2) A roofer props a ladder against a wall so that the base of the ladder is 4 feet away from the building. If the angle of elevation from the bottom of the ladder to the roof is 63°, how long is the ladder?
We have a right triangle formed by the ladder against the wall. The base of the triangle is 4 feet, while the angle of elevation from the bottom of the ladder to the roof is 63°. We'll call the side opposite to this angle as the height of the triangle h and the ladder as the hypotenuse L. We want to find L.
To solve this problem, we can use the sine function, which relates the angle, the opposite side, and the hypotenuse in a right triangle:
sin(angle) = opposite/hypotenuse
Let's plug in the values we know:
sin(63°) = h/L
We want to find L, so we need to get it alone on one side of the equation. To do this, we can multiply both sides by L:
L * sin(63°) = h
Now we need to find the height of the triangle h. We can use the tangent function, which relates the angle, the opposite side, and the adjacent side in a right triangle:
tan(angle) = opposite/adjacent
Plugging the values:
tan(63°) = h/4
To solve for h, we multiply both sides by 4:
4 * tan(63°) = h
Now that we have an expression for h, we can plug it back into the sine equation:
L * sin(63°) = 4 * tan(63°)
To solve for L, divide both sides by sin(63°):
L = (4 * tan(63°))/sin(63°)
Using a calculator to find the values:
L ≈ (4 * 1.9626)/0.8912
L ≈ 8.7934
Therefore, the ladder is approximately 8.79 feet long.
To find the length of the ladder, we can use trigonometry and apply the tangent function. The tangent of an angle is equal to the ratio of the length of the opposite side to the length of the adjacent side.
In this scenario, the adjacent side is the distance between the base of the ladder and the wall (4 feet) and the opposite side is the height of the building. Let's call the height of the building "h."
We can set up the equation as follows:
tan(63°) = h / 4
Now, we need to solve for "h." To do that, we can rearrange the equation:
h = 4 * tan(63°)
Calculate the value of tangent (tan) for the angle 63° using a calculator or trigonometric table. Then, multiply the result by 4 to find the height of the building (h).
Finally, the length of the ladder is equal to the hypotenuse of the right-angled triangle formed by the height of the building and the distance between the base of the ladder and the wall. Applying the Pythagorean theorem, we have:
length of the ladder = sqrt((4^2) + h^2)
Now, substitute the value of "h" that we just found into this equation and solve to get the final answer for the length of the ladder.
To find the length of the ladder, we can use trigonometric functions.
Let's denote the length of the ladder as "L".
The given angle of elevation is 63°, and the base of the ladder is 4 feet away from the building.
We can use the sine function:
sin(θ) = opposite/hypotenuse
In this case, the hypotenuse is the length of the ladder (L), and the opposite side is the height of the building.
Using the given angle of elevation:
sin(63°) = height of building/L
Now, we need to determine the height of the building.
Using the right triangle formed by the ladder, the base distance of 4 feet, and the height of the building, we can use the sine function again:
sin(90°) = height of building/4
Since sine of 90° is 1, we have:
1 = height of building/4
Solving for the height of the building:
height of building = 4 feet
Substituting this into our initial equation:
sin(63°) = 4/L
To find the value of L, we can rearrange the equation:
L = 4/sin(63°)
Using a calculator or trigonometric table to evaluate sin(63°):
L ≈ 4/0.891 = 4.49 feet
Therefore, the length of the ladder is approximately 4.49 feet.