Hi guys can you help me! Please teach me step by step.. I really need it. pls! Thank you.........
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?
Nope not a thing
The cross-section is a trapezoid of area
(1+3)/2 * 6 = 12 m^2
So, the volume is 12 * 12 = 144 m^3
with only 1 m of water, the cross-section of the water is a triangle of area 3/2 m^2, so its volume of water is 18 m^3
Thus the pool is 18/144 = 1/8 full
At depth y of water, the surface has width y/2, so
v = (1/2)(y)(y/2)(12) = 3y^2
dv/dt = 6y dy/dt
At y=1, then
1/4 = 6 dy/dt
and the height is rising at 1/24 m/s
As usual, check my arithmetic.
Of course, I can help you with that! Let's break down the problem step by step:
a.) To find out what percent of the pool is filled, we need to find the volume of water in the pool and divide it by the total volume of the pool.
Step 1: Calculate the volume of the pool:
Since the pool is trapezoidal in shape, we can use the formula for the volume of a trapezoidal prism:
V = (A1 + A2) * h * w
where V is the volume, A1 and A2 are the areas of the two bases, h is the height, and w is the width.
A1 = (1/2) * (a + b) * h1
where a and b are the lengths of the parallel bases and h1 is the height of the shallow end.
A2 = (1/2) * (a + b) * h2
where h2 is the height of the deep end.
In this case:
a = 6 meters (width of the pool)
b = 12 meters (length of the pool)
h1 = 1 meter (height of the shallow end)
h2 = 3 meters (height of the deep end)
Step 2: Calculate the area of the bases:
A1 = (1/2) * (6 + 12) * 1 = 9 square meters
A2 = (1/2) * (6 + 12) * 3 = 27 square meters
Step 3: Calculate the volume of the pool:
V = (9 + 27) * w = 36w cubic meters
b.) To determine at what rate the water level is rising, we need to calculate the derivative of the volume of water in the pool with respect to time.
Step 4: Calculate the derivative of the volume with respect to time:
dV/dt = w * dw/dt
Since the water is being pumped into the pool at a rate of 1/4 cubic meter per minute, we can substitute this value into the equation:
dV/dt = w * (1/4) = w/4 cubic meters per minute
Now, let's calculate the percentage of the pool that is filled and the rate at which the water level is rising:
a.) Percentage of the pool filled:
To find the percentage, divide the volume of water in the pool by the total volume of the pool, and multiply by 100:
Percentage = (Volume of water / Total volume) * 100
= (w / 36w) * 100
= 1/36 * 100
= 2.78% (rounded to two decimal places)
Therefore, approximately 2.78% of the pool is filled.
b.) Rate at which the water level is rising:
The rate at which the water level is rising is given by:
Rate = dV/dt = w/4 cubic meters per minute
So, the water level is rising at a rate of w/4 cubic meters per minute.
Note: To find the exact value of the rate of the water level rising, we would need to know the value of w.
Of course! I'll be glad to help you step by step with your questions.
a.) To find the percent of the pool that is filled, we need to determine the volume of water in the pool and compare it to the total capacity of the pool.
Step 1: Calculate the average depth of the pool.
The average depth is the sum of the shallow end depth and the deep end depth, divided by 2.
Average Depth = (1 meter + 3 meters) / 2 = 4 meters / 2 = 2 meters
Step 2: Calculate the cross-sectional area of the pool at the average depth.
The cross-sectional area of the pool is the product of the length and width.
Cross-sectional Area = 12 meters x 6 meters = 72 square meters
Step 3: Calculate the volume of water in the pool.
The volume of water is the cross-sectional area multiplied by the average depth.
Volume of Water = Cross-sectional Area x Average Depth = 72 square meters x 2 meters = 144 cubic meters
Step 4: Calculate the total capacity of the pool.
The total capacity is the product of the cross-sectional area and the maximum depth (deep end depth).
Total Capacity = Cross-sectional Area x Deep End Depth = 72 square meters x 3 meters = 216 cubic meters
Step 5: Calculate the percent of the pool that is filled.
Percent Filled = (Volume of Water / Total Capacity) x 100
Percent Filled = (144 cubic meters / 216 cubic meters) x 100 = 0.6667 x 100 ≈ 66.67%
So, approximately 66.67% of the pool is filled.
b.) To determine the rate at which the water level is rising, we need to calculate the change in volume over time.
Step 1: Determine the change in volume per minute.
The water is being pumped into the pool at a rate of 1/4 cubic meter per minute.
Step 2: Calculate the rate at which the water level is rising.
The rate is simply the change in volume per minute, which is 1/4 cubic meter.
Therefore, the water level is rising at a rate of 1/4 cubic meter per minute.
I hope this helps! Let me know if you have any further questions.