A 10 meter ladder is leaning against the side of a building. If the bottom of the ladder is 3 meters from the bottom of the building, find the measure of the acute angle that the ladder makes with the building.
well, you know that
sinθ = 3/10
17.5
To find the measure of the acute angle that the ladder makes with the building, we can use trigonometry.
Let's label the height of the building as 'h' and the angle that the ladder makes with the building as 'θ'.
Using the right triangle formed by the ladder, the building, and the ground, we can apply the trigonometric function tangent:
tan(θ) = opposite/adjacent
In this case, the opposite side is the height of the building 'h', and the adjacent side is the distance from the bottom of the ladder to the bottom of the building, which is 3 meters.
So, we have the equation:
tan(θ) = h/3
To solve for θ, we can take the inverse tangent (arctan) of both sides of the equation:
θ = arctan(h/3)
Now, we need to find the value of 'h'.
Using the Pythagorean theorem, we can express the relationship between the height of the building 'h', the distance from the bottom of the ladder to the building, and the length of the ladder:
h^2 + 3^2 = 10^2
Simplifying the equation:
h^2 + 9 = 100
h^2 = 100 - 9
h^2 = 91
Taking the square root of both sides, we can find the value of 'h':
h = √91
Now, we can substitute this value back into the equation for θ:
θ = arctan(√91/3)
Using a calculator to evaluate the arctan function, we find that:
θ ≈ 74.26 degrees
Therefore, the measure of the acute angle that the ladder makes with the building is approximately 74.26 degrees.
To find the measure of the acute angle that the ladder makes with the building, we can use trigonometry.
Let's consider the right triangle formed by the ladder, the building, and the ground. The ladder is the hypotenuse of the right triangle, while the distance from the bottom of the ladder to the bottom of the building is one of the legs.
Using the Pythagorean theorem, we can find the length of the other leg:
Leg^2 + Leg^2 = Hypotenuse^2
Let's call the length of the other leg "x". So, we have:
x^2 + 3^2 = 10^2
x^2 + 9 = 100
x^2 = 100 - 9
x^2 = 91
x = √91
Now that we know the lengths of the two legs of the right triangle, we can use trigonometry to find the measure of the acute angle. Specifically, we can use the inverse tangent function (tan^-1).
The tangent of an angle is defined as the ratio of the opposite side (the length of the ladder in this case) to the adjacent side (the length of the other leg, x). So, we have:
tan(angle) = opposite/adjacent
tan(angle) = 10/√91
To find the angle, we can take the inverse tangent (tan^-1) of both sides:
angle = tan^-1(10/√91)
Calculating this using a scientific calculator or an online calculator, we find that the measure of the acute angle is approximately 70.53 degrees.