Math Problem--> anybody?

Question

A dog is chained to a point of the side of a circular lighthouse of radius 1. The length of the dog's chain is pi. The dog may not go inside the lighthouse. How much area can the dog cover?

Answer 1

the dog's chain is pi and his area is pi x pi^2 minus the area of the light house: pi^3 - pi(1^2) =27.82

(The chain wraps around the lighthouse on each side to form a semi-circumference.)

Answer 2

Area of a Circle = pi*r^2

Answer 3

The answer is not as simple as the other answerers think.  The dog can wrap around the lighthouse to get to the exact opposite side.  That's a distance of 2 from the chain point.  Going the other direction he can get a distance of pi (full length of the chain).

The shape is more akin to an egg shape.  The unobstructed half is a semi-circle and the other half (around the lighthouse) is more like a cycloid shape.  The attached picture should help.  From that you would have to subtract the area of the lighthouse (pi sq. ft.)

This is going to require more than just geometry.

Let x be the angle formed by (1,0), the origin and the point where dog's chain leaves the edge of the lighthouse.  We assume the dog is extending its chain as far as possible causing the straight part of the chain to be tangent to the edge of the lighthouse.

Let's figure out the area to the right of x = -1 and above y = 0.

Since the radius of the lighthouse is 1 the length of the straight part of the chain is the same as x. Through a little trigonometry we can get the coordinates of the dog given x as (cos(x)+x*sin(x), sin(x)-x*cos(x)). The distance from (0,0) to the dog, given x, conveniently works out to √(1+x²), using the length of a line segment formula.

Let y be the angle formed by (1,0), the origin and the location of the dog. Through some more trigonometry we can solve for y in terms of x as y = x - arctan(x). Now we are ready to integrate.

The area in terms of polar coordinates is one half the integral over the range of the angle of the radius squared. We cannot simply take the integral from 0 to pi of 1+x² because this is not the radius formed by the angle x, but the radius formed by y. We could take the integral over the range of the angle y but finding the coordinates of the dog in terms of y is very hard. However we can take the integal as x goes from 0 to pi of 1+x² multiplied by the change in y given a change in x. Since y=x - arctan(x). dy/dx = 1-1/(1+x²)!

So now the area becomes the integral as x goes from 0 to pi of 1/2 the product of 1+x² and 1-1/(1+x²) which equals 1/2 the integral from 0 to pi of x² which equals pi^3/6.

This integral, however, does not cover the area bounded by the triangle (0,0), (-1,0) and (-1,pi). So we must add this area which is pi/2. Yet we must subtract the area inside the lighthouse which the dog may not go in, which is also pi/2. So these two modifications conventiently cancel each other out.

The area of the quarter circle to the left of x=-1 and above y=0 is pi^3/4, since the length of the chain is pi and the dog covers a semicircle to the left of the lighthouse. So the total area above y=0 is 5/12 * pi^3. Finally double this for the area under y=0 and the total area is 5/6 * pi^3

Answer 4

A= pi(radius) squared

so if the radius is one

you square that and multiply it by pi

so the area would be pi

that's how far he can go.  

Answer 5

very nice problem.

here goes

i'm assuming the dog is at the 'bottom' of the lighthouse.

the dog can just reach the other end (top) of the lighthouse (since the chain is pi, which is the circular length of half the lighthouse). This is  2 units in vertical displacement

he can also move left and right, a distance of pi. So now the area towards the lighthouse (above) looks like a parabola, with height 2 and breadth 2*pi. the area away from the light house (below)is accessible to him fully, i.e. a semi circle of radius pi.

So now the area he can cover is the area of the parabola + area of the semi cirle - area of the lighthouse

= 2/3 * breadth * height + (pi*r^2)/2 - pi*r^2

= (2/3 * 2*pi * 2) + (pi^3)/2 - pi

= 20.7391261

am i right?

Answer 6

the answer is 6.28

Answer 7

The dog can travel in a circle with radius of pi, th't go in the lighthousat circle will have an area of pi*pi^2 or pi^3

The lighthouse area is pi*1^2 or pi

The dog can't go in the lighthouse so subtract that

pi^3-pi

this is apprpox 37.86