Estimating the Change in Area

Oct 13, 2025 by windailys.com 82 views

Introduction

In mathematics, the area of a circle is a fundamental concept that is used to calculate the size of a circle. The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius of the circle. In this article, we will explore the change in area of a circle when its radius is increased from 10.00 to 10.05 m.

The Change in Area

To estimate the change in area, we can use the concept of differential calculus. The differential of the area function A(r) = πr^2 is given by dA = 2πr dr, where dr is the change in radius. In this case, the change in radius is 10.05 - 10.00 = 0.05 m.

import math
r_original = 10.00
dr = 0.05
dA = 2 * math.pi * r_original * dr
A_original = math.pi * (r_original ** 2)
r_new = r_original + dr
A_new = math.pi * (r_new ** 2)
delta_A = A_new - A_original
percentage_change = (delta_A / A_original) * 100

Estimating the Change in Area

Using the differential of the area function, we can estimate the change in area as follows:

dA = 2πr dr = 2π(10.00)(0.05) = 3.14 m^2

This is the estimated change in area when the radius of the circle is increased from 10.00 to 10.05 m.

Expressing the Estimate as a Percentage of the Circle's Original Area

To express the estimate as a percentage of the circle's original area, we can use the following formula:

Percentage change = (delta_A / A_original) × 100

where delta_A is the change in area and A_original is the original area.

# Calculate the percentage change in area
percentage_change = (delta_A / A_original) * 100

Calculating the Percentage Change in Area

Using the values calculated earlier, we can calculate the percentage change in area as follows:

Percentage change = (3.14 / (π(10.00)^2)) × 100 = (3.14 / 314.16) × 100 = 1.00%

Therefore, the estimated change in area is 1.00% of the circle's original area.

Conclusion

In this article, we have estimated the change in area of a circle when its radius is increased from 10.00 to 10.05 m. We have used the concept of differential calculus to calculate the differential of the area function and have expressed the estimate as a percentage of the circle's original area. The estimated change in area is 1.00% of the circle's original area.

References

Calculus, 3rd edition, Michael Spivak
Mathematics for Engineers and Scientists, 6th edition, Donald R. Hill

Introduction

In our previous article, we explored the change in area of a circle when its radius is increased from 10.00 to 10.05 m. We used the concept of differential calculus to calculate the differential of the area function and expressed the estimate as a percentage of the circle's original area. In this article, we will answer some frequently asked questions related to the change in area of a circle.

Q: What is the formula for the area of a circle?

A: The formula for the area of a circle is A = πr^2, where A is the area and r is the radius of the circle.

Q: How do you calculate the differential of the area function?

A: To calculate the differential of the area function, you can use the formula dA = 2πr dr, where dr is the change in radius.

Q: What is the estimated change in area when the radius of the circle is increased from 10.00 to 10.05 m?

A: The estimated change in area is 3.14 m^2.

Q: How do you express the estimate as a percentage of the circle's original area?

A: To express the estimate as a percentage of the circle's original area, you can use the formula: Percentage change = (delta_A / A_original) × 100, where delta_A is the change in area and A_original is the original area.

Q: What is the percentage change in area when the radius of the circle is increased from 10.00 to 10.05 m?

A: The percentage change in area is 1.00%.

Q: Can you provide an example of how to calculate the percentage change in area?

A: Yes, here is an example:

Suppose the original area of the circle is 314.16 m^2 and the change in area is 3.14 m^2. To calculate the percentage change in area, you can use the formula:

Percentage change = (3.14 / 314.16) × 100 = 1.00%

Q: What is the significance of the percentage change in area?

A: The percentage change in area is a measure of how much the area of the circle has changed when the radius is increased. In this case, the percentage change in area is 1.00%, which means that the area of the circle has increased by 1.00% when the radius is increased from 10.00 to 10.05 m.

Q: Can you provide a real-world example of how the change in area of a circle can be applied?

A: Yes, here is a real-world example:

Suppose you are designing a circular tank to hold a certain amount of liquid. If the radius of the tank is increased by 0.05 m, how much will the area of the tank increase? Using the formula for the area of a circle, you can calculate the original area of the tank and then use the differential of the area function to estimate the change in area. This will give you an idea of how much the area of the tank will increase when the radius is increased.

Conclusion

In this article, we have answered some frequently asked questions related to the change in area of a circle. We have used the concept of differential calculus to calculate the differential of the area function and expressed the estimate as a percentage of the circle's original area. We hope that this article has provided you with a better understanding of the change in area of a circle and how it can be applied in real-world scenarios.

References

Calculus, 3rd edition, Michael Spivak
Mathematics for Engineers and Scientists, 6th edition, Donald R. Hill