Chapter 1-9 Review: Basic Algebra, Geometry, and Circle Calculations

Basic Algebra and Writing Expressions

Multiplication and division can be written in many ways in algebra:
- If you see two letters like "2a" or "2(a)", that means multiplication: 2a = 2 \times a
- A number in front of parentheses also indicates multiplication: 2\, (a) = 2 \times a
- If both factors are in parentheses, e.g., $(2)(a)$ or $(2)(a)$, it’s still multiplication: (2)(a) = 2a
Division can be written in several forms:
- Triple slash or division symbol: \frac{3}{b}
- A slash: 3/b
- A horizontal "over" form: \frac{3}{b}
The main goal in word problems is to model the quantity you want using a formula, often combining multiplication/addition and sometimes division.

Nails cost model (an example of a real-world algebra word problem)

Scenario: Nails cost 2.42 per pound, shipping is a fixed 15 dollars, buy w pounds; total cost is c.
Define variables:
- w = number of pounds of nails
- c = total cost in dollars
Algebraic formula for the total cost:
- c = 2.42\,w + 15
Example: order 7 pounds; substitute to find cost:
- c = 2.42\times 7 + 15 = 31.94
- Result: c = 31.94\; \text{dollars}
Real-world note: unit consistency and rounding are important when presenting final costs.

Uber fare calculation (another real-world example)

Fare components:
- Base fare: 6.00
- Cost per minute: 0.15\$ per minute
- Ride time: 27\;\text{minutes}
- Cost per mile: 1.57\$ per mile
- Distance: 19.8\;\text{miles}
- Booking fee: 0.55
Convert all to dollars and write the total as a sum:
- C = 6 + 0.15\times 27 + 1.57\times 19.8 + 0.55
- Evaluate: C \approx 41.69
- Result: C \approx 41.69\;\text{USD}
Practical note: ensure consistent units (dollars) when summing; sometimes it helps to convert cents to dollars first (e.g., 15¢ = 0.15$).

Temperature conversion: Fahrenheit and Celsius

Relation between Fahrenheit (F) and Celsius (C):
- Common form: F = 1.8\,C + 32 = \tfrac{9}{5}C + 32
Examples:
- If C = 29^\circ C, then F = 1.8\times 29 + 32 = 84.2^\circ F
- If C = 37^\circ C, then F = 1.8\times 37 + 32 = 98.6^\circ F
Practical note: conversion sheets simplify quick checks when traveling or reading foreign weather reports.

Perimeter and area of polygons: key concepts

Definitions:
- Perimeter: the distance around a shape (sum of side lengths)
- Polygon: a closed plane figure with straight sides
- Quadrilateral: a polygon with four sides
Shape categories and formulas:
- Rectangle: sides length and width
- Perimeter: P = 2\ell + 2w = 2(\ell + w)
- Area: A = \ell w
- Square: special rectangle with all sides equal
- Perimeter: P = 4s
- Area: A = s^2
- Parallelogram: opposite sides are parallel; height is the perpendicular distance between bases
- Perimeter: P = 2a + 2b (where a and b are adjacent side lengths)
- Area: A = b \cdot h (base times height)
- Trapezoid: two bases parallel, other sides not necessarily parallel
- Perimeter: sum of all four sides (no single fixed formula)
- Area: A = \frac{(b1 + b2)}{2} \cdot h (mean of bases times height)
Important notes:
- Areas use square units (e.g., cm^2, m^2, ft^2)
- Perimeters use linear units (e.g., cm, m, ft)
- Some problems let you compute the perimeter without knowing every single side, using symmetry or measurements of opposite sides

Examples from the transcript: perimeter and area calculations

Example 1 (simple quad): sides are 8, 9, 5, 9 inches
- Perimeter: P = 8 + 9 + 5 + 9 = 31\;\text{inches}
Example 2 (trapezoid with parallel bases 14 yd and 7 yd): two bases given, assume the non-parallel sides equal due to symmetry
- Perimeter: P = 2\times 14 + 2\times 7 = 42\;\text{yards}
Example 3 (rectangle in cm): dimensions 95 cm by 14 cm
- Perimeter: P = 2(95 + 14) = 218\;\text{cm}
- Area: A = 95 \times 14 = 1330\;\text{cm}^2 (note: this example focuses on perimeter but area is also standard for rectangles)
Area of a composite shape (rectangle + triangle): given a rectangle 16 by 6 and a right triangle with base 16 and height 6
- Rectangle area: A_{rect} = 16 \times 6 = 96\;\text{(sq units)}
- Triangle area: A_{tri} = \frac{1}{2} \cdot 16 \cdot 6 = 48\;\text{sq units}
- Total area: A = 96 + 48 = 144\;\text{sq units}
Unit conversion note: 1\,\text{yard}^2 = 9\;\text{ft}^2; conversion between square yards and square feet can affect material quantities (e.g., vinyl siding) when budgeting

Circles: circumference and area

Circle basics:
- Radius (r): distance from center to edge
- Diameter (d): distance across the circle through the center, with d = 2r
- Circumference (C): distance around the circle
- C = 2\pi r or C = \pi d
- Area (A): interior area
- A = \pi r^2 (radius form) or A = \pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4}
Why pi appears: the ratio of circumference to diameter is a constant ~3.14159 for all circles
Example calculations from the transcript:
- Circle with radius 78 cm:
- C = 2\pi r = 2\pi \cdot 78 \approx 490\text{ cm}
- A = \pi r^2 = \pi \cdot 78^2 \approx 19099\;\text{cm}^2
- Circle with diameter 18 cm (radius 9 cm):
- C = \pi d = \pi \cdot 18 = 18\pi \approx 56.55\text{ cm}
- A = \pi r^2 = \pi \cdot 9^2 = 81\pi \approx 254.47\text{ cm}^2
Rounding practice: round circumference and area to a reasonable precision (e.g., two decimals) depending on problem requirements

Part of a circle: how to handle arcs in real-world problems

When a problem involves only a portion of a circle, scale the full-circle circumference by the fraction of the circle involved
General arc length formula (in radians):
- If a central angle is (\theta) radians, arc length is L = r\theta
- If the angle is given in degrees ((\theta{deg})), arc length is L = \frac{\theta{deg}}{360^\circ} \cdot (2\pi r)
Real-world examples from the transcript:
- Conduit bending example (quarter circle): radius for centerline matters
- Bender radius: 2 inches; pipe diameter: 1 inch; centerline radius for bending is 2.5 inches (2 + 0.5) because bend happens along the centerline of the material
- Arc length for 90° (quarter circle):
  - L = \frac{90^\circ}{360^\circ} \cdot (2\pi \cdot 2.5) = \frac{1}{4} \cdot (5\pi) = 1.25\pi \approx 3.927\;\text{inches}
- Total piece length to cut: straight segment 8 in + bend arc 3.927 in + straight segment 12 in ≈ 23.927 in
- Practical note: in practice you’d cut to 24 inches so that bending yields the correct post-bend lengths
- Another example: three-quarter circle with centerline radius 9 inches (or another radius as given)
- Arc length for 3/4 of a circle: L = \frac{3}{4} \cdot (2\pi r) = \frac{3}{2}\pi r
- If r = 9 in, then L = \frac{3}{2}\pi \cdot 9 = 13.5\pi \approx 42.41\text{ in}
- Another example (centerline radius nuance): for a 1-inch conduit, centerline radius is the bend center, not the outside/edge radius
Practical takeaways for engineers and tradespeople:
- Always use the centerline radius when calculating lengths for bent pieces
- Convert fractions of a circle into arc lengths using the appropriate formula
- Be aware of measurement granularity (sixteenths of an inch on typical tape measures) and round to the nearest feasible measurement

Quick tips on measurement, units, and real-world budgeting

Unit consistency is crucial when mixing measurements (e.g., feet, inches, centimeters, meters)
When converting units:
- 1 meter = 100 cm; 1 ft = 12 in; 1 yd = 3 ft
- To convert cm to m, move the decimal two places to the left: 1\text{ m} = 100\text{ cm}
Area and volume units require squared or cubed units, respectively: A\text{ (area)}: \text{units}^2, \quad V\text{ (volume)}: \text{units}^3
In construction and engineering, small rounding decisions can have large material or cost implications (e.g., rounding lengths up to avoid material shortfalls or waste)
Real-world implications of math literacy:
- Accurate cost estimation (nails, concrete, siding, etc.) reduces waste and saves money
- Understanding arc lengths helps in fabricating bent conduits and other curved components efficiently
- Clear grasp of unit conversions is essential when reading plans or working in different countries with different measurement systems

Summary of core formulas to remember

Algebra/word problems:
- Basic cost model: c = p\,w + s where p = price per unit, w = quantity, s = fixed shipping
Nails example:
- c = 2.42\,w + 15
- For w = 7: c = 2.42\times 7 + 15 = 31.94
Uber fare model:
- C = 6 + 0.15 \times t + 1.57 \times d + 0.55 where t = \text{minutes}, d = \text{miles}
- For the given values: C \approx 41.69
Temperature conversion:
- F = \frac{9}{5}C + 32
- Examples: C=29\Rightarrow F \approx 84.2^\circ F; \quad C=37\Rightarrow F \approx 98.6^\circ F
Perimeter and area basics:
- Rectangle: P = 2\ell + 2w = 2(\ell + w),\quad A = \ell w
- Square: P = 4s,\quad A = s^2
- Parallelogram: P = 2a + 2b,\quad A = b\cdot h
- Trapezoid: A = \frac{(b1 + b2)}{2} \cdot h
Circle basics:
- C = 2\pi r = \pi d
- A = \pi r^2
Partial circles (arc length):
- For a fraction f of a full circle: L = f\cdot(2\pi r) = (\text{fraction})\times 2\pi r
Centerline radius concept (bending): use radius of bend plus material radius to get true centerline radius

Real-world connections and implications

The lecture shows how algebra underpins everyday cost calculations, travel fares, and measurements in construction
It emphasizes the importance of unit management, rounding decisions, and converting between systems (metric vs customary)
It highlights practical problem-solving strategies: break complex shapes into known shapes (rectangle + triangle), use symmetry to reduce measurements, and apply arc-length formulas for bent components
Ethical/practical takeaway: accurate modeling avoids waste, underbidding, or safety issues in trades; clear communication of units and results is essential in professional settings