krit.club logo

Introduction to Polynomials - Model linear growth and decay in practical scenarios using equations

Grade 9CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

A polynomial of degree 1 is called a linear polynomial, represented as P(x)=ax+bP(x) = ax + b, where a≠0a \neq 0.

•

Linear Growth occurs when a quantity increases by a constant amount per unit of time or distance. In the equation y=mx+cy = mx + c, the rate of growth is m>0m > 0.

•

Linear Decay occurs when a quantity decreases by a constant amount per unit of time or distance. In the equation y=mx+cy = mx + c, the rate of decay is m<0m < 0.

•

The constant term cc represents the initial value (the value of yy when x=0x = 0).

•

Practical scenarios often involve a fixed cost and a variable cost, which can be modeled using a linear equation of the form y=ax+by = ax + b.

📐Formulae

y=mx+cy = mx + c

Rate of Change (m)=y2−y1x2−x1\text{Rate of Change } (m) = \frac{y_2 - y_1}{x_2 - x_1}

P(x)=ax+bP(x) = ax + b

💡Examples

Problem 1:

A taxi driver charges a fixed fare of ₹50₹50 for the first kilometer and ₹12₹12 per kilometer for every subsequent kilometer. If the distance covered is xx km and the total fare is ₹y₹y, write a linear equation for this information.

Solution:

The fare for the first kilometer = ₹50₹50. The remaining distance = (x−1)(x - 1) km. The fare for the remaining distance = 12(x−1)12(x - 1). Total fare y=50+12(x−1)y = 50 + 12(x - 1). Simplifying: y=50+12x−12⇒y=12x+38y = 50 + 12x - 12 \Rightarrow y = 12x + 38.

Explanation:

We identify the fixed component and the variable component to form a linear polynomial representing the total cost.

Problem 2:

A water tank initially contains 10001000 liters of water. Due to a small leak, it loses 2525 liters of water every hour. Write an equation for the volume of water VV remaining in the tank after tt hours. Also, find the volume after 1212 hours.

Solution:

Initial volume c=1000c = 1000. Rate of decay m=−25m = -25. The equation is V=1000−25tV = 1000 - 25t. For t=12t = 12: V=1000−25(12)V = 1000 - 25(12) V=1000−300V = 1000 - 300 V=700V = 700 liters. Calculation: 1000−300700\begin{array}{r} 1000 \\ -300 \\ \hline 700 \end{array}

Explanation:

This is a case of linear decay where the initial value is 10001000 and it decreases at a constant rate of 2525 per unit of time tt.