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# The Fish Pond

Extracts from this document...

Introduction

The Fish Pond

In this activity I will be using the information given to me about this specific fish pond, to create and solve an equation that will represent the amount of fish I will raise and afterwards sell at the local market. In order to do this in an efficient and cost effective manner, I will need to determine the pond’s carrying capacity which can also be referred to as the maximum population that the pond is capable of sustaining in the long run.  I will do this by using the scaled map of the pond and the given information:

• An average depth of 1.5 meters
• One fish requires 37  to grow to its maximum size and does so in one year.
1. The first step in solving for the maximum population would be to estimate the surface area of  the pond by each of the following methods::
1. Use the sum of the areas of three different triangles:

By calculating the area of each of these triangles and then adding them together we can estimate the pond’s surface area.

Area of a triangle = ½ bh, where b is base and h is  height

Area of : Given that AC=1220 m it can be used as the base of the triangle

Height can be calculated by finding a line perpendicular to the base passing through a vertex of

the triangle, B.

AC → B = 3.3 cm

3.3cm100 = 330 m

Middle

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

1. 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

1. 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.

Conclusion

Logistic Model Graph

A

t

Equation Value Chart

 t= A= 3 14,085 6 116,796 9 229,042 12 244,856 15 245,998 18 246,075 21 246,080 24 246,081 27 246,081 30 246,081
1. 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.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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