What angle does a 3.8m ladder make with the ground if it reaches 2.1 m up the wall?
How far is the foot of the ladder from the wall?
sinθ = 2.1/3.8
distance from wall can be found using either
x = 3.8 cosθ
or
x^2 = 3.8^2 - 2.1^2
To find the angle that the ladder makes with the ground, we can use the trigonometric function called inverse tangent (also known as arctan).
The opposite side of the triangle is the height the ladder reaches up the wall, which is given as 2.1 m. The adjacent side of the triangle is the distance from the foot of the ladder to the wall. We'll denote this distance as x. And the hypotenuse of the triangle is the length of the ladder, which is given as 3.8 m.
Using the formula for tangent:
tan(angle) = opposite/adjacent
We can rearrange this formula to solve for the angle:
angle = arctan(opposite/adjacent)
Plugging in the given values:
angle = arctan(2.1/3.8)
angle ≈ 30.3 degrees
Therefore, the ladder makes an angle of approximately 30.3 degrees with the ground.
To find the distance from the foot of the ladder to the wall, we can use another trigonometric function called cosine.
Using the formula for cosine:
cos(angle) = adjacent/hypotenuse
We can rearrange this formula to solve for the adjacent side:
adjacent = hypotenuse * cos(angle)
Plugging in the given values:
adjacent = 3.8 * cos(30.3)
adjacent ≈ 3.8 * 0.866
adjacent ≈ 3.29 m
Therefore, the foot of the ladder is approximately 3.29 meters away from the wall.
To find the angle the ladder makes with the ground, we can use trigonometric functions. In this case, we can use the tangent function.
First, let's visualize the problem. We have a ladder leaning against a wall, making an angle with the ground. The ladder reaches 2.1 meters up the wall and has a length of 3.8 meters.
To find the angle, we use the tangent function: tangent(angle) = opposite/adjacent.
In this case, the opposite side is the height the ladder reaches up the wall, which is 2.1 meters. The adjacent side is the distance from the foot of the ladder to the wall, which we have to find.
Let's denote the angle as theta and the distance from the foot of the ladder to the wall as x. The equation becomes:
tangent(theta) = opposite/adjacent
tangent(theta) = 2.1/x
To solve for x, we rearrange the equation:
x = 2.1 / tangent(theta)
Now, we can plug the values into the equation and solve for x.
If you are using a scientific calculator, follow these steps:
1. Enter 2.1.
2. Press the "divide" button (/).
3. Enter the tangent of the angle (theta). Assuming you have the value of theta, enter it in degrees.
4. Press the "equals" button (=).
The result will be the distance from the foot of the ladder to the wall (x).
Now, to find the angle itself, you can use inverse tangent (also known as arc tangent or atan). However, this step is optional for this specific question as it only asks for the distance.
If you want to find the angle using a scientific calculator, follow these steps:
1. Enter the tangent value you calculated (2.1/x).
2. Press the inverse tangent button (usually labeled as "INV" or "2nd" followed by "tan" or "tan^(-1)").
3. Press the "equals" button.
The result will be the angle (theta) in degrees.
Remember to round your answers to the appropriate decimals if necessary.