Understanding the Cylinder Problem

The question asks us to find the total surface area (TSA) of a solid cylinder. We are given the volume (V) of the cylinder as $55,440 \text{ cm}^3$ and its height (h) as 40 cm. We are also told to use the value $\pi = \frac{22}{7}$.

Key Formulas for Cylinders

To solve this problem, we need two main formulas related to cylinders:

• Volume of a Cylinder (V): $V = \pi r^2 h$, where $r$ is the radius and $h$ is the height.
• Total Surface Area of a Cylinder (TSA): $TSA = 2\pi r (r+h)$, where $r$ is the radius and $h$ is the height.

We have the volume and height, so we can use the volume formula first to find the radius ($r$). Once we have the radius, we can use the TSA formula to calculate the total surface area.

Step-by-Step Calculation

Step 1: Finding the Radius (r)

We start with the formula for the volume of a cylinder: $V = \pi r^2 h$

Now, substitute the given values: $V = 55,440$, $h = 40$, and $\pi = \frac{22}{7}$. $55,440 = \frac{22}{7} \times r^2 \times 40$

To find $r^2$, we rearrange the equation: $r^2 = \frac{55,440 \times 7}{22 \times 40}$

Let's simplify the calculation: $r^2 = \frac{55,440}{880} \times 7$ First, divide 55,440 by 880: $ \frac{55440}{880} = \frac{5544}{88} = 63 $ So, the equation becomes: $r^2 = 63 \times 7$ $r^2 = 441$

Now, find the radius $r$ by taking the square root of 441: $r = \sqrt{441}$ $r = 21 \text{ cm}$ So, the radius of the cylinder is 21 cm.

Step 2: Calculating the Total Surface Area (TSA)

Now that we have the radius ($r = 21 \text{ cm}$) and the height ($h = 40 \text{ cm}$), we can calculate the total surface area using the formula: $TSA = 2\pi r (r+h)$

Substitute the values: $\pi = \frac{22}{7}$, $r = 21$, and $h = 40$. $TSA = 2 \times \frac{22}{7} \times 21 \times (21 + 40)$

First, simplify the term $2 \times \frac{22}{7} \times 21$: $2 \times 22 \times \frac{21}{7} = 2 \times 22 \times 3 = 132$

Next, calculate the term in the parenthesis: $(21 + 40) = 61$

Now, multiply these results together: $TSA = 132 \times 61$ Performing the multiplication: $132 \times 61 = 8052$

Therefore, the total surface area of the solid cylinder is $8052 \text{ cm}^2$.