September 2012

Solved Problem 01 | Cube

Problem 01
Show that (a) the total surface of the cube is twice the square of its diagonal, (b) the volume of the cube is $\frac{1}{9}\sqrt{3}$ times the cube of its diagonal.
 

Solution 01
Space diagonal $s = a\sqrt{3}$, thus, $a = \dfrac{s}{\sqrt{3}}$
 

(a) Show that A = 2s2
$A = 6a^2$

$A = 6\left( \dfrac{s}{\sqrt{3}} \right)^2$

$A = 6\left( \dfrac{s^2}{3} \right)$

$A = 2s^2$       okay!
 

Solved Problem 02 | Cube

Problem 02
How much material was used in the manufacture of 24,000 celluloid dice, if each die has an edge of 1/4 inch?
 

Solved Problem 03 | Cube

Problem 03
What is the weight of a block of ice 24 in. by 24 in. by 24 in., if ice weighs 92 per cent as much as water, and water weighs 62.5 lb per cu. ft.?
 

Solution 03
Unit weight of water
$\gamma_{water} = 62.5 \, \text{ lb/ft}^3$
 

Unit weight of ice
$\gamma_{ice} = 92\% \, \gamma_{water}$

$\gamma_{ice} = 0.92(62.5)$

$\gamma_{ice} = 57.5 \, \text{ lb/ft}^3$
 

Volume of ice block
$V_{ice} = (24/12)^3$

$V_{ice} = 8 \, \text{ ft}^3$
 

Weight of 8 ft3 ice block

Solved Problem 04 | Cube

Problem 04
04-cube.jpgFind the volume and total area of the largest cube of wood that can be cut from a log of circular cross section whose radius is 12.7 inches. See figure.
 

Solved Problem 05 | Cube

Problem 05
A vegetable bin built in the form of a cube with an edge of 6 ft. is divided by a vertical partition which passes through two diagonally opposite edges. Find the lateral surface of either compartment.
 

Solved Problem 06 | Cube

Problem 06
The plane section ABCD shown in the figure is cut from a cube of edge a. Find the area of the section if D and C are each at the midpoint of an edge.
 

06-cube-diagonal-plane.gif

 

Solved Problem 07 | Cube

Problem 07
Find the area of the triangle whose vertex is at the midpoint of an upper edge of a cube of edge a and whose base coincides with the diagonally opposite edge of the cube.
 

Solved Problem 08 | Cube

Problem 08
If a cube has an edge equal to the diagonal of another cube, find the ratio of their volumes.
 

Solved Problem 09 | Cube

Problem 09
One cube has a face equivalent to the total area of another
cube. Find the ratio of their volumes.
 

Solved Problem 10 | Cube

Problem 10
Pass a plane through a cube so that the section formed will be a regular hexagon. If the edge of the cube is 2 units, find the area of this section.
 

Solved Problem 01 | Rectangular Parallelepiped

Problem 1
Counting 38 cu. ft. of coal to a ton, how many tons will a coal bin 19 ft. long, 6 ft. wide, and 9 ft. deep contain, when level full?
 

Solved Problem 02 | Rectangular Parallelepiped

Problem 2
Compute the cost of lumber necessary to resurface a footbridge 16 ft. wide and 150 ft. long with 2-in. planks, if lumber is \$40 per 1000 board feet. Neglect waste. (One board foot = 1 ft. by 1 ft. by 1 in.)
 

Solved Problem 03 | Rectangular Parallelepiped

Problem 3
Building bricks are closely stacked in a pile 7 ft. high, 36 ft. long, and 12 ft. wide. If the bricks are 2 in. by 4 in. by 9 in., how many bricks are in the pile?
 

Solved Problem 04 | Rectangular Parallelepiped

Problem 4
A packing box 2.2 ft. by 4.9 ft. by 5.5 ft. is to be completely covered with tin. How many square feet of the metal are needed? (Neglect waste for seams, etc.)
 

Solved Problem 05 | Rectangular Parallelepiped

Problem 5
How many cubic yard of material are needed for the foundation of barn 40 by 80 ft., if the foundation is 2 ft thick and 12 ft. high.
 

05-rectangular-barn.gif

 

Solved Problem 06 | Rectangular Parallelepiped

Problem 6
A tank, open at the top, is made of sheet iron 1 in. thick. The internal dimensions of the tank are 4 ft. 8 in. long; 3 ft. 6 in. wide; 4 ft. 4 in. deep. Find the weight of the tank when empty and find the weight when full of salt water. (Salt water weighs 64 lb/ft3 and iron is 7.2 times as heavy as salt water.
 

Solved Problem 07 | Rectangular Parallelepiped

Problem 7
The edges of the trunk are 3 ft., 4 ft., 6 ft. A second trunk is twice as long; the other edges are 3 ft., 4 ft. How do their volumes compare?
 

Solved Problem 08 | Rectangular Parallelepiped

Problem 8
An electric refrigerator is built in a form of rectangular parallelepiped. The inside dimensions are 3 ft. by 2.6 ft. by 1.8 ft. A freezing unit (1.1 ft. by 0.8 ft. by 0.7 ft.) subtracts from the storage room of the box. Find the capacity of the refrigerator.
 

Solved Problem 09 | Rectangular Parallelepiped

Problem 9
A solid concrete porch consists of 3 steps and a landing. The steps have a tread of 11 in., a rise of 7 in., and a length of 7 ft.; the landing is 6 ft. by 7 ft. How much material was used in its construction?
 

09-concrete-proch.gif

 

Solved Problem 12 | Rectangular Parallelepiped

Problem 12
In the figure is shown a rectangular parallelepiped whose dimensions are 2, 4, 6. Points A, B, C, E, F, and L are each at the midpoint of an edge. Find the area of each of the sections ABEF, ABC, and MNL.
 

12-rectangular-parallelepiped.gif

 

Solution 12

Solved Problem 13 | Rectangular Parallelepiped

Problem 13
The figure represents a rectangular parallelepiped; AD = 20 in., AB = 10 in., AE = 15 in.
(a) Find the number of degrees in the angles AFB, BFO, AFO, BOF, AOF, OFC.
(b) Find the area of each of the triangles ABO, BOF, AOF.
(c) Find the perpendicular distance from B to the plane AOF.
 

Area, angle, and distance in rectangular parallelepiped.

 

Solution 13

Solved Problem 14 | Rectangular Parallelepiped

Problem 14
Find the angles that the diagonals of the rectangular parallelepiped 2 in. by 3 in. by 4 in. makes with the faces.