A 6 m ladder is leaning against the wall of a house. The angle between the ladder and the ground is 68 degrees. How far away is the ladder from the wall.? How high up the house does the ladder reach ? Give your answers accurate to two decimal places.
check the related questions below
The distance the latter from the wall is 6*cos(68)=2.247~2.25m
The height the latter reaches to the house is 6*sin(68)=5.563~5.56m
Answer(1) 2.25m
Answer(2) 5.56m
To find the distance between the ladder and the wall, we can use some trigonometry.
The given angle between the ladder and the ground is 68 degrees. Let's call the distance between the ladder and the wall "x".
Using the trigonometric function cosine (cos), we can relate the angle and the distances as follows:
cos(angle) = adjacent side / hypotenuse
In this case, the adjacent side is the distance between the ladder and the wall (x), and the hypotenuse is the length of the ladder (6 m).
Therefore,
cos(68 degrees) = x / 6
To find the value of x, we can rearrange the equation:
x = 6 * cos(68 degrees)
Using a scientific calculator, we can find:
x ≈ 2.41 meters
So, the ladder is approximately 2.41 meters away from the wall.
To find how high up the ladder reaches on the house, we can use the trigonometric function sine (sin).
sin(angle) = opposite side / hypotenuse
In this case, the opposite side is the height up the ladder reaches on the house. We'll call it "h".
Therefore,
sin(68 degrees) = h / 6
To find the value of h, we can rearrange the equation:
h = 6 * sin(68 degrees)
Using a scientific calculator, we can find:
h ≈ 5.77 meters
So, the ladder reaches approximately 5.77 meters high up the house.
To find the distance the ladder is from the wall, you can use trigonometry. In this case, the ladder acts as the hypotenuse of a right triangle, with the ground and the wall as the other two sides.
First, let's label the sides of the triangle:
- The ladder is the hypotenuse and we'll call it 'h' (which is 6 meters in this case).
- The distance from the ladder to the wall is the adjacent side, so we'll label it 'x'.
- The height of the ladder on the wall is the opposite side, so we'll label it 'y'.
We are given the angle between the ladder and the ground, which is 68 degrees.
Now, we can use the trigonometric function cosine to find the adjacent side:
cos(68) = x/h
x = h * cos(68)
x = 6 * cos(68)
To find the height up the wall, we can use the trigonometric function sine:
sin(68) = y/h
y = h * sin(68)
y = 6 * sin(68)
Calculating these values using a calculator or a programming language, we find that:
x ≈ 2.55 meters (distance from the ladder to the wall)
y ≈ 5.67 meters (height up the wall)
Therefore, the ladder is approximately 2.55 meters away from the wall, and it reaches approximately 5.67 meters up the house.