To create a sketch of the new Ferris wheel with the specified conditions, we need to focus on the following details:

1. Arc length of 25π/3 feet for part of the Ferris wheel.
2. The center must be at least 60 feet above the ground.
3. We need to formulate the equation of the Ferris wheel.

Step 1: Relationship Between Arc Length and Radius

The arc length ($s$) of a circle is related to the radius ($r$) and the central angle ($\theta$) by the formula: $s = r \cdot \theta$ where $s$ is the arc length, $r$ is the radius, and $\theta$ is the central angle in radians.

From the problem, we are given: $s = \frac{25\pi}{3} \, \text{feet}.$

Step 2: Solving for the Radius of the Ferris Wheel

If we assume that the arc length given corresponds to part of the circle, we would need to make an assumption about the angle to solve for the radius. For simplicity, let’s assume the arc corresponds to half of the Ferris wheel, meaning the central angle is $\pi$ radians (half a circle). This means the arc length would cover half the circumference of the wheel. Therefore, we can set up the equation as: $\frac{25\pi}{3} = r \cdot \pi$

Solving for $r$: $r = \frac{25\pi}{3\pi} = \frac{25}{3} \, \text{feet}.$

Thus, the radius of the Ferris wheel is $\frac{25}{3}$ feet.

Step 3: Setting the Equation for the Ferris Wheel

The general equation for a Ferris wheel (or any circle) with center $(h, k)$ and radius $r$ is given by: $(x - h)^2 + (y - k)^2 = r^2$

Here:

• $r = \frac{25}{3}$,
• $h$ represents the horizontal center, and
• $k$ represents the vertical center.

Step 4: Center of the Ferris Wheel

We are told that the center of the Ferris wheel must be at least 60 feet above the ground. Therefore, the vertical coordinate of the center $k$ must be at least 60 feet.

If the center is exactly 60 feet from the ground, then $k = 60$. Thus, the equation of the Ferris wheel becomes: $(x - h)^2 + (y - 60)^2 = \left( \frac{25}{3} \right)^2$

Step 5: Simplifying the Equation

Simplify the radius squared: $\left( \frac{25}{3} \right)^2 = \frac{625}{9}$

Thus, the equation of the Ferris wheel is: $(x - h)^2 + (y - 60)^2 = \frac{625}{9}$

This is the equation of the Ferris wheel with radius $\frac{25}{3}$ feet and the center 60 feet above the ground.

Conclusion:

The equation of the new Ferris wheel is: $(x - h)^2 + (y - 60)^2 = \frac{625}{9}$ where $h$ represents the horizontal center of the Ferris wheel, and the center of the Ferris wheel is 60 feet above the ground.

Would you like further clarification on any of the steps?

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Here are 5 related questions for practice:

1. What is the full circumference of the Ferris wheel based on its radius?
2. How would the equation change if the center were 70 feet above the ground?
3. If the Ferris wheel makes a full revolution in 5 minutes, what is the angular velocity in radians per minute?
4. How long would it take for the Ferris wheel to complete a quarter turn?
5. What is the maximum height a rider on this Ferris wheel would reach?

Tip: When solving for arc length, always ensure you know the angle in radians to use the correct formula.