An ice cream shop currently sells soft-serve ice cream in cones that have a radius of 3 inches and a height of 6 inches. The shop wants to offer a wider cone for their hard-serve ice cream that has the same volume as their existing cone but a wider radius of 4 inches.

What will be the height, to the nearest inch, of the new cone? (1 point)
• 4 inches
• 14 inches
• 3 inches
• 18 inches

1 answer

To find the height of the new cone that has the same volume as the existing cone but a wider radius, we first need to calculate the volume of the existing cone.

The formula for the volume \( V \) of a cone is given by:

\[ V = \frac{1}{3} \pi r^2 h \]

Where:

  • \( r \) is the radius of the cone,
  • \( h \) is the height of the cone.

For the existing soft-serve ice cream cone:

  • Radius (\( r \)) = 3 inches
  • Height (\( h \)) = 6 inches

Calculating the volume of the existing cone:

\[ V = \frac{1}{3} \pi (3^2) (6) = \frac{1}{3} \pi (9) (6) = \frac{1}{3} \pi (54) = 18\pi \]

Now, we want to find the height of the new cone with a radius of 4 inches that has the same volume of \( 18\pi \).

Let \( h' \) be the height of the new cone. Using the volume formula:

\[ V = \frac{1}{3} \pi (r')^2 h' \]

Where \( r' = 4 \) inches, and we set this equal to the volume of the existing cone:

\[ 18\pi = \frac{1}{3} \pi (4^2) h' \]

Simplifying this equation:

\[ 18\pi = \frac{1}{3} \pi (16) h' \]

Cancel \( \pi \) from both sides:

\[ 18 = \frac{16}{3} h' \]

To find \( h' \), multiply both sides by \( \frac{3}{16} \):

\[ h' = 18 \cdot \frac{3}{16} = \frac{54}{16} = \frac{27}{8} \]

Now, convert \( \frac{27}{8} \) into a decimal:

\[ \frac{27}{8} = 3.375 \]

The nearest inch to \( 3.375 \) is \( 3 \) inches.

Therefore, the height of the new cone is 3 inches.