Calculating the Area of the Playground

Oct 13, 2025 by leolavanderia.com.br 22 views

Introduction

In this problem, we are given the diameter of a roller, which is $12 \, \text{m}$ , and the length of the playground, which is $100 \, \text{m}$ . We are also told that it takes 400 complete revolutions to level the playground. Our goal is to find the area of the playground and then calculate the cost of leveling it at a rate of $$250 per 100 , \text{sq m}$$.

Calculating the Area of the Playground

To find the area of the playground, we need to first calculate the circumference of the roller. The formula for the circumference of a circle is given by:

C = 2\pi r

where $r$ is the radius of the circle. Since we are given the diameter of the roller, which is $12 \, \text{m}$ , we can find the radius by dividing the diameter by 2:

r = \frac{12}{2} = 6 \, \text{m}

Now, we can calculate the circumference of the roller using the formula:

C = 2\pi r = 2\pi(6) = 12\pi \, \text{m}

Next, we need to find the area of the playground. Since the roller takes 400 complete revolutions to level the playground, we can assume that the area of the playground is equal to the area covered by the roller in 400 revolutions. The area covered by the roller in one revolution is equal to the circumference of the roller multiplied by the length of the playground:

A = Cl = (12\pi)(100) = 1200\pi \, \text{sq m}

Since the roller takes 400 complete revolutions to level the playground, the total area of the playground is equal to 400 times the area covered by the roller in one revolution:

A_{\text{total}} = 400(1200\pi) = 480,000\pi \, \text{sq m}

Calculating the Cost of Leveling the Playground

Now that we have found the area of the playground, we can calculate the cost of leveling it at a rate of $$250 per 100 , \text{sq m}$$. To do this, we need to first convert the area of the playground from square meters to hundreds of square meters:

A_{\text{total}} = \frac{480,000\pi}{100} = 4800\pi \, \text{hundreds of sq m}

Next, we can multiply the area of the playground in hundreds of square meters by the cost per hundred square meters to find the total cost:

\text{Cost} = (4800\pi)(250) = 1,200,000\pi \, \text{dollars}

Conclusion

In this problem, we found the area of the playground by calculating the circumference of the roller and multiplying it by the length of the playground. We then calculated the cost of leveling the playground at a rate of $$250 per 100 , \text{sq m}$$. The final answer is:

Area of the playground: $480,000\pi \, \text{sq m}$

Cost of leveling the playground: $1,200,000\pi \, \text{dollars}$

Introduction

In the previous article, we calculated the area of the playground and the cost of leveling it at a rate of $$250 per 100 , \text{sq m}$$. However, we received many questions from readers who wanted to know more about the problem. In this article, we will answer some of the most frequently asked questions.

Q: What is the formula for calculating the area of the playground?

A: The formula for calculating the area of the playground is:

A_{\text{total}} = 400(1200\pi) = 480,000\pi \, \text{sq m}

However, this formula assumes that the roller takes 400 complete revolutions to level the playground. If the number of revolutions is different, you will need to adjust the formula accordingly.

Q: How do I calculate the circumference of the roller?

A: The formula for calculating the circumference of a circle is:

C = 2\pi r

where $r$ is the radius of the circle. Since we are given the diameter of the roller, which is $12 \, \text{m}$ , we can find the radius by dividing the diameter by 2:

r = \frac{12}{2} = 6 \, \text{m}

Now, we can calculate the circumference of the roller using the formula:

C = 2\pi r = 2\pi(6) = 12\pi \, \text{m}

Q: What if the length of the playground is different from 100 m?

A: If the length of the playground is different from 100 m, you will need to adjust the formula accordingly. The area covered by the roller in one revolution is equal to the circumference of the roller multiplied by the length of the playground:

A = Cl = (12\pi)(x) = 12\pi x \, \text{sq m}

where $x$ is the length of the playground. Since the roller takes 400 complete revolutions to level the playground, the total area of the playground is equal to 400 times the area covered by the roller in one revolution:

A_{\text{total}} = 400(12\pi x) = 4800\pi x \, \text{sq m}

Q: How do I calculate the cost of leveling the playground if the cost per 100 sq m is different from $250?

A: If the cost per 100 sq m is different from $250, you will need to adjust the formula accordingly. The cost of leveling the playground is equal to the area of the playground in hundreds of square meters multiplied by the cost per hundred square meters:

\text{Cost} = (4800\pi x)(y) = 4800\pi xy \, \text{dollars}

where $y$ is the cost per 100 sq m.

Q: What if the roller takes a different number of revolutions to level the playground?

A: If the roller takes a different number of revolutions to level the playground, you will need to adjust the formula accordingly. The area covered by the roller in one revolution is equal to the circumference of the roller multiplied by the length of the playground:

A = Cl = (12\pi)(x) = 12\pi x \, \text{sq m}

Since the roller takes $n$ complete revolutions to level the playground, the total area of the playground is equal to $n$ times the area covered by the roller in one revolution:

A_{\text{total}} = n(12\pi x) = 12\pi nx \, \text{sq m}

Conclusion

In this article, we answered some of the most frequently asked questions about the problem. We hope that this article has been helpful in clarifying any confusion. If you have any further questions, please don't hesitate to ask.

Q&A Summary:

The formula for calculating the area of the playground is: $A_{\text{total}} = 400(1200\pi) = 480,000\pi \, \text{sq m}$
The formula for calculating the circumference of the roller is: $C = 2\pi r$
If the length of the playground is different from 100 m, you will need to adjust the formula accordingly.
If the cost per 100 sq m is different from $250, you will need to adjust the formula accordingly.
If the roller takes a different number of revolutions to level the playground, you will need to adjust the formula accordingly.