Question

an ice cream shop offers two types of ice cream cones that hold the same volume. the waffle cone holds 12 ounces and is 5 inches tall. the sugar cone holds 12 ounces and is 8 inches tall. which cone has a larger radius? waffle cone sugar cone they have the same radius

Explanation:

The volume of a cone is given by the formula \( V = \frac{1}{3}\pi r^2 h \), where \( r \) is the radius and \( h \) is the height. Since both cones have the same volume (\( V = 12 \) ounces, assuming volume is proportional to the amount they hold), we can set up the equation for each cone and solve for \( r \).

Step 1: Recall the volume formula for a cone

The volume \( V \) of a cone is \( V = \frac{1}{3}\pi r^2 h \). We can rearrange this formula to solve for \( r \):
\[ r = \sqrt{\frac{3V}{\pi h}} \]

Step 2: Calculate the radius of the waffle cone

For the waffle cone, \( V = 12 \) (ounces, volume) and \( h = 5 \) (inches). Plugging these values into the formula for \( r \):
\[ r_{\text{waffle}} = \sqrt{\frac{3 \times 12}{\pi \times 5}} = \sqrt{\frac{36}{5\pi}} \approx \sqrt{\frac{36}{15.708}} \approx \sqrt{2.292} \approx 1.514 \]

Step 3: Calculate the radius of the sugar cone

For the sugar cone, \( V = 12 \) (ounces, volume) and \( h = 8 \) (inches). Plugging these values into the formula for \( r \):
\[ r_{\text{sugar}} = \sqrt{\frac{3 \times 12}{\pi \times 8}} = \sqrt{\frac{36}{8\pi}} \approx \sqrt{\frac{36}{25.133}} \approx \sqrt{1.432} \approx 1.197 \]

Step 4: Compare the radii

Comparing \( r_{\text{waffle}} \approx 1.514 \) and \( r_{\text{sugar}} \approx 1.197 \), we see that the waffle cone has a larger radius.

Answer:

Waffle cone