A 20-ft-by-30-ft swimming pool is filled with water. The depth is measured at 5-ft intervals, starting at one corner of the pool, and the values are recorded in the table. Estimate the volume of water using the Midpoint Rule with m = 2 and n = 3.
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A 20-ft-by-30-ft swimming pool is filled with water. The depth is measured at 5-ft intervals, starting
at one corner of the pool, and the values are recorded in the table. Estimate the volume of water in the
pool.
Well, it looks like someone is really diving into the depths of this pool question! Let's see if we can swim through the calculations together.
Using the Midpoint Rule, we estimate the volume of water by dividing the pool into smaller rectangular sections and approximating the height in each section.
First, let's find the width and length of each rectangular section. With the given measurements, the width between each interval is 20 ft divided by 3, and the length between each interval is 30 ft divided by 2. This gives us a width of 6.67 ft and a length of 15 ft.
Next, we can find the volume of each section by multiplying the width, length, and midpoint height. The midpoint height of each section is the average of the heights at both ends of the section.
Let's calculate the volume of each section:
Section 1: (6.67 ft) * (15 ft) * ((0 ft + 0 ft) / 2) = 0 ft^3
Section 2: (6.67 ft) * (15 ft) * ((0 ft + 5 ft) / 2) = 250.8333 ft^3
Section 3: (6.67 ft) * (15 ft) * ((5 ft + 5 ft) / 2) = 500 ft^3
Section 4: (6.67 ft) * (15 ft) * ((5 ft + 10 ft) / 2) = 750 ft^3
Section 5: (6.67 ft) * (15 ft) * ((10 ft + 10 ft) / 2) = 1000 ft^3
Now, let's add up the volumes of all the sections:
Total volume ≈ 0 ft^3 + 250.8333 ft^3 + 500 ft^3 + 750 ft^3 + 1000 ft^3 = 2500.8333 ft^3
So, using the Midpoint Rule with m = 2 and n = 3, we estimate the volume of water in the pool to be approximately 2500.8333 cubic feet.
I hope that calculation didn't make too big of a splash!
To estimate the volume of water in the swimming pool using the Midpoint Rule, we first need to understand the concept behind the rule.
The Midpoint Rule is a method of approximating the area under a curve by dividing it into smaller rectangles. The width of each rectangle is determined by the number of subintervals chosen, and the height is determined by the function values at the midpoint of each subinterval. By summing up the areas of these rectangles, we can get an estimate of the total area.
In this case, we have a 20-ft-by-30-ft swimming pool. The depth is measured at 5-ft intervals, starting at one corner of the pool. We are given the following depth values:
Depth: 3ft 4ft 6ft
6ft 5ft 4ft
We need to estimate the volume of water using the Midpoint Rule with m = 2 and n = 3.
Step 1: Determine the width and height of each rectangle.
The width of each rectangle can be calculated by dividing the length of the pool by the number of subintervals. In this case, the length is 30 ft, and we have 3 subintervals, so the width of each rectangle is 30 ft / 3 = 10 ft.
The height of each rectangle is the difference between two adjacent depth values. For example, the height of the first rectangle is 4 ft - 3 ft = 1 ft.
Step 2: Calculate the area of each rectangle.
To calculate the area of each rectangle, multiply the width by the height. For example, the area of the first rectangle is 10 ft * 1 ft = 10 ft^2.
Step 3: Sum up the areas of all rectangles.
Add up the areas of all the rectangles to get the total estimated volume of water in the pool. In this case, we have 6 rectangles, so we need to sum up the areas of all 6 rectangles.
10 ft^2 + 10 ft^2 + 1 ft^2 + 10 ft^2 + 9 ft^2 + 16 ft^2 = 56 ft^2.
Therefore, the estimated volume of water using the Midpoint Rule with m = 2 and n = 3 is 56 ft^2.
Divide the surface of the pool into 4x6 = 24 rectangular columns.
Consider the depth of each column to be the value at the center of the x- and y-intervals.
So, picking a column where all four corners are at different depths, to illustrate finding the depth at the center of the rectangular column,
Pick the x-interval (10,15) and the y-interval (10,15). Now, we do not know what the depth is at (12.5,12.5), and the four corners on the bottom do not necessarily all lie in the same plane, but we can use the average of the depths at the centers of the diagonals, which would be
((6+5)/2 + (8+4)/2)/2 = (13/2 + 12/2)/2 = 25/4
So, calculate the depth at the center of each rectangle and add them up, and then multiply the result by 25, the area of each rectangle.