To find the height of the new cone with a radius of 4 inches that has the same volume as the existing cone with a radius of 3 inches and a height of 6 inches, we can use the formula for the volume of a cone:
\[ V = \frac{1}{3} \pi r^2 h \]
Step 1: Calculate the volume of the existing cone.
For the original cone:
- Radius \( r_1 = 3 \) inches
- Height \( h_1 = 6 \) inches
Calculating the volume \( V_1 \):
\[ V_1 = \frac{1}{3} \pi (3)^2 (6) \] \[ V_1 = \frac{1}{3} \pi (9) (6) \] \[ V_1 = \frac{1}{3} \pi (54) \] \[ V_1 = 18\pi \text{ cubic inches} \]
Step 2: Set up the equation for the new cone.
For the new cone:
- Radius \( r_2 = 4 \) inches
- Height \( h_2 \) (unknown)
The volume \( V_2 \) is given as:
\[ V_2 = \frac{1}{3} \pi (4)^2 h_2 \] \[ V_2 = \frac{1}{3} \pi (16) h_2 \] \[ V_2 = \frac{16}{3} \pi h_2 \]
Step 3: Set the volumes equal to each other.
Since both cones have the same volume:
\[ 18\pi = \frac{16}{3} \pi h_2 \]
Step 4: Solve for \( h_2 \).
Divide both sides by \( \pi \):
\[ 18 = \frac{16}{3} h_2 \]
Multiply both sides by 3 to eliminate the fraction:
\[ 54 = 16 h_2 \]
Now, divide both sides by 16:
\[ h_2 = \frac{54}{16} \] \[ h_2 = 3.375 \text{ inches} \]
Rounding to the nearest inch, we get:
\[ h_2 \approx 3 \text{ inches} \]
Conclusion: The height of the new cone will be 3 inches.
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