Directions: Use π = 22/7.

A conical vessel has a volume of 3234 cm³ and a base diameter of 28 cm. A cylindrical vessel has the same height as the cone, and its radius is $\frac{5}{7}$ of the cone’s radius. What is the volume of the cylindrical vessel?

The correct answer is 4950 cm³.

Step-by-step explanation:

Step 1: Find the radius and height of the conical vessel.

We are given that the base diameter of the conical vessel is 28 cm.

Radius of the cone ($r_{\text{cone}}$) = $\frac{\text{Diameter}}{2}$ = $\frac{28}{2}$ = 14 cm

The formula for the volume of a cone is:

$V_{\text{cone}}=\frac{1}{3}\pi r_{\text{cone}}^{2}h$

Given that $V_{\text{cone}}=3234$ cm³ and $\pi =\frac{22}{7}$, we substitute these values into the formula to calculate the height ($h$):

$3234=\frac{1}{3}\times \frac{22}{7}\times {14}^2\times h$

$3234=\frac{1}{3}\times \frac{22}{7}\times 196\times h$

Simplifying $\frac{196}{7}=28$:

$3234=\frac{1}{3}\times 22\times 28\times h$

$3234=\frac{616}{3}\times h$

$h=\frac{3234\times 3}{616}$

$h=\frac{9702}{616}=15.75$ cm

Step 2: Find the radius of the cylindrical vessel.

The problem states that the radius of the cylindrical vessel ($r_{\text{cyl}}$) is $\frac{5}{7}$ of the cone's radius:

$r_{\text{cyl}}=\frac{5}{7}\times r_{\text{cone}}$

$r_{\text{cyl}}=\frac{5}{7}\times 14=10$ cm

Step 3: Calculate the volume of the cylindrical vessel.

The cylindrical vessel has the same height as the cone, so $h=15.75$ cm.

The formula for the volume of a cylinder is:

$V_{\text{cyl}}=\pi r_{\text{cyl}}^{2}h$

Substituting $\pi =\frac{22}{7}$, $r_{\text{cyl}}=10$ cm, and $h=15.75$ cm:

$V_{\text{cyl}}=\frac{22}{7}\times {10}^2\times 15.75$

$V_{\text{cyl}}=\frac{22}{7}\times 100\times 15.75$

Dividing 15.75 by 7 gives $\frac{15.75}{7}=2.25$:

$V_{\text{cyl}}=22\times 100\times 2.25$

$V_{\text{cyl}}=22\times 225=4950$ cm³

Thus, the volume of the cylindrical vessel is 4950 cm³.