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# Assignment 2 Pond area

Extracts from this document...

Introduction

## 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 area

### 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

Middle

The formula

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

Conclusion

### Find the area of the pond by integration

Area =

=

=[(-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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