Sure, let's work through this step-by-step:
### Part (a): Finding the Volume of the Cone
The formula for the volume [tex]\( V \)[/tex] of a cone is given by:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height of the cone.
Given:
- Radius [tex]\( r = 39 \)[/tex] cm
- Height [tex]\( h = 52 \)[/tex] cm
Substituting these values into the formula, we get:
[tex]\[ V = \frac{1}{3} \pi (39)^2 (52) \][/tex]
Now, let's simplify the expression:
[tex]\[ 39^2 = 1521 \][/tex]
[tex]\[ 39^2 \cdot 52 = 1521 \cdot 52 = 79132 \][/tex]
So,
[tex]\[ V = \frac{1}{3} \pi \cdot 79132 = \frac{79132}{3} \pi \][/tex]
This simplifies to:
[tex]\[ V = 26377.333 \pi \][/tex]
Therefore, the volume of the cone in terms of [tex]\( \pi \)[/tex] is:
[tex]\[ V = 26377.333 \pi \][/tex]
### Part (b): Volume of the Cone with Half the Radius
When the radius is halved, the new radius [tex]\( r' \)[/tex] is:
[tex]\[ r' = \frac{39}{2} = 19.5 \][/tex] cm
The height remains the same, [tex]\( h = 52 \)[/tex] cm.
Now, using the same volume formula for cones:
[tex]\[ V' = \frac{1}{3} \pi (r')^2 h \][/tex]
Substituting the new radius:
[tex]\[ V' = \frac{1}{3} \pi (19.5)^2 (52) \][/tex]
Let's calculate [tex]\( (19.5)^2 \)[/tex]:
[tex]\[ 19.5^2 = 380.25 \][/tex]
Next, we find:
[tex]\[ 380.25 \cdot 52 = 19773 \][/tex]
So,
[tex]\[ V' = \frac{1}{3} \pi \cdot 19773 = \frac{19773}{3} \pi \][/tex]
This simplifies to:
[tex]\[ V' = 6591 \pi \][/tex]
Now, let's compare the two volumes:
- Original volume [tex]\( V = 26377.333 \pi \)[/tex]
- New volume [tex]\( V' = 6591 \pi \)[/tex]
To see how the volume has changed, we take the ratio of the new volume to the original volume:
[tex]\[ \text{Ratio} = \frac{V'}{V} = \frac{6591 \pi}{26377.333 \pi} = \frac{6591}{26377.333} \approx \frac{1}{4} \][/tex]
Therefore, when the radius is halved and the height remains the same, the volume of the cone is reduced to approximately one-fourth of its original volume.
