Emilia is riding a horse tied to a pole with a 5 m long rope. If she travels around the circular path twice, how far has she traveled?
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2. A basketball has a circumference of 29.5 inches. What is the radius?
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3. If the basketball above is placed into a box that is 12 in x 12 in x 12 in, how much space is left within the box?

### Problem 1: Emilia Riding a Horse

To determine how far Emilia has traveled, we need to calculate the circumference of the circular path and then multiply it by the number of times she travels around the path.

The formula for the circumference of a circle is:

\[ C = 2\pi r \]

where \( r \) is the radius of the circle. Given that the rope is 5 meters long, the radius \( r \) is 5 meters.

\[ C = 2\pi \times 5 = 10\pi \text{ meters} \]

Since she travels around the path twice, the total distance traveled is:

\[ \text{Total Distance} = 2 \times 10\pi = 20\pi \text{ meters} \]

Approximating \(\pi\) as 3.14:

\[ \text{Total Distance} \approx 20 \times 3.14 = 62.8 \text{ meters} \]

So, Emilia has traveled approximately 62.8 meters.

### Problem 2: Radius of the Basketball

The circumference \( C \) of the basketball is given as 29.5 inches. The formula for the circumference of a circle is:

\[ C = 2\pi r \]

Solving for the radius \( r \):

\[ r = \frac{C}{2\pi} = \frac{29.5}{2\pi} \]

Approximating \(\pi\) as 3.14:

\[ r \approx \frac{29.5}{2 \times 3.14} \approx \frac{29.5}{6.28} \approx 4.7 \text{ inches} \]

So, the radius of the basketball is approximately 4.7 inches.

### Problem 3: Space Left in the Box

First, calculate the volume of the basketball. The formula for the volume \( V \) of a sphere is:

\[ V = \frac{4}{3}\pi r^3 \]

Using the radius \( r \approx 4.7 \) inches:

\[ V \approx \frac{4}{3} \pi (4.7)^3 \approx \frac{4}{3} \pi (103.823) \approx 435.06 \text{ cubic inches} \]

Next, calculate the volume of the box. Since the box is a cube with side length 12 inches, its volume \( V_{\text{box}} \) is:

\[ V_{\text{box}} = 12 \times 12 \times 12 = 1728 \text{ cubic inches} \]

The space left within the box is the difference between the volume of the box and the volume of the basketball:

\[ \text{Space Left} = V_{\text{box}} - V \approx 1728 - 435.06 \approx 1292.94 \text{ cubic inches} \]

So, approximately 1292.94 cubic inches of space is left within the box.

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Would you like more details or have any questions about these solutions? 

Here are 5 additional questions you might find interesting:

1. What is the area of the circular path Emilia travels?
2. How many complete rotations will Emilia need to travel 100 meters?
3. What is the volume of a cylinder with a radius of 4.7 inches and a height of 12 inches?
4. If the rope Emilia's horse is tied to is lengthened to 10 meters, how far would she travel in two rotations?
5. How would the remaining space change if the box dimensions were increased to 15 in x 15 in x 15 in?

**Tip:** When dealing with geometry problems, always identify and use the correct formulas for areas, volumes, and perimeters to find accurate solutions.

Learn how to calculate the total distance Emilia traveled on a circular path tied to a pole and the remaining space in a box after placing a basketball inside. Solutions include calculating circumference, radius, and volume using basic geometric formulas. Suitable for students in Grades 9-12.

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