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Question 1159719: A water storage tank has the shape of a cylinder with diameter 22 ft. It is mounted so that the circular cross-sections are vertical. If the depth of the water is 15 ft, what percentage of the total capacity is being used? (Round your answer to one decimal place.)
Found 2 solutions by greenestamps, jim_thompson5910:
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
72.6%.
The answer is of little use to you, unless you just want to answer the question without learning anything.
So I will tell you one way you can get that answer and let you have the experience of working the problem yourself.
Draw a sketch of the cross section of the tank. It will be a circle with radius 11 with a horizontal chord 4 units above the middle of the circle. (The depth of the water is 15ft; the radius of the tank is 11ft; the chord representing the water level is 4ft above the center).
Draw the radii to the two endpoints of the chord.
A radius of the tank to one end of the chord forms a right triangle, allowing you to determine the length of the chord. (In that triangle, one leg is half of the chord; the other leg is 4ft and the hypotenuse is 11ft.)
The area of the cross section of the tank that is water can be viewed as composed of two parts: (a) the area of the triangle formed by the chord and the two radii to the ends of the chord; and (b) the sector of the circle "below" those two radii.
At this point you know the measures of the legs of the triangle, so you can easily determine its area.
For the sector of the circle, observe that the angle above the midline of the cross section is  . So the angle of the sector of the circle containing water is 180 degrees plus twice that angle.
The next-to-last step is to add the areas of the triangle and the sector of the circle; the last step is to convert that to a percentage by dividing that area by the area of the whole circle.
Answer by jim_thompson5910(35256)  (Show Source):
You can put this solution on YOUR website!
Here's how the diagram would look like if you were to take a 2D cross section of the cylinder
Point A = center of the circle
Points B and C form the horizontal waterline, BC is a chord of the circle
D is the midpoint of segment BC
Point E is directly below point A. We know that the radius is 11 ft (half the diameter of 22 ft), so AE = 11. The depth of the water is 15 ft, so 15-11 = 4 ft must be the length of segment AD so we have the proper water height.
Triangle ADB and triangle ADC are right triangles. This allows you to use the cosine trig ratio to determine angle CAD and DAB, which combine to angle BAC.
The cosine trig ratio is
cos(angle) = adjacent/hypotenuse
For a triangle like ADC, we have AD = 4 as the adjacent side and AC = 11 as the hypotenuse. Use the inverse cosine or arccosine function to determine angle DAC, which will lead to angle BAC.
Subtract this from 360 to find the reflex angle of angle BAC. For instance, if BAC was 120 degrees (it may or may not be), then the reflex angle of BAC is 360-120 = 240 degrees.
Once you know angle BAC and its reflex angle counterpart, you can determine the area of the green triangle and the blue circular sector region. The formulas to use are
Area of triangle = (1/2)*(side1)*(side2)*sin(included angle)
Area of sector = (angle/360)*(area of circle) = (angle/360)*(pi*r^2)
Keep in mind that the sector will use the reflex angle, while the triangle uses angle BAC itself.
After finding the combined area of the green and blue regions, you would divide this over the total circular area (formula is pi*r^2) to find the percentage that is taken up by water.
Note: The volume of a cylinder is found by multiplying the area of the base by the depth. The same can be applied to finding the volume of the water in the tank. You find the area of the green and blue regions combined, then multiply that with the depth of the tank to get the total volume of water. You'll find cancellations will happen when you divide the water volume over the entire tank volume. So in summary, you only need to worry about the 2D representation as explained above.
Hopefully this explanation makes sense. If not, then let me know and I'll try another approach to explain. Thank you.
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