A trough is 11 feet long and has ends that are isosceles triangles that are 1 foot high and 2 feet wide. If the trough is being filled at a rate of 9 cubic feet per minute, how fast is the height of the water increaseing when the height is 7 inches?
The amount of paint needed to cover the walls of a room varies jointly as the perimeter of the
room and the height of the wall. If a room with a perimeter of 65 feet and 8-foot walls requires
5.2 quarts of paint, find the amount of paint needed to cover the walls of a room with a
perimeter of 80 feet and 10-foot walls
To find how fast the height of the water is increasing, we need to calculate the rate of change of the height with respect to time. This can be done using related rates.
Let's define the variables:
h = height of the water in inches
t = time in minutes
We are given that the trough is 11 feet long, which is equivalent to 11 * 12 = 132 inches. The ends of the trough are isosceles triangles with a height of 1 foot (12 inches) and a base of 2 feet (24 inches).
From the given information, we can determine the volume of water in the trough as a function of height:
Volume (V) = (base area of the trough) * (height of the water)
The base area of the trough can be calculated as the sum of the areas of the two isosceles triangles at the ends:
Base area = (1/2) * base * height = (1/2) * 24 * 12 = 144 square inches.
Therefore, the volume of water in the trough is given by:
V = 144h
We know that the rate of change of volume with respect to time is 9 cubic feet per minute. We need to convert this to cubic inches per minute:
9 cubic feet = 9 * 12 * 12 * 12 cubic inches.
The rate of change of volume with respect to time is given by:
dV/dt = 9 * 12 * 12 * 12 cubic inches per minute.
Since V = 144h, we can differentiate both sides of the equation with respect to time to find the rate of change of height with respect to time:
dV/dt = d(144h)/dt
9 * 12 * 12 * 12 = 144 * dh/dt
dh/dt = (9 * 12 * 12 * 12) / 144
dh/dt = 9 * 12 * 12 cubic inches per minute.
Now, we can substitute the given value of h (7 inches) to find the rate at which the height is increasing when h = 7 inches:
dh/dt = 9 * 12 * 12 = 1296 cubic inches per minute.
Therefore, the height of the water is increasing by 1296 cubic inches per minute when the height is 7 inches.