Estimated area of the pond = 422,300 m
- Divide the pond into rectangles and sum the areas.
By using the given information we can determine the lengths of the rectangles we use to divide the pond.
After measuring each rectangle for their width we can also determine that they are all have a width of 1 cm, or 100 m.
After finding the widths we can multiply by the lengths we obtain by measuring in centimeters.
1st rectangle = 100 • 750 m = 75,000 m
2nd rectangle = 100 m • 1070 m= 107,000 m
3rd rectangle = 100 m • 1030 m = 103,000 m
4th rectangle = 100 m • 940 m = 94,000 m
5th rectangle = 100 m • 1050 m = 105,000 m
6th rectangle = 100 m • 520 m = 52,000 m
7th rectangle = 50 m • 450 m = 22,500 m
When all areas are added, the estimated surface area of the pond = 558,500 m
- Use the area of trapezoids.
After measuring the lengths of the parallel sides and the heights of the all the trapezoids, the values are plugged into the equation to solve for their areas.
Area of a trapezoid = ½ (a + b) h where a and b are the parallel sides, h is the height
1st trapezoid = ½ (500 m + 1010 m) 100 = 75,500 m
2nd trapezoid = ½ (1010 m + 1070 m) 100 = 104,000 m
3rd trapezoid = ½ (1070 m + 1030 m) 100 = 105,000 m
4th trapezoid = ½ (1030 m + 1050 m) 100 = 104,000 m
5th trapezoid = ½ (1050 m + 1050 m) 100 = 105,000 m
6th trapezoid = ½ (1050 m + 500 m) 100 = 77,500 m
7th trapezoid = ½ (520 m + 200 m) 100 = 36,000 m
After getting the area of each trapezoid and adding the areas up, the estimated surface area of the pond = 607,000 m
- Which of the three above methods do you consider to be the most accurate? Why?
After looking at the results of the three different methods of finding the total surface area of the pond, the most accurate way seems to be the trapezoids. One reason would be that its shape fits best to fit into the overall shape of the pond. The areas that the trapezoid may go out of the pond, compensates for those areas that are not covered in the equation. Also the values used are mostly those that are given, and therefore can be more accurate since we know that they must be true.
- What could be done in order to increase the accuracy of the approximation in #2?
In order to increase the accuracy of the approximation that the trapezoid is the best way to solve for the area of the pond, would be to use more than one shape. For instance with the use of trapezoids, triangles and rectangles all at once, it may be easier to fit into the shape of the pond and more accurately find the area.
- a) Find the volume of water in the pond using your most accurate estimate.
Volume = SA •depth
It is given that the average depth is 1.5 meters.
V= 607,000 m • 1.5 m =
V = 910,500
b) What is the carrying capacity of your pond?
37 = space required for fish to grow to maximum size and does so in one year
Carrying capacity = V/ 37
Capacity = 910,500 / 37
Capacity = 246,081.0811 fish
-
Let p represent the initial population of the pond, m represent the carrying capacity, k=0.9/m represent the growth constant and t represent time in years.
- Given the pond is stocked with 1000 fish initially, and using the carrying capacity obtained in #2 the exponential function that models the fish population of the pond in terms of time would is as follows,
To set up this equation we use a logarithmic growth model equation:
m
1+ Be
246081.0811
1+ Be
To find B, we set t = 0, resulting in
A = m
1 + B
Since the pond is being stocked with 1,000 fish initially, we plug this in for A and then solve to find our B.
1000= 246081.0811/ 1+ B =
1000 + 1000B = 246081.0811
B= 245.0810811
The reason we particularly choose a logistic growth model in this instance is the fact that it is a population growth that takes into account limitations on food and the environment. The initial population growth resembles exponential growth, but then at some point, due perhaps to food or space limitations, the growth slows down and eventually levels off. When this happens the population approaches an equilibrium level.
-
After approximately how many years will it take for your pond to reach its carrying capacity?
a) 246,081.0811
1+ 245.0810811e
246081.0811
1 + 2.29337635…
246081.0811
1
A = 246081.0811
This proves that it will take approximately 30 years for the pond to reach its carrying capacity.
b) What would you do to halve the time in part (a)?
In order to halve the time in part (a) you would set up the equation so that t is divided by two, as seen in the following equation:
246081.0811
1 + 245.0810811e
246081.0811
1 + 245.0810811e
A = 245998.4266
In order to halve the time in part (a), you would have to initially start out with 245,998 fish. This would lessen the amount of time would take to reach the carrying capacity.
- Can you harvest 1000 fish each year for the first three years?
To figure this out you need to take the amount of fish that will be present after the first year and subtract 1,000 fishes.
A = 26081.0811 ÷ (1 + 245.0810811e ) t = 1
A = 2445 – 1000 = 1445
Therefore for the 2nd year the equation would be 2445 + 1445 = 3890
A =3890 – 1000 = 2890
3rd year would be 2445 +3890+2890 = 9225
A= 9225 – 1000 = 8225
Yes you would be able to harvest 1000 fish each year for the first three years.
d) Can you harvest 2000 fish each year for the first three years?
1st year A= 2445 – 2000 = 445
2nd year A = 2445+445 = 2890
2890 – 2000 = 890
3rd year A = 2445 + 2890 + 890 = 6225
62225 – 2000 = 4225
Yes although barely making it at the beginning, you would be able to harvest 2000 fishes each year the first three years.
Logistic Model Graph
A
t
Equation Value Chart
- At what time do you think the fish population will be growing the fastest?
I would have to say that the fish population would be growing the fastest before it reaches its carrying capacity at 246,081. One reason being that the growth rate of the fishes is increasing, as it is getting closer to the carrying capacity, and when it finally reaches it limited resources will inhibit further growth. As seen in the graph as time goes by the population continues to increase, until the limitations start to make an impact.
8. Some of the methods I will use to efficiently and effectively conduct business would be to uphold a certain minimum number of fish in the pond at all times. This way I will never be too low on fish and lose profit when there is not enough. I could do this by figuring out when I will have to restock to maintain a good amount of fish. Another way would be to make sure there is enough harvesting as well so that the fish pond does not meet its capacity too soon and overpopulate as well. Above all would be to pay attention to the patterns in the fish population and the factors that my cause it to change, like weather for example. If I want to continue to harvest fish and be successful, only through these processes will I be able to do so.
