate the distance from the bottom of the ladder to the building.
b)if the distance from the bottom of the ladder to the buliding is increased by 2feet, approx how far does the top of the ladder move down the building?
Look at your trig functions
d/24 = cos 25°
height H/24 = sin 25°
for the 2nd part,
(d+2)^2 + h^2 = 24^2
ladder slid down H-h feet
21.75
To solve this problem, we can use trigonometric functions. Let's denote the height of the building as "h" and the distance from the bottom of the ladder to the building as "d."
a) To find the distance from the bottom of the ladder to the building, we can use the sin function. The sin of an angle is equal to the opposite side divided by the hypotenuse. In this case, the opposite side is the height of the building, and the hypotenuse is the length of the ladder.
sin(25°) = h / 24
Rearranging the equation, we get:
h = sin(25°) * 24
Using a calculator, we find:
h ≈ 9.96 feet
Therefore, the distance from the bottom of the ladder to the building is approximately 9.96 feet.
b) If we increase the distance from the bottom of the ladder to the building by 2 feet, the new distance would be d + 2.
To determine how far the top of the ladder moves down the building, we need to find the change in height, which we'll call "Δh."
Using the same trigonometric expression as before:
sin(25°) = Δh / 24
Rearranging the equation, we get:
Δh = sin(25°) * 24
Using a calculator, we find:
Δh ≈ 9.96 feet
Therefore, if the distance from the bottom of the ladder to the building is increased by 2 feet, the top of the ladder would move down the building by approximately 9.96 feet.
To approximate the distance from the bottom of the ladder to the building, we can use trigonometry. In this case, since we know the length of the ladder and the angle between the ladder and the building, we can use the sine function. Here's how:
a) To find the distance from the bottom of the ladder to the building:
1. Use the sine function: sine(angle) = opposite/hypotenuse
In this case, the opposite side is the distance from the bottom of the ladder to the building, and the hypotenuse is the length of the ladder.
2. Plug in the values: sine(25°) = distance/hypotenuse
Since the length of the ladder is 24 feet, we have: sine(25°) = distance/24
3. Rearrange the equation to solve for the distance: distance = sine(25°) * 24
4. Calculate the value: distance ≈ 10.24 feet (rounded to two decimal places)
Therefore, the approximate distance from the bottom of the ladder to the building is 10.24 feet.
b) To approximate how far the top of the ladder moves down the building when the distance from the bottom of the ladder to the building is increased by 2 feet:
1. We can consider the original distance from the bottom of the ladder to the building as the opposite side of a right-angled triangle, and the top of the ladder as the adjacent side.
2. Now, when the distance is increased by 2 feet, the new distance becomes 10.24 + 2 = 12.24 feet.
3. Using the tangent function: tangent(angle) = opposite/adjacent
4. Plug in the values: tangent(25°) = 10.24/adjacent
5. Rearrange the equation to solve for the adjacent: adjacent = 10.24 / tangent(25°)
6. Calculate the value: adjacent ≈ 22.18 feet (rounded to two decimal places)
Therefore, when the distance from the bottom of the ladder to the building is increased by 2 feet, the top of the ladder moves down approximately 22.18 feet.