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)

Responses

18 inches
18 inches

4 inches
4 inches

14 inches
14 inches

3 inches

1 answer

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

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

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

Where:

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

For the original cone:

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

Calculating the volume of the original cone:

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

Now we need to find the height of the new cone with a radius of 4 inches, which must have the same volume of \( 18 \pi \) cubic inches.

Let \( h' \) be the height of the new cone. We can set up the equation:

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

Setting this equal to the volume of the original cone:

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

We can simplify this equation:

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

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

\[ h' = 18 \times \frac{3}{16} \] \[ h' = \frac{54}{16} \] \[ h' = 3.375 \]

Rounding \( 3.375 \) to the nearest whole number gives us \( 3 \) inches. Thus, the height of the new cone is:

3 inches.