joi, 6 aprilie 2017

Geomerty - an Aquarium a right prism

The figure is represented schematically an Aquarium in the form of a right prism with square floor, which features 8 dm floor side and the lateral edge of Aquarium of 5 dm. The faces of the Aquarium are made of glass. The Aquarium floor is made from a different material. Aquarium is not covered. Into Aquarium is located water up to a height of 4dm (glass thickness neglect).

a) Calculate how many litres of water are in the Aquarium,
b) Calculate how many square metres of glass are needed to into 100 pcs. of Aquarium, having the dimensions given in the hypotesis,
c) Show that, at any time, the distance between two fish is less than or equal to 12 dm.


Solution: 
Remember the concept of Prism:
A prism is a solid figure that has two parallel congruent sides (polygons) that are called bases. These bases are conected by lateral faces that are parallelograms. At the same time the prism's edges are parallel to each other. Prism's height is equal to the distance between bases. 
When we determine the lateral surface areas of a prism we take the sum of the lateral area for each lateral faces (without the bases).

When the Prism is right, it is a particular case, the lateral faces are rectangles, and these are perpendicular planes to the prism's bases. The Prism's height is equal to the straight edge of the prism's side.

The lateral area is equal to the perimeter of the base multiplied by the height of the prism's (= lateral edge of the prism's). 

Prism's volume is equal to the area of the base multiplied by the prism's height.
a) Prism's base is a square of side 8 dm.
A ABCD = 82 dm2 = 64 dm2.

Side edge of the Aquarium is H = 5 dm but the water in the Aquarium rises up to h = 4 dm. 

The volume of the Aquarium's water :
V = AABCD h= 644 = 256 dm3 . Since 1 dm3 = 1 litre the water volume (in liters) = 256 litres.

b) To calculate how many square metres of glass are needed for making an Aquarium, we need the lateral area of this Prism (as the hypotesis says that the basis of the Aquarium is made of another material, and the Aquarium is not covered, we will not consider the bases). 

The lateral area of the prism = 
PABCD ∙H = (8∙4) 5 dm2 = 325 dm2 = 160 dm2  of glass.

For 100 pcs. of Aquarium is needed = 
100 160 dm2 = 16000 dm2 = 160 m2  (use the relationship of conversion of the unit of measurement 1 dm2 = 0.01 dm2).

c) At this point we are required to show that at all times, the distance between two fish is less than or equal to 12 dm. We understand that the distance between the two fish may not be greater than 12 dm. So how far can stay two fish in the Aquarium? The greatest distance between two points of the water prism is water Prism's diagonal.
The height of the water in the Aquarium)is h = 4 dm.

The diagonal
AM2 = AB2 + BC2 + h2 = 82 + 82 + 42 = 144 dm2, where by extracting the square root AM = 12 dm. 
Distance between two fish is always less than or equal to 12 dm.

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