A swimming pool is 12 meters long, 6 meters wide, 1 meter deep at the shallow end, and 3 meters deep at the deep end. Water is being pumped into the pool at ¼ cubic meter per minute, and there is 1 meter of water at the deep end.
a. What percent of the pool is filled?
b. At what rate is the water level rising?
now the volume is another story because now we will have to make some assumption, for example constant slope up from 3 m to 1 m. In that case we have a trapezoid and
V = 6 * 12 * average depth
= 6 * 12 * 2 - 144 m^3
To solve these problems, we need to find the total volume of the pool and the rate of change of the water level.
a. To find the percent of the pool filled, we need to find the volume of the pool when it is completely filled and compare it to the current level of water in the pool.
The shape of the pool can be considered as a trapezoidal prism, where the shallow end is the smaller base, the deep end is the larger base, and the distance between them is the height of the prism.
1. First, let's calculate the volume of the pool when it is completely filled.
- Volume of a trapezoidal prism = (1/2) * (b1 + b2) * h * L
- Where b1 is the length of the shallow end, b2 is the length of the deep end, h is the height of the prism, and L is the length of the pool.
- Plugging in the values: b1 = 6m, b2 = 12m, h = 3m, L = 6m
- Volume of the completely filled pool = (1/2) * (6 + 12) * 3 * 6 = 108 cubic meters
2. Now, let's find the current volume of water in the pool.
- The shape of the current volume of water can be considered as a rectangular prism, with the same length and width as the pool, and the depth changes from 1m to 3m.
- Volume of a rectangular prism = length * width * depth
- Plugging in the values: length = 12m, width = 6m, depth = 3m - 1m = 2m
- Volume of the current water level = 12 * 6 * 2 = 144 cubic meters
3. Finally, let's calculate the percentage of the pool filled.
- Percentage = (Volume of current water level / Volume of completely filled pool) * 100
- Percentage = (144 / 108) * 100 = 133.33%
Therefore, the pool is filled approximately 133.33%.
b. To find the rate at which the water level is rising, we need to calculate the rate at which the volume of water in the pool is increasing.
1. The water is being pumped into the pool at 1/4 cubic meter per minute.
2. The rate at which the water level is rising is the same as the rate at which the volume of water in the pool is increasing.
Therefore, the rate at which the water level is rising is 1/4 cubic meter per minute.
a. To find the percentage of the pool that is filled, we need to compare the volume of the water in the pool to the total volume of the pool.
First, we need to calculate the average depth of the pool. The pool has a shallow end depth of 1 meter and a deep end depth of 3 meters. To find the average, we add the shallow and deep end depths and divide by 2:
(1 + 3) / 2 = 2
So, the average depth of the pool is 2 meters.
Next, we calculate the volume of the pool using the formula for the volume of a rectangular prism:
Volume = length * width * depth
Volume = 12 meters * 6 meters * 2 meters
Volume = 144 cubic meters
Now, to calculate the volume of water in the pool, we need to determine how long it takes to fill the pool. The water is being pumped into the pool at a rate of ¼ cubic meter per minute. Therefore, to fill the pool with 144 cubic meters of water, we divide the total volume of water by the rate of pumping:
Time = Volume / Rate
Time = 144 cubic meters / 1/4 cubic meters per minute
Time = 144 cubic meters * 4 minutes/cubic meter
Time = 576 minutes
So, it would take 576 minutes to fill the pool completely.
To find the volume of water that has been pumped into the pool in a certain amount of time, we multiply the pumping rate by the time:
Volume of water pumped = Rate * Time
Volume of water pumped = 1/4 cubic meters per minute * Time
Now, we can calculate the percentage of the pool that is filled by dividing the volume of water pumped into the pool by the volume of the pool and multiplying by 100:
Percentage filled = (Volume of water pumped / Volume of pool) * 100
Percentage filled = (Volume of water pumped / 144 cubic meters) * 100
b. The rate at which the water level is rising can be determined by calculating the change in water level over time.
Since the pool is rectangular and the water level is rising uniformly, we only need to consider the change in the length of the pool that is submerged.
The maximum depth of the pool is 3 meters, and at the current depth, 1 meter, the pool has 1 meter of water. So the water level is rising by 2 meters.
The time it takes for the water level to rise by 2 meters can be determined by dividing the change in water level by the rate of pumping:
Time = Change in level / Rate
Time = 2 meters / 1/4 cubic meters per minute
Time = 8 minutes
Therefore, the water level is rising at a rate of 2 meters every 8 minutes.
Part b is easy and requires no assumptions about the shape of the pool bottom.
The rate of change of height = volume added per second / surface area of water.
(draw a picture)
or
dh/dt = .25 m^3/min / (12m*6m)
= .00347 m/min or 21 cm/hour