Understanding the Cylinder Problem

We are given the volume ($V$) and height ($h$) of a solid cylinder and asked to find its total surface area (TSA). We need to use the provided value of pi ($\pi$) and round our final answer to the nearest integer.

• Given Volume ($V$): 5852 cm³
• Given Height ($h$): 38 cm
• Value of Pi ($\pi$): $\frac{22}{7}$
• Goal: Calculate Total Surface Area (TSA)

Calculating Cylinder Radius from Volume

First, we need to find the radius ($r$) of the cylinder. The formula for the volume of a cylinder is:

$V = \pi r^2 h$

Let's substitute the given values into the formula:

$5852 = \frac{22}{7} \times r^2 \times 38$

To find $r^2$, we rearrange the equation:

$r^2 = \frac{5852 \times 7}{22 \times 38}$

First, calculate the denominator: $22 \times 38 = 836$.

Now, divide the numerator by the denominator:

$r^2 = \frac{5852 \times 7}{836}$

Let's perform the division: $5852 \div 836 = 7$.

So, $r^2 = 7 \times 7$.

$r^2 = 49$

Now, find the radius by taking the square root of $r^2$:

$r = \sqrt{49}$

$r = 7$ cm

Calculating Cylinder Total Surface Area

Now that we have the radius ($r = 7$ cm) and the height ($h = 38$ cm), we can calculate the total surface area (TSA) of the cylinder. The formula for the TSA of a cylinder is:

$TSA = 2\pi r (h + r)$

Substitute the values of $\pi$, $r$, and $h$ into the formula:

$TSA = 2 \times \frac{22}{7} \times 7 \times (38 + 7)$

Simplify the expression:

$TSA = 2 \times 22 \times (45)$ (since $\frac{7}{7}$ cancels out)

$TSA = 44 \times 45$

Perform the multiplication:

$44 \times 45 = 1980$

The total surface area of the cylinder is 1980 cm². Since the question asks to round to the nearest integer, and our answer is already an integer, no further rounding is needed.

Final Answer Check

The calculated total surface area is 1980 cm², which matches one of the options provided.