Let's break down each of the descriptions provided, identify the quantities involved, formulate linear equations, and analyze their slope and y-intercept.
A. Identification of Quantities
-
Distance Traveled by a Car:
- Quantities: Distance (d), Time (t)
-
Total Cost of Buying Bottles of Water:
- Quantities: Total Cost (C), Number of Bottles (n)
-
Area of a Rectangle:
- Quantities: Area (A), Length (l), Width (w)
-
Volume of a Rectangular Prism:
- Quantities: Volume (V), Length (l), Width (w), Height (h)
B. Linear Equations
-
Distance Traveled by a Car:
- Equation: \( d = 40t \)
-
Total Cost of Buying Bottles of Water:
- Equation: \( C = 2.5n + 20 \)
-
Area of a Rectangle (assuming a fixed width):
- Equation: \( A = l \times w \). If \( w \) is constant: \( A = kw \), where \( k \) is the width.
-
Volume of a Rectangular Prism (assuming a specific height):
- Equation: \( V = l \times w \times h \). If \( h \) is constant: \( V = k \times (l \times w) \).
C. Explanation of Slope and y-intercept
-
Distance Traveled by a Car \( d = 40t \):
- Slope: 40 (miles per hour) – it represents the speed of the car.
- y-intercept: 0 (the car starts at 0 distance when time is 0).
-
Total Cost of Buying Bottles of Water \( C = 2.5n + 20 \):
- Slope: 2.5 (dollars per bottle) – it represents the cost of each bottle of water.
- y-intercept: 20 (the cost of the water jug) – this is the fixed cost before buying any bottles.
-
Area of a Rectangle \( A = lw \) with constant width \( w \):
- Slope: \( w \) (constant) – the area increases linearly as the length increases.
- y-intercept: 0 (when length is 0, the area is 0).
-
Volume of a Rectangular Prism \( V = lwh \) with constant height \( h \):
- Slope: \( wh \) (product of width and height) – it represents how the volume increases with more length.
- y-intercept: 0 (when there is no length, the volume is 0).
D. Creating Tables and Graphs
1. Distance Traveled by a Car: d = 40t
| Time \( t \) (hours) | Distance \( d \) (miles) | |-----------------------|--------------------------| | 0 | 0 | | 1 | 40 | | 2 | 80 | | 3 | 120 | | 4 | 160 |
Graph: A straight line through (0,0) with a slope of 40.
2. Total Cost of Buying Bottles of Water: C = 2.5n + 20
| Number of Bottles \( n \) | Total Cost \( C \) (dollars) | |----------------------------|-------------------------------| | 0 | 20 | | 1 | 22.50 | | 2 | 25 | | 3 | 27.50 | | 4 | 30 |
Graph: A straight line starting from (0,20) with a slope of 2.5.
3. Area of a Rectangle (Assuming width \( w = 2 \)): A = 2l
| Length \( l \) (units) | Area \( A \) (square units) | |-------------------------|------------------------------| | 0 | 0 | | 1 | 2 | | 2 | 4 | | 3 | 6 | | 4 | 8 |
Graph: A straight line through (0,0) with a slope of 2.
4. Volume of a Rectangular Prism (Assuming width \( w = 2 \) and height \( h = 3 \)): V = 6l
| Length \( l \) (units) | Volume \( V \) (cubic units) | |-------------------------|-------------------------------| | 0 | 0 | | 1 | 6 | | 2 | 12 | | 3 | 18 | | 4 | 24 |
Graph: A straight line through (0,0) with a slope of 6.
Summary
- Each description relates two quantities in a linear way.
- The slope typically represents the rate of change, while the y-intercept indicates the starting value when the independent variable (time, number of bottles, length, etc.) is zero.
- Values were tabulated to facilitate graphing, showing the linear relationship visually.
1 answer