A rope hangs from the ceiling. It reaches the floor exactly. It's 4 feet from the wall. When it's pulled to the wall, the end of the rope is 4 inches above the floor. How high is the ceiling?
L(1-cosT) = 1/3 foot
sin T = 4 feet/L so L sin T =4
1-cosT = 1/3L
L sinT = 4
cosT = (3L-1)/3L
so cos^2T = (9L^2 -6L+1)/9L^2
so 1-cos^2T = (6L-1)/9L^2
which is sin^2 T
but
sin^2T =16/L^2
so
16/L^2 = (6L-1)/9L^2
144 = 6L-1
6L = 145
L = 24.2 feet
check my arithmetic!
4" is 1/3 foot. So,
(h - 1/3)^2 + 4^2 = h^2
Damon is correct. 24'2"
To find the height of the ceiling, we can use the concept of similar triangles. Let's assume that the distance from the floor to the ceiling is represented by the variable "h" (in feet).
We can create two similar triangles: one formed by the rope, the wall, and the floor, and the other formed by the rope, the wall, and the ceiling.
In the first triangle, the distance from the wall to the rope is given as 4 feet, and the distance from the rope to the floor is also 4 feet.
In the second triangle, the distance from the wall to the rope is still 4 feet, and the distance from the rope to the ceiling is represented by "h" feet.
Since the two triangles are similar, their corresponding sides are proportional. Therefore, we can set up the following equation:
(4 feet / 4 feet) = (4 inches / h feet)
Simplifying this equation, we get:
1 = (4 inches / h feet)
To solve for "h," we can cross-multiply:
h feet = 4 inches / 1
Converting the units, we have:
h feet = 4/12 feet
Simplifying further, we get:
h feet = 1/3 feet
Therefore, the height of the ceiling is 1/3 feet, or 4 inches.