A Waterwheel Has A Radius Of 4 Feet, And The Center Of The Wheel Is 1 Foot Under The Waterline. You Notice A Yellow Mark At The Top Of The Wheel. If The Wheel Rotates $\frac{\pi}{3}$ Radians, How Far Above The Water Is The Yellow Mark?A. 1

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Introduction

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A waterwheel is a device that uses the energy of flowing water to perform various tasks. In this problem, we are given a waterwheel with a radius of 4 feet and the center of the wheel is 1 foot under the waterline. We need to find the height of the yellow mark at the top of the wheel when it rotates by $\frac{\pi}{3}$ radians.

Understanding the Problem

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To solve this problem, we need to understand the concept of circular motion and the relationship between the radius of a circle and its circumference. The waterwheel is rotating, which means that the yellow mark is moving in a circular path. We can use the concept of arc length to find the distance traveled by the yellow mark.

Calculating the Arc Length

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The arc length of a circle is given by the formula:

$$s = r\theta$$

where $s$ is the arc length, $r$ is the radius of the circle, and $\theta$ is the angle of rotation in radians.

In this problem, the radius of the waterwheel is 4 feet, and the angle of rotation is $\frac{\pi}{3}$ radians. Plugging these values into the formula, we get:

$$s = 4 \times \frac{\pi}{3} = \frac{4\pi}{3}$$

Finding the Height of the Yellow Mark

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Now that we have the arc length, we need to find the height of the yellow mark above the waterline. To do this, we need to use the concept of right triangles and the Pythagorean theorem.

Let's draw a right triangle with the yellow mark as the vertex opposite the angle of rotation. The radius of the waterwheel is the hypotenuse of the triangle, and the distance from the center of the wheel to the waterline is the other leg of the triangle.

Using the Pythagorean theorem, we can write:

$$a^2 + b^2 = c^2$$

where $a$ and $b$ are the legs of the triangle, and $c$ is the hypotenuse.

In this problem, the radius of the waterwheel is 4 feet, and the distance from the center of the wheel to the waterline is 1 foot. Plugging these values into the equation, we get:

$$1^2 + b^2 = 4^2$$

Simplifying the equation, we get:

$$1 + b^2 = 16$$

Subtracting 1 from both sides, we get:

$$b^2 = 15$$

Taking the square root of both sides, we get:

$$b = \sqrt{15}$$

Calculating the Height of the Yellow Mark

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Now that we have the value of $b$, we can use it to find the height of the yellow mark above the waterline. The height of the yellow mark is equal to the distance from the center of the wheel to the waterline plus the value of $b$.

So, the height of the yellow mark is:

$$h = 1 + \sqrt{15}$$

Conclusion

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In this problem, we used the concept of circular motion and the Pythagorean theorem to find the height of the yellow mark above the waterline. We calculated the arc length of the waterwheel and used it to find the value of $b$. Finally, we used the value of $b$ to find the height of the yellow mark.

The final answer is $\boxed{1 + \sqrt{15}}$.

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Introduction

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In our previous article, we solved a problem involving a waterwheel with a radius of 4 feet and the center of the wheel 1 foot under the waterline. We found the height of the yellow mark at the top of the wheel when it rotates by $\frac{\pi}{3}$ radians. In this article, we will answer some frequently asked questions related to the problem.

Q&A

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Q: What is the formula for calculating the arc length of a circle?

A: The formula for calculating the arc length of a circle is:

$$s = r\theta$$

where $s$ is the arc length, $r$ is the radius of the circle, and $\theta$ is the angle of rotation in radians.

Q: How do I calculate the height of the yellow mark above the waterline?

A: To calculate the height of the yellow mark above the waterline, you need to use the concept of right triangles and the Pythagorean theorem. Let's draw a right triangle with the yellow mark as the vertex opposite the angle of rotation. The radius of the waterwheel is the hypotenuse of the triangle, and the distance from the center of the wheel to the waterline is the other leg of the triangle.

Using the Pythagorean theorem, you can write:

$$a^2 + b^2 = c^2$$

where $a$ and $b$ are the legs of the triangle, and $c$ is the hypotenuse.

In this problem, the radius of the waterwheel is 4 feet, and the distance from the center of the wheel to the waterline is 1 foot. Plugging these values into the equation, you get:

$$1^2 + b^2 = 4^2$$

Simplifying the equation, you get:

$$1 + b^2 = 16$$

Subtracting 1 from both sides, you get:

$$b^2 = 15$$

Taking the square root of both sides, you get:

$$b = \sqrt{15}$$

The height of the yellow mark is equal to the distance from the center of the wheel to the waterline plus the value of $b$.

Q: What is the value of $b$ in the problem?

A: The value of $b$ is $\sqrt{15}$.

Q: How do I find the height of the yellow mark above the waterline?

A: To find the height of the yellow mark above the waterline, you need to add the value of $b$ to the distance from the center of the wheel to the waterline.

So, the height of the yellow mark is:

$$h = 1 + \sqrt{15}$$

Q: What is the final answer to the problem?

A: The final answer to the problem is $\boxed{1 + \sqrt{15}}$.

Conclusion

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In this article, we answered some frequently asked questions related to the problem involving a waterwheel with a radius of 4 feet and the center of the wheel 1 foot under the waterline. We calculated the arc length of the waterwheel and used it to find the value of $b$. Finally, we used the value of $b$ to find the height of the yellow mark above the waterline.

Additional Resources

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If you want to learn more about circular motion and the Pythagorean theorem, you can check out the following resources:

• Circular Motion
• Pythagorean Theorem
• Right Triangle

Final Thoughts

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We hope this article has been helpful in answering your questions about the problem involving a waterwheel with a radius of 4 feet and the center of the wheel 1 foot under the waterline. If you have any more questions or need further clarification, please don't hesitate to ask.