The Carousel on the National Mall in Washington, DC, has 4 rings of horses. Andre is riding on the inner ring, which has a radius of 9 ft . Lin is riding on the outer ring, which is 8 ft farther from the center than the inner ring. a In one rotation of the carousel, how much farther does Lin travel than Andre?

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UpStudy AI · Accepted Answer

To find out how much farther Lin travels than Andre in one rotation of the carousel, we need to calculate the circumference of both the inner and outer rings and then find the difference between the two circumferences.

1. **Identify the radii:**
   - The radius of the inner ring (Andre's position) is $$ r_A = 9 $$ ft.
   - The radius of the outer ring (Lin's position) is $$ r_L = r_A + 8 $$ ft.

2. **Calculate the radius of the outer ring:**
   $$
   r_L = 9 + 8 = 17 \text{ ft}
   $$

3. **Calculate the circumference of the inner ring (Andre's path):**
   The formula for the circumference $$ C $$ of a circle is given by:
   $$
   C = 2 \pi r
   $$
   Therefore, the circumference of the inner ring is:
   $$
   C_A = 2 \pi r_A = 2 \pi (9)
   $$

4. **Calculate the circumference of the outer ring (Lin's path):**
   Similarly, the circumference of the outer ring is:
   $$
   C_L = 2 \pi r_L = 2 \pi (17)
   $$

5. **Find the difference in distance traveled:**
   The difference in distance traveled by Lin and Andre is:
   $$
   \Delta C = C_L - C_A
   $$

Now, let's perform the calculations.

I'll calculate the circumferences and the difference.
Calculate the value by following steps:
- step0: Calculate:
  $$2\pi \times 17$$
- step1: Multiply the terms:
  $$34\pi $$
Calculate or simplify the expression $$ 2 * \pi * 9 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$2\pi \times 9$$
- step1: Multiply the terms:
  $$18\pi $$
Calculate or simplify the expression $$ 106.81415 - 56.548668 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$106.81415-56.548668$$
- step1: Subtract the numbers:
  $$50.265482$$
Now that we have calculated the circumferences:

1. **Circumference of the inner ring (Andre's path):**
   $$
   C_A \approx 56.55 \text{ ft}
   $$

2. **Circumference of the outer ring (Lin's path):**
   $$
   C_L \approx 106.81 \text{ ft}
   $$

3. **Difference in distance traveled:**
   $$
   \Delta C \approx 106.81 - 56.55 \approx 50.27 \text{ ft}
   $$

Thus, in one rotation of the carousel, Lin travels approximately **50.27 feet farther** than Andre.
In one rotation, Lin travels about 50.27 feet farther than Andre.

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