Question

A 12 foot ladder leans against a building and reaches a window 10.5 feet above ground. What is the measure of the angle, to the nearest degree, that the ladder forms with the ground

Answer 1

Okay, let's solve this step-by-step:

* The ladder is 12 feet long

* The window is 10.5 feet above the ground

* So the ladder extends from the ground up to the window, a total vertical distance of 10.5 feet

* Using the Pythagorean theorem: a^2 + b^2 = c^2 (where c is the hypotenuse)

* In this case: 10.5^2 + x^2 = 12^2 (where x is the length of the ladder on the ground)

* Solving for x: x = sqrt(12^2 - 10.5^2) = sqrt(143) = 11.83

* Therefore, the ladder extends 11.83 feet along the ground

* To find the angle:

Angle = arcsin(10.5/12) = 65.97 degrees (approx. 66 degrees)

So the angle that the ladder forms with the ground is 66 degrees, rounded to the nearest degree.

Answer 2

Explanation:

We can use trigonometry to find the measure of the angle that the ladder forms with the ground.

Let θ be the angle that the ladder makes with the ground. Then, we can use the tangent function to relate the opposite side (the height of the window, 10.5 feet) to the adjacent side (the distance from the base of the ladder to the wall):

tan(θ) = opposite / adjacent

tan(θ) = 10.5 / x

where x is the distance from the base of the ladder to the wall.

We can solve for x by using the Pythagorean theorem, which relates the sides of a right triangle:

x^2 + 10.5^2 = 12^2

x^2 = 144 - 10.5^2

x^2 = 42.25

x = √42.25

x ≈ 6.5

Substituting this value into the equation for tangent, we get:

tan(θ) = 10.5 / 6.5

tan(θ) ≈ 1.6154

To find θ, we can take the inverse tangent (or arctangent) of both sides:

θ = tan^-1(1.6154)

θ ≈ 58.27°

Therefore, the measure of the angle that the ladder forms with the ground, to the nearest degree, is 58°.