A 3.5 m long ladder is resting against a wall making an angle of 70° to the ground.
How far is the foot of the ladder from the wall?
How far up the wall is the top of the ladder?
foot: x/3.5 = cos70°
top: y/3.5 = sin70°
To find the distance of the foot of the ladder from the wall, we can use the sine function.
1. Identify the given information:
- Length of the ladder, L = 3.5 m.
- Angle between the ladder and the ground, θ = 70°.
2. Apply the sine function:
- sin(θ) = opposite/hypotenuse.
- In this case, the opposite side is the distance of the foot of the ladder from the wall, and the hypotenuse is the length of the ladder.
3. Substitute the values:
- sin(70°) = opposite/3.5 m.
4. Solve for the distance of the foot of the ladder from the wall:
- Multiply both sides by 3.5 m to isolate the variable.
- opposite = sin(70°) * 3.5 m.
Therefore, the distance of the foot of the ladder from the wall is approximately sin(70°) * 3.5 m.
To find the distance up the wall to the top of the ladder, we can use the cosine function.
1. Apply the cosine function:
- cos(θ) = adjacent/hypotenuse.
- In this case, the adjacent side is the distance up the wall to the top of the ladder, and the hypotenuse is the length of the ladder.
2. Substitute the values:
- cos(70°) = adjacent/3.5 m.
3. Solve for the distance up the wall to the top of the ladder:
- Multiply both sides by 3.5 m to isolate the variable.
- adjacent = cos(70°) * 3.5 m.
Therefore, the distance up the wall to the top of the ladder is approximately cos(70°) * 3.5 m.
To find the distance between the foot of the ladder and the wall, you can use trigonometric functions. In this case, you will use the cosine function, as the adjacent side and the hypotenuse are known.
1. Identify the known values:
Length of the ladder (hypotenuse) = 3.5 m
Angle between the ladder and ground = 70°
2. Apply the cosine function:
cos(angle) = adjacent / hypotenuse
In this case, the adjacent side is the distance between the foot of the ladder and the wall, which we'll call "x". The hypotenuse is 3.5 m.
cos(70°) = x / 3.5
3. Solve the equation for x:
Multiply both sides of the equation by 3.5 to isolate x:
x = 3.5 * cos(70°)
Using a calculator, evaluate cos(70°) and then multiply it by 3.5. The result will give you the distance between the foot of the ladder and the wall.
To find the height up the wall of the top of the ladder, you can use the sine function. Since the opposite side and the hypotenuse are given, you can express the relationship as follows:
1. Identify the known values:
Length of the ladder (hypotenuse) = 3.5 m
Angle between the ladder and the ground = 70°
2. Apply the sine function:
sin(angle) = opposite / hypotenuse
In this case, the opposite side is the height of the top of the ladder up the wall. We'll call it "h". The hypotenuse is 3.5 m.
sin(70°) = h / 3.5
3. Solve the equation for h:
Multiply both sides of the equation by 3.5 to isolate h:
h = 3.5 * sin(70°)
Using a calculator, evaluate sin(70°) and then multiply it by 3.5. The result will give you the height up the wall of the top of the ladder.