To find the volume of the perfume bottle using the formula for the volume of a cone, we'll first identify the dimensions we have:
- The height (h) is the measurement "up and down," which is 12.4 cm.
- The diameter (side to side) is 6.2 cm, so the radius (r) will be half of this diameter.
First, we calculate the radius:
\[ r = \frac{\text{diameter}}{2} = \frac{6.2 , \text{cm}}{2} = 3.1 , \text{cm} \]
Now we can use the volume formula:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting the values and using \( \pi \approx 3.14 \):
\[ V = \frac{1}{3} \times 3.14 \times (3.1 , \text{cm})^2 \times 12.4 , \text{cm \]
First, calculate \( r^2 \):
\[ r^2 = (3.1 , \text{cm})^2 = 9.61 , \text{cm}^2 \]
Now, substitute \( r^2 \) into the volume formula:
\[ V = \frac{1}{3} \times 3.14 \times 9.61 , \text{cm}^2 \times 12.4 , \text{cm} \]
Calculating \( 3.14 \times 9.61 \):
\[ 3.14 \times 9.61 \approx 30.1584 \]
Now, substitute this value back into the volume formula:
\[ V = \frac{1}{3} \times 30.1584 , \text{cm}^2 \times 12.4 , \text{cm} \]
Calculating \( 30.1584 \times 12.4 \):
\[ 30.1584 \times 12.4 \approx 373.96176 \]
Now, divide by 3 to find the volume:
\[ V \approx \frac{373.96176}{3} \approx 124.65392 , \text{cm}^3 \]
Thus, the volume of the perfume bottle is approximately:
\[ \boxed{124.65 , \text{cm}^3} \]