Prep 2 - Mathematics
Unit 2: Linear Functions, Slope & Real Numbers Operations
Mathematical Modeling: Velocity, Depreciation, and Rates of Change in Everyday Life
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Mathematics is not merely abstract symbols; it is the universal language describing the living world around us! In this applied lesson, we discover how 'Linear Functions and Rate of Change' model real-world phenomena with astonishing precision. We examine uniform velocity where a vehicle travels at a constant speed: the relation between distance d and time t is modeled by d = v * t, where the slope of the distance-time graph represents the uniform speed v. We also explore economic depreciation models (such as the linear loss of value of consumer equipment or machinery over years) and rates of filling water reservoirs. Learning to read, construct, and interpret linear graphs equips you with invaluable mathematical reasoning for science, business, and everyday decision-making.
1. Uniform Speed & Distance-Time Graphs
Kinematic modeling with linear equations:
- Uniform Motion Model:
Distance = Speed × Time → d = v × t. - Slope as Velocity: In a distance-time graph (d on vertical axis, t on horizontal axis):
• Slope = Δd / Δt = Speed (v). - Interpreting Graph Features:
• Steeper line → higher speed.
• Horizontal line (Δd = 0) → object is stationary (at rest, speed = 0).
• Negative slope → object returning back toward the starting point.
2. Economic Depreciation & Cost Models
Financial modeling with linear relations:
- Linear Cost Model:
Total Cost C = (Fixed Cost) + (Unit Cost × Quantity) → C = c&sub0; + m × x.
• Here, the y-intercept c&sub0; represents fixed startup overhead, and slope m is the cost per manufactured unit. - Linear Depreciation: An asset loses value steadily each year:
•Value V = Initial Price - (Annual Rate × Time).
• The slope is negative, reflecting loss in value over service lifetime.
Real-World Modeling Workshop:
Q1: A car travels on a highway. After 2 hours it has covered 160 km, and after 5 hours it has covered 400 km. Verify that the speed is uniform and find its value.
Solution:
1. Points on graph: (t&sub1;, d&sub1;) = (2, 160), (t&sub2;, d&sub2;) = (5, 400).
2. Slope = (400 - 160) / (5 - 2) = 240 / 3 = 80 km/h.
3. Speed is constant at 80 km/h; linear equation: d = 80t.
Solution:
1. Points on graph: (t&sub1;, d&sub1;) = (2, 160), (t&sub2;, d&sub2;) = (5, 400).
2. Slope = (400 - 160) / (5 - 2) = 240 / 3 = 80 km/h.
3. Speed is constant at 80 km/h; linear equation: d = 80t.
Q2: A factory purchased a machine for 50,000 EGP. If its value depreciates linearly by 4,000 EGP every year, write the equation for its value V after t years, and find its value after 6 years.
Solution:
1. Equation: V = 50,000 - 4,000t.
2. For t = 6: V = 50,000 - 4,000(6) = 50,000 - 24,000 = 26,000 EGP.
Solution:
1. Equation: V = 50,000 - 4,000t.
2. For t = 6: V = 50,000 - 4,000(6) = 50,000 - 24,000 = 26,000 EGP.
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mathematical model
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النموذج الرياضي
📖 Def: a description of a system using mathematical concepts and language
uniform velocity
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السرعة المنتظمة
📖 Def: motion covering equal distances in equal intervals of time
depreciation
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الإهلاك المحاسبي
📖 Def: the steady reduction in the monetary value of an asset over time
fixed cost
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التكلفة الثابتة
📖 Def: expenses that do not change as production volume fluctuates
unit rate
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معدل الوحدة
📖 Def: a comparison of two measurements in which one of the terms has a value of 1
distance-time graph
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منحنى المسافة والزمن
📖 Def: a plot showing how distance changes as time elapses
stationary
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ساكن وفي حالة سكون
📖 Def: remaining in one place without moving (speed = 0)
initial value
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القيمة الابتدائية
📖 Def: the starting value of a dependent variable when independent variable is 0
extrapolation
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الاستقراء الرياضي
📖 Def: estimating an unknown value beyond the range of known data
interpolation
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الاستكمال الداخلي
📖 Def: estimating a value within two known values in a data sequence
independent variable
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المتغير المستقل
📖 Def: a variable whose variation does not depend on that of another
dependent variable
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المتغير التابع
📖 Def: a variable whose value depends on that of another variable
Click on an option to check your answer and view the detailed explanation:
1
In a distance-time graph, what does the slope of the straight line represent?
2
If the line on a distance-time graph is completely horizontal, what does this indicate about the object?
3
A water tank contains 200 liters of water and is being emptied at a constant rate of 25 liters per minute. Which equation models the remaining volume V after t minutes?
4
A smartphone was purchased for 12,000 EGP. If its market value depreciates by 1,500 EGP per year, after how many years will its value become 6,000 EGP?
5
A taxi company charges a fixed base fee of 15 EGP plus 5 EGP per kilometer driven. What is the total fare for an 8 km journey?
6
In a linear cost equation C = 200 + 15x, what does the number 200 represent?
7
A train covers 150 km in 2 hours and 375 km in 5 hours at a constant rate. What is its speed?
8
Which of the following graphs indicates a constant rate of decrease (negative slope)?
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