Question 169790: At a local shopping, a business was allowed to construct a building over a pond. concrete pillars were used to keep the building above the water line. The pond is the shape of a perfect circle and the building is a rectangle. The corners are labeled A,B,C, and D. The ratio from side AB to side BC is 4:3 and the perimeter is 280 units. The edge of the circular pond is 4 units beyond each corner of the building on the diagonal when viewed in an aerial photo. What is the are of the pond that is not covered by the building?
Answer by gonzo(654)   (Show Source):
You can put this solution on YOUR website! "The ratio from side AB to side BC is 4:3 and the perimeter is 280 units."
let L = length of rectangle
let W = width of rectangle
let P = perimeter.
since 2 times the length plus 2 times the width equals the perimeter, formula for perimeter becomes:
P = 2L + 2W
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i labeled ABCD starting form the top left corner and working around the rectangle clockwise.
A is top left
B is top right
C is bottom right
D is bottom left
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"The ratio from side AB to side BC is 4:3 and the perimeter is 280 units."
AB / BC = 4 / 3 (ratio from above statement)
multiply both sides of equation by BC:
AB = (4/3)*BC
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let AB be the length of the rectangle.
then:
L = AB
W = BC
L = (4/3)*W
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substituting (4/3)*W for L, equation for perimeter becomes:
P = 2L + 2W
P = 2*(4/3)*W + 2W
simplifying:
P = 2*4/3*W + 2W
P = 8/3*W + 2W
multiply both sides of equation by 3:
3P = 8W + 2*3W
3P = 8W + 6W
3P = 14W
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now P = 280 (given)
so:
3(280) = 14W
840 = 14W
W = 60
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if W = 60, then L = 80 because:
2*60 + 2*L = 280
120 + 2L = 280
subtract 120 from both sides of equation:
2L = 280 - 120 = 160
divide both sides of equation by 2:
L = 80
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we have:
W = 60
L = 80
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we need to solve for the diagonal of the rectangle.
formula for that is
L^2 + W^2 = D^2
where D is the diagonal of the rectangle (either one will do since this is a rectangle and the diagonals are equal in a rectangle).
L^2 + W^2 = D^2 becomes:
60^2 + 80^2 = D^2
3600 + 6400 = D^2
10000 = D^2
D = 100 because 100 * 100 = 10000
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problem states that the perimeter of the pond is 4 feet from all corners of the
rectangle.
that means you have to add 8 to each diagonal.
measurement of each diagonal becomes 100 + 8 = 108.
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the center of the rectangle will be the center of the circle.
the center of the rectangle is half the distance of the diagonal.
that then becomes the radius of the circle.
the radius of the circle is 1/2 of 108 = 54 feet.
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the area of the circle is pi*r^2 = pi*54^2 = 9160.884178
the area of the rectangle is L*W = 60*80 = 4800
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subtract the area of the rectangle from the area of the circle:
you get:
4360.884178 square feet.
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that answers the question of what is the area of the pond not covered by the building.
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your other problem also asked for how many plants can be put in this area.
the answer to that would be 4360.884178 / 30 since each plant requires 30 square feet.
that answer is 145.3628059 which rounds off to 145 plants.
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i will post a picture of the problem setup on the following website no later than 1/2 hour from now (it is now 5:30 eastern time on november 26th)
if you're interested you can go there to see.
look for "169790 picture"
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the website is:
www.geocities.com/gonzo89p
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