Question

A refill bottle of dish soap was purchased to replenish an existing bottle. The existing bottle has a diameter of 6 cm and a height of 1623 cm . The refill bottle is 24 cm tall with a radius of 5 cm. Determine the number of times the existing bottle can be filled by the refill.(1 point) Responses 2.4 2.4 3 3 4 4 7

Answer 1

To determine how many times the existing bottle can be filled by the refill bottle, we need to calculate the volumes of both bottles and then divide the volume of the refill bottle by the volume of the existing bottle.

1. Volume of the existing bottle: The existing bottle is a cylinder, and its volume \( V \) can be calculated using the formula: \[ V = \pi r^2 h \] where \( r \) is the radius and \( h \) is the height.

   • The diameter of the existing bottle is 6 cm, so the radius \( r \) is: \[ r = \frac{6}{2} = 3 \text{ cm} \]
   • The height \( h \) is 1623 cm.

   Now, plug in the values: \[ V_{existing} = \pi (3^2)(1623) = \pi (9)(1623) = 14607\pi \text{ cm}^3 \]

2. Volume of the refill bottle: The refill bottle is also a cylinder, so we use the same volume formula.

   • The radius \( r \) of the refill bottle is 5 cm, and its height \( h \) is 24 cm.

   Now, calculate the volume: \[ V_{refill} = \pi (5^2)(24) = \pi (25)(24) = 600\pi \text{ cm}^3 \]

3. Calculate the number of times the existing bottle can be filled: To find this, divide the volume of the refill bottle by the volume of the existing bottle: \[ \text{Number of fillings} = \frac{V_{refill}}{V_{existing}} = \frac{600\pi}{14607\pi} = \frac{600}{14607} \] Simplifying: \[ \frac{600}{14607} \approx 0.0411 \]

4. Result: This result indicates that the refill bottle doesn't provide enough volume to fill the existing bottle even once.

It appears there was a misunderstanding in the casing of heights or the measurements. Please double-check the height of the existing bottle, as a value of 1623 cm seems unlikely for a dish soap container.

Answer 2

2.4
3
4
7

Answer 3

Let's double-check the calculations to ensure we're determining the number of times the existing bottle can be filled based on the provided options.

1. Volume of the existing bottle:

   • Diameter of the existing bottle = 6 cm, therefore radius \( r = 3 \) cm.
   • Height \( h = 1623 \) cm.

   Now calculate the volume: \[ V_{existing} = \pi r^2 h = \pi (3^2)(1623) = \pi (9)(1623) = 14607\pi \text{ cm}^3 \]

2. Volume of the refill bottle:

   • Radius of the refill bottle = 5 cm, height = 24 cm.

   Now calculate the volume: \[ V_{refill} = \pi r^2 h = \pi (5^2)(24) = \pi (25)(24) = 600\pi \text{ cm}^3 \]

3. Finding how many times the refill bottle can fill the existing bottle: \[ \text{Number of fillings} = \frac{V_{existing}}{V_{refill}} = \frac{14607\pi}{600\pi} \]

   • The \(\pi\) cancels out, so: \[ \text{Number of fillings} = \frac{14607}{600} \]

4. Calculate the number of fillings: \[ \frac{14607}{600} \approx 24.345 \]

This means the refill bottle can fill the existing bottle approximately 24 times.

Given the provided response choices of 2.4, 3, 4, and 7, those all seem too low given your dimensions. Please verify the dimensions provided because an existing height of 1623 cm for a dish soap bottle appears unusually large. If that height is indeed correct, then the refill can fill the existing bottle significantly more than any of the provided options. If the height is different (for example, if it's about 16.23 cm instead of 1623 cm), please let me know to re-evaluate.