To find the volume of a cone after applying a scale factor, we can use the formula for the volume of a cone and the property of dilations.
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 base and \( h \) is the height of the cone.
When a cone is dilated by a scale factor \( k \), the new radius \( r' \) and height \( h' \) of the cone will be: \[ r' = k \cdot r \] \[ h' = k \cdot h \] The volume of the dilated cone \( V' \) then will be: \[ V' = \frac{1}{3} \pi (r')^2 (h') = \frac{1}{3} \pi (k \cdot r)^2 (k \cdot h) = \frac{1}{3} \pi k^3 r^2 h \] This shows that the volume increases by the cube of the scale factor \( k^3 \).
Assuming you know the original volume \( V \) of the cone, the volume after dilation will be: \[ V' = k^3 \cdot V \]
In your case, the scale factor \( k = 13 \), hence \( k^3 = 13^3 = 2197 \).
Now, if you provide the original volume of the cone \( V \), you can calculate the new volume as follows: \[ V' = 2197 V \]
Without the original volume value, I cannot provide a numerical answer. If you provide the original radius and height, or the original volume of the cone, I can compute it for you.
It looks like you provided a list of options (30.86 π cm³, 41.72 π cm³, etc.). After calculating the volume based on the original dimensions or volume, you can then check which option corresponds to \( V' \).
If you'd like further assistance, please provide the original dimensions of the cone, or its volume prior to dilation.
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