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You work for Georgia Department of Natural Resources. They have asked you to estimate the surface area and volume of pond on Cumberland Island, Georgia.
You know that the surface area of a circle is:
You know that the volume of a sphere is:
You begin with surface area. You find the center of the pond and measure from the center point to outer points on the bank of the pond. The more of these you choose the more accurate your estimated radius. All measurements are in feet.
You choose 16 points to measure from the center to the bank. You measure the lengths in feet. Here is what you got.
Now you must take the average of all of these points and this is your estimate for r.
Therefore, your estamite for surface area of this pond is 3129.628 square feet.
Now, you look at volume. You can use a sonigram machine to measure the depth at any point on the pond. You realize you have to look at cross sections of the pond. Again, all measurements are in feet.
You use points on the bank that you already found lengths for when you found surface area.
Starting with the cross section of AB, you know that AP = BP = 25. You pick two points between A and P, points C and D. Starting with point C, you measure from P to C. Then you use the sonigram machine to see how deep it is from C to the bottom of the pond.
You know from here you can use the Pythagorean Theorem. You can do this also with PD, PE, and PF.
Now you know 4 lengths from the bottom of the pond to the center point. You can repeat this with a different cross section. You look at GH. You know that GP = PH =40. You choose 2 points on GP and PH.
Again, you use the Pythagorean Theorem.
You have 4 more lengths from the bottom of the pond to the center point. You repeat this with cross section MN. You know that MP = PN = 37.
You do one more cross section. You know that UP = PV = 27.
Now, to estimate the volume you have to take all 24 lengths you got from the cross section and average them. This will be your r. Then you can find an estimate of volume of a sphere. After this you must half it to get the volume of the pond.
It is important to note that the more points you take, the more accurate your estimates will be.
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