a ladder is leaning up against a wall. The ladder is 8 feet long and is 3 feet away from the base of the wall. find the angle the ladder makes with the wall.
The ladder, ground, and wall form a rt. triangle:
r = 8Ft = hyp.
X = 3Ft = hor. side.
Y = Wall = ver. side.
cosA = X/r = 3/8 = 0.375.
A = 68 Deg.
B = 90 - 68 = 22 Deg. = Angle ladder makes with wall.
To find the angle the ladder makes with the wall, we can use trigonometry. Specifically, we can use the tangent function, which relates the angle to the sides of a right triangle.
Let's label the angle as θ (theta), the distance from the base of the wall to the top of the ladder as h, and the distance from the base of the wall to the ladder as d.
Given that the ladder is 8 feet long and 3 feet away from the base of the wall, we have:
The hypotenuse (ladder length) = 8 ft
Adjacent side (distance from base to ladder) = 3 ft
Using the tangent function:
tan(θ) = opposite / adjacent
tan(θ) = h / d
Since we are looking for the angle θ, we need to solve for it. Rearranging the formula, we have:
θ = arctan(h / d)
However, we don't have the values of h and d directly. To find these values, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Applying the Pythagorean theorem, we have:
h^2 + d^2 = 8^2
Substituting the known values, this becomes:
h^2 + 3^2 = 8^2
h^2 + 9 = 64
h^2 = 55
Taking the square root of both sides, we find:
h = √55
Now that we know h, we can substitute it back into the equation for θ:
θ = arctan(√55 / 3)
Using a calculator to evaluate this expression, we find that θ is approximately 74.74 degrees.
Therefore, the ladder makes an angle of approximately 74.74 degrees with the wall.