Bot GPT 4-o mini The volume \( V \) of a cone can be calculated using the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.
Given the original cone has a height \( h = 35 \) cm and a radius \( r = 15 \) cm, we can calculate its volume:
\[ V = \frac{1}{3} \pi (15)^2 (35) \]
First, calculate \( (15)^2 \):
\[ (15)^2 = 225 \]
Now substituting this back into the volume equation:
\[ V = \frac{1}{3} \pi (225) (35) \]
Now, calculate \( 225 \times 35 \):
\[ 225 \times 35 = 7875 \]
Now we can find the volume:
\[ V = \frac{1}{3} \pi (7875) = 2625 \pi \]
Now, when we dilate the cone with a scale factor of \( \frac{1}{3} \), both the radius and the height are multiplied by \( \frac{1}{3} \). Therefore, the new radius and new height will be:
- New radius \( r' = \frac{1}{3} \times 15 = 5 \) cm
- New height \( h' = \frac{1}{3} \times 35 \approx 11.67 \) cm
Now we calculate the volume of the new cone. Using the same volume formula:
\[ V' = \frac{1}{3} \pi (r')^2 (h') \]
Substituting in the new radius \( r' = 5 \) and new height \( h' \approx 11.67 \):
\[ V' = \frac{1}{3} \pi (5)^2 \left(11.67\right) \]
Calculating \( (5)^2 \):
\[ (5)^2 = 25 \]
Now substituting this back into the volume equation:
\[ V' = \frac{1}{3} \pi (25) (11.67) \]
Calculating \( 25 \times 11.67 \):
\[ 25 \times 11.67 \approx 291.75 \]
Now, substituting this back to find \( V' \):
\[ V' \approx \frac{1}{3} \pi (291.75) \approx 97.25 \pi \]
Rounding to the nearest tenth, the volume of the new similar cone is:
\[ V' \approx 97.3 \pi , \text{cm}^3 \]
So the volume of the new similar cone is:
\[ \boxed{97.3 \pi} , \text{cm}^3 \]
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