B) Mr. Uprety Bought A Plot Of 5 Aana Land In A Municipality. Now, He Has Built A House Covering The Space Of 42-* Aana And The Kitchen Garden Covers Remaining 4-1 Aana Of His Plot. (i) Make The Exponential Equation According To The Given Context. (ii)

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Understanding the Relationship Between Land Area and House Construction

In this article, we will explore the concept of exponential equations and how they can be applied to real-world scenarios. We will use the context of a plot of land and its usage to create an exponential equation. This will help us understand the relationship between the land area and the construction of a house.

Mr. Uprety bought a plot of 5 Aana land in a municipality. He has built a house covering the space of 42 Aana and the kitchen garden covers the remaining 4 Aana of his plot. We need to create an exponential equation based on this information.

Step 1: Understanding the Given Information

  • The total area of the plot is 5 Aana.
  • The house covers an area of 42 Aana.
  • The kitchen garden covers an area of 4 Aana.

Step 2: Creating the Exponential Equation

Let's assume that the area of the plot is represented by the variable 'x'. We know that the total area of the plot is 5 Aana, so we can write an equation:

x = 5

Now, let's consider the area of the house and the kitchen garden. We know that the house covers an area of 42 Aana and the kitchen garden covers an area of 4 Aana. We can represent the area of the house as 42x and the area of the kitchen garden as 4x.

The Exponential Equation

We can now create an exponential equation based on the given information. Let's assume that the area of the plot is represented by the variable 'x'. We know that the total area of the plot is 5 Aana, so we can write an equation:

x = 5

We also know that the area of the house is 42x and the area of the kitchen garden is 4x. We can write an equation based on this information:

42x + 4x = 5

Simplifying the equation, we get:

46x = 5

Dividing both sides by 46, we get:

x = 5/46

However, this is not an exponential equation. To create an exponential equation, we need to use the concept of exponential growth or decay.

Exponential Growth and Decay

Exponential growth and decay are two types of exponential relationships. Exponential growth occurs when a quantity increases at a rate proportional to its current value. Exponential decay occurs when a quantity decreases at a rate proportional to its current value.

In the context of the plot of land, we can assume that the area of the house and the kitchen garden increases or decreases at a rate proportional to their current values. Let's assume that the area of the house and the kitchen garden increases at a rate proportional to their current values.

The Exponential Equation

We can now create an exponential equation based on the given information. Let's assume that the area of the plot is represented by the variable 'x'. We know that the total area of the plot is 5 Aana, so we can write an equation:

x = 5

We also know that the area of the house is 42x and the area of the kitchen garden is 4x. We can write an equation based on this information:

42x + 4x = 5

Simplifying the equation, we get:

46x = 5

Dividing both sides by 46, we get:

x = 5/46

However, this is not an exponential equation. To create an exponential equation, we need to use the concept of exponential growth or decay.

Let's assume that the area of the house and the kitchen garden increases at a rate proportional to their current values. We can write an exponential equation based on this information:

A(t) = A0 * e^(kt)

where A(t) is the area of the house or kitchen garden at time t, A0 is the initial area, e is the base of the natural logarithm, and k is the growth rate.

Solving the Exponential Equation

We can now solve the exponential equation to find the growth rate 'k'. We know that the area of the house is 42 Aana and the area of the kitchen garden is 4 Aana. We can write an equation based on this information:

A(t) = A0 * e^(kt)

Substituting the values, we get:

42 = 5 * e^(k * t)

Dividing both sides by 5, we get:

8.4 = e^(k * t)

Taking the natural logarithm of both sides, we get:

ln(8.4) = k * t

Dividing both sides by t, we get:

k = ln(8.4) / t

However, we don't know the value of 't'. We can assume that 't' is a constant value.

In this article, we created an exponential equation based on the context of a plot of land and its usage. We used the concept of exponential growth and decay to create the equation. We also solved the equation to find the growth rate 'k'. However, we didn't find the value of 't' because it is a constant value.

In the future, we can use this exponential equation to model the growth of the house and the kitchen garden over time. We can also use this equation to predict the future growth of the house and the kitchen garden.

In our previous article, we created an exponential equation based on the context of a plot of land and its usage. We used the concept of exponential growth and decay to create the equation. In this article, we will answer some frequently asked questions related to exponential equations and land area.

Q: What is an exponential equation?

A: An exponential equation is a mathematical equation that involves an exponential function. Exponential functions are functions that involve the exponentiation of a variable or a constant. In the context of land area, an exponential equation can be used to model the growth or decay of the area over time.

Q: How do I create an exponential equation?

A: To create an exponential equation, you need to identify the variables and the constants involved in the problem. You also need to determine the type of exponential function that is involved (e.g. exponential growth or decay). Once you have identified the variables and the constants, you can write the exponential equation using the formula:

A(t) = A0 * e^(kt)

where A(t) is the area at time t, A0 is the initial area, e is the base of the natural logarithm, and k is the growth rate.

Q: What is the difference between exponential growth and decay?

A: Exponential growth occurs when a quantity increases at a rate proportional to its current value. Exponential decay occurs when a quantity decreases at a rate proportional to its current value. In the context of land area, exponential growth can be used to model the growth of a house or a garden over time, while exponential decay can be used to model the decay of a plot of land over time.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable (e.g. t) on one side of the equation. You can do this by using the properties of exponents and logarithms. For example, if you have the equation:

A(t) = A0 * e^(kt)

You can solve for t by taking the natural logarithm of both sides:

ln(A(t)) = ln(A0 * e^(kt))

Simplifying the equation, you get:

ln(A(t)) = ln(A0) + kt

Dividing both sides by k, you get:

t = (ln(A(t)) - ln(A0)) / k

Q: What are some real-world applications of exponential equations?

A: Exponential equations have many real-world applications, including:

  • Modeling population growth and decay
  • Modeling the growth of a company or a business
  • Modeling the decay of a substance or a material
  • Modeling the growth of a house or a garden over time
  • Modeling the decay of a plot of land over time

Q: What are some common mistakes to avoid when working with exponential equations?

A: Some common mistakes to avoid when working with exponential equations include:

  • Not identifying the variables and the constants involved in the problem
  • Not determining the type of exponential function that is involved (e.g. exponential growth or decay)
  • Not using the correct formula for the equation
  • Not isolating the variable (e.g. t) on one side of the equation
  • Not using the properties of exponents and logarithms correctly

In this article, we answered some frequently asked questions related to exponential equations and land area. We discussed the concept of exponential equations, how to create and solve them, and some real-world applications of exponential equations. We also discussed some common mistakes to avoid when working with exponential equations.