Here’s a detailed breakdown of the math concepts and solutions for the scenarios in the workbook, section by section.?
Measurement and Calculation
1. Fencing Around a Site
- Formula: Perimeter = ( 2 * ({Length} + {Width}) ).
- Given: Length = 47.8m, Width = 30.3m.
- Solution:
( 2 * (47.8 + 30.3) = 2 * 78.1 = 156.2 \, {m} ).
Answer: 157m (rounded up).
2. Studs for Framing Walls
- Formula: Studs = ( {Wall Length (in feet)} / {Stud Spacing (in feet)} ).
- Example: If wall length = 20 ft. and stud spacing = 2 ft.:
( {Studs} = 20 / 2 = 10 ).
(Calculate individually for each wall and sum totals.)
3. Bonding Product for Vinyl Flooring
- Formula: Coverage = ( {Floor Area} / {Coverage per Container} ).
- Given: Room area = ( 9.2 * 7.8 = 71.76 \, {m}^2 ). Coverage = ( 45 - 60 \, {m}^2 / \, {L} ).
Assume average coverage = 52.5 ( {m}^2 / \, {L} ).
( 71.76 / 52.5 \approx 1.37 ).
Answer: 2 containers (rounded up).
4. Cost of Rubber Tiles for Playground
- Formula: Tiles = ( {Area of Playground} / {Area per Tile} ).
Cost = ( {Number of Tiles} * {Cost per Tile} ).
- Example: Radius = ( r \, {ft.} ), Tile Area = ( 2 \, {ft.}^2 ).
( {Playground Area} = \pi r^2 ). Calculate area, divide by tile size, and round up.
5. Sand for a Border
- Formula: Sand Volume = ( {Border Depth} * {Perimeter} * {Width} ).
Use the dimensions of the square and ensure all units are consistent.
6. Volume of a Cone-shaped Pile
- Formula: Volume = ( \frac{1}{3} \pi r^2 h ).
Use ( r = \frac{3}{4} * {Height} ). Convert all units to consistent measures (e.g., yards).
7. Insulation Material for a Pipe
- Formula: Surface Area = ( \pi d h ).
- Given: Diameter ( d = 2 \, {m} ), Height ( h = 6 \, {m} ).
( \pi * 2 * 6 = 37.68 \, {m}^2 ).
Answer: 37.68m².
Using Formulas for Construction Tasks
Stairs Stringer Length
- Formula: ( c = \sqrt{({Rise}^2 + {Run}^2)} ).
Example: Rise = 9 in., Run = 12 in.
( c = \sqrt{(9^2 + 12^2)} = \sqrt{225} = 15 \, {in.} ).
Transition Elbow Diagonal
- Formula: Diagonal = ( \sqrt{(a^2 + b^2)} ).
Use measurements provided for dimensions ( a ) and ( b ).
Water Pressure
- Formula: Water Pressure = ( h * 0.433 ).
Example: Height ( h = 28 \, {ft.} ).
( 28 * 0.433 = 12.124 \, {psi} ).
Answer: 12.1 psi.
Airflow Volume in a Duct
a. Cross-sectional Area:
( A = ({Height} * {Width}) / 144 \, {ft}^2 ).
b. Volume: ( V = A * v ).
Working with Quantities
Sewer Line Fall
- Formula: Total Fall = ( {Length} * {Grade} ).
- Example: ( 30 \, {ft.} * \frac{1}{4} \, {in./ft.} ).
Convert inches to feet before calculating.
Kilometers to Miles
- Formula: ( {Miles} = {Kilometers} * 0.6214 ).
- Example: ( 35,000 * 0.6214 = 21,747 \, {miles} ).
Pancake Recipe Adjustment
- Use proportions to scale each ingredient.
- Example: Original = ( 30 ), Desired = ( 75 ). Scale factor = ( 75/30 = 2.5 ). Multiply each ingredient by ( 2.5 ).
Estimating Weights
Steel Plate Weight
- Formula: ( {Weight} = {Area} * {Weight per Unit Area} ).
Convert dimensions and thickness to match weight per unit (e.g., lb./ft²).
Concrete Panel Weight
- Formula: ( {Weight} = {Volume} * {Density} ).
Use ( 150 \, {lb./ft}^3 ) for density.
Here’s a detailed breakdown of the tasks and solutions based on the questions provided:
Ingredient Measurements
1. Scaling Ingredients
Using proportions to scale up from 30 to 75 servings:
- Scale Factor: ( 75/30 = 2.5 ).
| Ingredient | Amount for 30 | Amount for 75 |
|-----------------------|-------------------|-------------------|
| Flour | 625 g | ( 625 * 2.5 = 1562.5 \, g ) |
| Sugar | 60 g | ( 60 * 2.5 = 150 \, g ) |
| Baking Powder | 30 g | ( 30 * 2.5 = 75 \, g ) |
| Eggs | 4 | ( 4 * 2.5 = 10 \, eggs ) |
| Milk | 1 L | ( 1 * 2.5 = 2.5 \, L ) |
| Melted Butter | 125 g | ( 125 * 2.5 = 312.5 \, g ) |
2. Concrete Calculations
- Fence Posts:
-
1 bag per post 9 posts = 9 bags.
Answer: 9 bags.
-
Slab:
-
Volume ( V = L * W * H ). Convert dimensions to feet:
( L = 5', W = 2 \frac{4}{12} = 2.33', H = \frac{6}{12} = 0.5' ).
( V = 5 * 2.33 * 0.5 = 5.825 \, {ft}^3 ).
Bags required: Check product label for coverage.
Answer: ( \approx 5.825 \, {bags} ) (round appropriately).
-
Tube Form:
- Volume ( V = \pi r^2 h ), ( r = \frac{8}{2} = 4 \, {in} = \frac{4}{12} \, {ft} ), ( h = 15 \, {ft} ).
( V = \pi * (0.333)^2 * 15 \approx 5.236 \, {ft}^3 ).
Bags required: Use product label and multiply by water ratio ( (2.5 \, {L per bag}) ).
Answer: ( \approx 5.236 \, {bags}, {Water: } 2.5 * 5.236 \approx 13.09 \, {L} ).
Money Math
1. Selling Prices
Markup formula: ( {Selling Price} = ({Wholesale Price}) * (1 + {Markup\%}) ).
| Wholesale Price | Markup | Selling Price |
|----------------------|------------|----------------------------------|
| $97.25 | 10% | ( 97.25 * 1.10 = 106.98 ) |
| $249.99 | 15% | ( 249.99 * 1.15 = 287.49 )|
| $6.50 | 8% | ( 6.50 * 1.08 = 7.02 ) |
2. After-Tax Costs
Formula: ( {Total Cost} = {Cost} * (1 + {Tax\%}) ).
| Cost | Tax | Total |
|-------------|---------|----------------------------|
| $73.50 | 12% | ( 73.50 * 1.12 = 82.32 ) |
| $1,847.00 | 13% | ( 1,847.00 * 1.13 = 2,087.11 ) |
| $86.75 | 8% | ( 86.75 * 1.08 = 93.69 ) |
3. Sale Prices
Formula: ( {Sale Price} = {Price} * (1 - {Discount\%}) ).
| Price | Discount | Sale Price |
|--------------|--------------|----------------------------|
| $85.40 | 10% | ( 85.40 * 0.90 = 76.86 ) |
| $1,348.00 | 33% | ( 1,348.00 * 0.67 = 903.16 ) |
| $459.75 | 40% | ( 459.75 * 0.60 = 275.85 ) |
4. Dishwasher Repair Invoice
- Labour: ( 1.2 * 88.25 = 105.90 ), GST = ( 105.90 * 0.05 = 5.30 ), Total = ( 105.90 + 5.30 = 111.20 ).
- Parts: ( 2.80 * 1.11 = 3.11 ).
- Total Bill: ( 111.20 + 3.11 = 114.31 ).
- Change: ( 120.01 - 114.31 = 5.70 ).
Data Analysis
Tolerance Table
| Specification | Measure | Within Tolerance? (v/X) |
|--------------------|-------------|-----------------------------|
| ( 22.5° \pm 2° ) | ( 20.1° ) | ( v ) |
| ( 0.850 \pm 0.020 \, {m} ) | ( 0.827 \, {m} ) | ( X ) |
| ( 0.750 \pm 0.015 \, {L} ) | ( 0.761 \, {L} ) | ( X ) |