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Assignment 2 Pond area

Extracts from this document...

Introduction

AS Use of Maths

Working with Calculus

Assignment 2: Pond Area

Introduction:

As part of the ‘Mathematical Garden’ a lake has been built, whose edges can be modelled by  y =10COS(0.08x)+15  and y =10SIN(0.08x)+25 between x = 0 and x = 60, which gives the picture below:

Finding the areaimage00.png

By numerical methods

To find an area between 2 lines, you find the area under one line and then subtract the area under the other.

As there are 2 methods of finding the approximate

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Middle

The formula

image03.png Area =½ width(h1+2h2+ .. +2hn-1+ hn)

= ½ x 20 (25 + 2 x 35 + 2 x25 + 15)

= ½ x 20 x 160 = 1600m2

As this curve is mostly concave, the area will be an underestimate, but it would have been more accurate if it had been divided up into more strips

But by the calculator the integral is 1614m2, so the error% is (1614-1600)/1614 x 100 =0.87% which is very small and in part due to the last 3rd of the curve being convex

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Conclusion

So the area of Trig Pond is 1600 – 758.6 = 841.4m2

Find the area of the pond by integration

Area = image06.png

= image07.png

=[(-125cos(4.8) + 1500) – (-125cos(0) ] – [(125sin(4.8) + 900)- (125sin(0))]

=(1489.06 + 125) – (775.48 – 0) = 1614.06 – 775.48 = 838.58 m2

Conclusion

Both numerical method gave a fairly close result to the calculus one, even though the number of divisions was not very large.

LMcG: Braintree College                807616.doc

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