Question 1019529: A swimming pool is 50m long and 25m wide. It is 1meter deep at the shallow end and 8 meter deep at the other end. Find the volume of water in the pool when it is full and the total area of the pool which is in contact with the water
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you can break the pool up into sections where you can easily calculate the volume and surface area from.
add the sections up together and you get the total volume and surface area that comes in contact with the water.
the first section, called S1, sits on top of the bottom section, called S2.
it is a rectangular prism.
the second section, called S2, sits below the top section, called S1.
it is a triangular prism.
you place S1 on top of S2 and you have your pool.
S1 and S2 are shown in the following diagram.
S1 on top of S2 together, placed together, forming the pool, are shown in the following diagram.
S1 is a rectangular prism whose volume is equal to length * width * height.
50 is the length.
25 is the width.
1 is the height.
the volume of S1 is therefore 50 * 25 * 1 = 1250 cubic meters *****
-----
S2 is a triangular prism whose volume is equal to area of the base * height of the prism.
area of the base is equal to 1/2 * base * height of the triangle formed by the base.
base of the triangle is equal to 7.
height of the triangle is equal to 50.
area of the base is therefore equal to 1/2 * 7 * 50 = 175 square meters.
height of the prism is equal to 25 meters.
volume of S2 is equal to area of the base * the height which is equal to 175 * 25 = 4375 cubic meters. *****
-----
volume of the water in the pool is therefore equal to the volume of S1 plus the volume of S2 which is equal to 1250 + 4375 = 5625 cubic meters. *****
-----
the surface area of the pool that is in contact with the water is:
sides of S1.
sides of S2.
bottom of S2.
the top of S1 is not included because it is open to the air above.
the bottom of S1 is not included because it is open to the water below.
the sides of S1 that are in contact with the water are:
the rectangle formed by A1, A2, B2, B1.
the area of this is 50 * 1 = 50 square meters.
the rectangle formed by A4, A3, B3, B4.
the area of this is 50 * 1 = 50 square meters.
the rectangle formed by A1, B1, B4, A4.
the area of this is 25 * 1 = 25 square meters.
the rectangle formed by A2, B2, B3, A3.
the area of this is 25 * 1 = 25 square meters.
total surface area of S1 that is in contact with the water is equal to 2 * 50 + 2 * 25 = 100 + 50 = 150 square meters.*****
-----
the sides of S2 that are in contact with the water are:
the rectangle formed by B1, C1, C4, B4.
the area of this is 25 * 7 = 175 square meters.
the triangle formed by B4, B3, C4.
the area of this is 1/2 * 7 * 50 = 175 square meters.
the triangle formed by B1, B2, C1.
the area of this is 1/2 * 7 * 50 = 175 square meters.
the rectangle formed by C1, B2, B3, C4.
the area of this is 25 * sqrt(2549) = 1262.190556 square meters.
this is the bottom of S2.
note that the length of the line formed by the points C4, B3, and the points B1, B2, is equal to sqrt(2549).
this is because those lines are the hypotenuse of the right triangles formed by the points B4, B3, C4, and the points B1, B2, C1.
as such they are equal to sqrt(7^2 + 50^2) = sqrt(49 + 2500) = sqrt(2549).
-----
total surface area of S2 that is in contact with the water is equal to 3 * 175 + 1262.190556 = 1787.190556 square meters. *****
-----
the total surface area of the pool that is in contact with the water is equal to the surface area of S1 that is in contact with the water plus the surface area of S2 that is in contact with the water.
that would be equal to 150 + 1787.190556 = 1937.190556 square meters.
in summary:
when the pool is full, .....
the volume of the water in the pool is 5625 cubic meters, and .....
the surface area of the pool that is in contact with the water is 1937.190556 square meters.
|
|
|
