if a ladder 15 meters long is placed so as to reach the bottom of a window 10 meters high, what distance on the ground does the ladder adjusted if it is placed on the top of a two meter square window and what is the new angle of elevation of the top of the window with respect on the ground?
X = ?
Y = 10+2 = 12 m.
L = 15 m.
Sin A = Y/L = 12/15 = 0.80, A = 53.1o
X = L*Cos A = 15*Cos53.1 = 9 m.
To find the distance on the ground when the ladder is adjusted, we can use the Pythagorean theorem. Let's call the distance on the ground x.
According to the Pythagorean theorem:
(hypotenuse)^2 = (base)^2 + (height)^2
In this case, the hypotenuse is the ladder length, which is 15 meters, and the height is 10 meters. The base is the distance on the ground, which is x. So we have:
15^2 = x^2 + 10^2
Simplifying the equation gives:
225 = x^2 + 100
Rearranging the equation, we have:
x^2 = 225 - 100
x^2 = 125
To find the value of x, we take the square root of both sides:
x = sqrt(125)
x = 11.18 meters (rounded to two decimal places)
Therefore, when the ladder is adjusted and placed on the top of a two-meter square window, the distance on the ground is approximately 11.18 meters.
Now let's find the new angle of elevation of the top of the window with respect to the ground. We can use trigonometry to solve this. Let's call the new angle of elevation θ.
Using the tangent function:
tan(θ) = (height of the window) / (distance on the ground)
In this case, the height of the window is 2 meters, and the distance on the ground is 11.18 meters. Plugging in these values, we have:
tan(θ) = 2 / 11.18
To find the angle θ, we take the inverse tangent (or arctan) of both sides:
θ = arctan(2 / 11.18)
θ ≈ 10.10 degrees (rounded to two decimal places)
Therefore, the new angle of elevation of the top of the window with respect to the ground is approximately 10.10 degrees.