page 6 of 7

1. Angles and Lines
2. Intersecting Angles
3. Triangles
4. Circles
5. Perimeters & Areas
6. Solids
7. Coordinate Geometry

Solids

    There are two solids that interest us in preparing for the test. They are the rectangular solid (a box) and a circular cylinder. A cube is a special rectangular box whose sides are all equal. The volume of a box is the product of its three sides: V = bwh. The volume of a circular cylinder is the area of the base times the height: V = rh.
     The surface area of a cube is 6 x (area of one side) since all six sides have the same area. The surface area of a cylinder is composed of the top and bottom circular areas and the area around the cylinder. The top and bottom are simply 2 x x r . The area of the side is 2 x x radius x height.

Box V = bhw

Cylinder V =rh

 

Example I

How many liters does it take to fill a box that is 2 m by 20 cm by 20 mm?

Solution
There are 1000 liters in a cubic meter. Hence, we find the volume in cubic meters and multiply by 1000. The volume is

V = bwh
   = 2 x 0.2 x 0.02 = 0.008m

   1000 liters/m x  0.008m = 8 liters

Example 2

A large box measuring 20 ft x 10 ft and 8 ft high is to be painted, including the top and excluding the bottom. If one quart covers 100 ft of area, how many quarts are required?



Solution
The surface area includes 5 rectangles: top (20 x 10), 2 sides (20 x 8), and 2 more sides (10 x 8). The total area is 680. The number of required quarts is: 680/100= 6.8 or 7 quarts. NOTE: it is assumed the bottom is not painted. This assumption doesn't follow either logically or contextually from the way the problem is stated. If a box were to be painted, I would assume every side would have to be painted. "Including the top" seems redundant, but "excluding the bottom" seems relevant.

 

Example 3

It takes about 7.5 gallons to fill a volume of one cubic foot. How many gallons are needed to fill a cylinder 2 ft high and 28 inches in radius(=22/7)?




Solution
The volume of a cylinder is the area of its circular base times its height:
V =rh
=22/7
r = 28 inches, or 28/12 feet (divide 28 by 12 to convert) or 7/3 (factor out the 4), r = 49/9
h = 2, height equals 2.

Now let's plug the numbers into the equation: V =rh
= 22/7 x (49/9) x 2 =
cancel out the 7 under 22 and the 49 above 9 and multiply..

= 308 / 9ft

Since it takes 7.5 gallons to fill one cubic foot, multiply the cubic area by 7.5.

= 308/9 * 7.5 gallons = 256 2/3 gallons

Example 4

A gallon of paint covers 400 ft of wall area. How many gallons are required to paint the walls of a building with perimeter 200 ft and height 10 ft (assuming there are no windows)?

Solution
The perimeter is the distance around the building, that is, the length of the rectangles that make up its sides. Since each rectangular side is assumed to be 10 ft high, the total area is

Thus,
A = bh = 200 x 10 = 2000ft

Number of gallons required = 2000/400 = 5

Note: the shape of the building makes no difference here. The building could be circular (think of taking the wrapper off of a bottle to make a rectangle).

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