# Solved Problem 04 | Cube

**Problem 04**

Find 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 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 ft^{3} ice block

# 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 01 | Cube

# Smallest number for given remainders

# Arithmetic, geometric, and harmonic progressions

**Elements**

a_{1} = value of the first term

a_{m} = value of any term after the first term but before the last term

a_{n} = value of the last term

n = total number of terms

m = m^{th} term after the first but before n^{th}

d = common difference of arithmetic progression

r = common ratio of geometric progression

S = sum

## Arithmetic Progression, AP

Arithmetic progression is a sequence of numbers in which the difference of any two adjacent terms is constant. The constant difference is commonly known as common difference and is denoted by d. Examples of arithmetic progression are as follows:

# Example 04 - Simultaneous Non-Linear Equations of Three Unknowns

**Problem**

Solve for x, y, and z from the following system of equations.

$x(y + z) = 12$ → Equation (1)

$y(x + z) = 6$ → Equation (2)

$z(x + y) = 10$ → Equation (3)

# Example 03 - Simultaneous Non-Linear Equations of Three Unknowns

**Problem**

Find the value of x, y, and z from the given system of equations.

$x(x + y + z) = -36$ → Equation (1)

$y(x + y + z) = 27$ → Equation (2)

$z(x + y + z) = 90$ → Equation (3)

# Example 02 - Simultaneous Non-Linear Equations of Three Unknowns

**Problem**

Find the value of x, y, and z from the following equations.

$xy = -3$ → Equation (1)

$yz = 12$ → Equation (2)

$xz = -4$ → Equation (3)