To determine the ratio of the diameter and height of a cylindrical pillar, we need to use the given curved surface area and volume. Let's denote the radius of the cylindrical pillar as $\text{r}$ and its height as $\text{h}$.

Cylindrical Pillar Formulas

First, recall the standard formulas for a cylinder:

• Curved Surface Area (CSA): The formula for the curved surface area of a cylinder is $\text{CSA} = 2\pi\text{rh}$.
• Volume (V): The formula for the volume of a cylinder is $\text{V} = \pi\text{r}^2\text{h}$.
• Diameter (d): The diameter of a cylinder is twice its radius, so $\text{d} = 2\text{r}$.

Given Data for the Cylindrical Pillar

From the question, we are provided with the following information:

• Curved Surface Area (CSA) $= 264 \, \text{m}^2$
• Volume (V) $= 924 \, \text{m}^3$

Formulating Equations

Using the given data and the formulas, we can set up two equations:

1. $\text{Equation 1: } 2\pi\text{rh} = 264$
2. $\text{Equation 2: } \pi\text{r}^2\text{h} = 924$

Calculating the Radius of the Pillar

To find the radius $\text{r}$, we can divide Equation 2 by Equation 1. This step helps eliminate the height $\text{h}$ and $\pi$ from the equations, allowing us to solve directly for $\text{r}$.

$\frac{\text{Volume}}{\text{Curved Surface Area}} = \frac{\pi\text{r}^2\text{h}}{2\pi\text{rh}}$

Substitute the given values:

$\frac{924}{264} = \frac{\pi\text{r}^2\text{h}}{2\pi\text{rh}}$

Simplify the right side of the equation:

$\frac{924}{264} = \frac{\text{r}}{2}$

Now, solve for $\text{r}$:

$\text{r} = 2 \times \frac{924}{264}$

To simplify the fraction $\frac{924}{264}$, we can divide both the numerator and the denominator by common factors. Both are divisible by 12:

$924 \div 12 = 77$

$264 \div 12 = 22$

So, the fraction becomes $\frac{77}{22}$. This can be further simplified by dividing both by 11:

$\frac{77}{22} = \frac{7}{2}$

Now, substitute this simplified fraction back into the equation for $\text{r}$:

$\text{r} = 2 \times \frac{7}{2}$

$\text{r} = 7 \, \text{m}$

So, the radius of the cylindrical pillar is $7 \, \text{m}$.

Calculating the Height of the Pillar

Next, we use Equation 1 (the curved surface area formula) and the value of $\text{r}$ we just found to calculate the height $\text{h}$.

$2\pi\text{rh} = 264$

Substitute $\text{r} = 7 \, \text{m}$ and use $\pi = \frac{22}{7}$:

$2 \times \frac{22}{7} \times 7 \times \text{h} = 264$

$2 \times 22 \times \text{h} = 264$

$44\text{h} = 264$

Now, solve for $\text{h}$:

$\text{h} = \frac{264}{44}$

$\text{h} = 6 \, \text{m}$

So, the height of the cylindrical pillar is $6 \, \text{m}$.

Determining the Diameter of the Pillar

The diameter $\text{d}$ is twice the radius $\text{r}$:

$\text{d} = 2\text{r}$

Substitute $\text{r} = 7 \, \text{m}$:

$\text{d} = 2 \times 7$

$\text{d} = 14 \, \text{m}$

So, the diameter of the cylindrical pillar is $14 \, \text{m}$.

Ratio of Diameter and Height

Finally, we need to find the ratio of the diameter to the height ($\text{d} : \text{h}$).

$\text{d} : \text{h} = 14 : 6$

To simplify the ratio, divide both numbers by their greatest common divisor, which is 2:

$\frac{14}{2} : \frac{6}{2} = 7 : 3$

Thus, the ratio of the diameter and its height is $7 : 3$.