The sum of the curved surface area and total surface area of a solid cylinder is 2068 cm 2. If radius of its base is 7 cm, then what is the volume of this cylinder ? (use π = 22/7)
Calculate Cylinder Volume: Problem Analysis
The problem asks us to find the volume of a solid cylinder. We are given the radius of the base and the sum of its curved surface area and total surface area. We need to use the relevant formulas for the surface areas and volume of a cylinder to solve this problem.
Given information:
- Sum of Curved Surface Area (CSA) and Total Surface Area (TSA) = 2068 cm2
- Radius of the base (r) = 7 cm
- Use $\pi = \frac{22}{7}$
To find:
- Volume of the cylinder
Cylinder Formulas Used
Here are the formulas we will use to solve this problem:
- Curved Surface Area (CSA) of a cylinder = $2\pi rh$
- Total Surface Area (TSA) of a solid cylinder = $2\pi rh + 2\pi r^2$ (CSA + Area of two circular bases)
- Volume of a cylinder = $\pi r^2h$
Where 'r' is the radius of the base and 'h' is the height of the cylinder.
Step-by-Step Solution to Find Cylinder Volume
Let the height of the cylinder be 'h' cm.
We are given that the sum of the curved surface area and the total surface area is 2068 cm2.
Sum of CSA and TSA = CSA + TSA
Substitute the formulas:
$(2\pi rh) + (2\pi rh + 2\pi r^2) = 2068$
Combine the terms:
$4\pi rh + 2\pi r^2 = 2068$
Now, substitute the given values for radius ($r=7$ cm) and $\pi = \frac{22}{7}$ into the equation:
$4 \times \frac{22}{7} \times 7 \times h + 2 \times \frac{22}{7} \times (7)^2 = 2068$
Simplify the terms:
$4 \times 22 \times h + 2 \times \frac{22}{7} \times 49 = 2068$
$88h + 2 \times 22 \times 7 = 2068$
$88h + 308 = 2068$
Now, we need to solve for the height 'h'. Subtract 308 from both sides of the equation:
$88h = 2068 - 308$
$88h = 1760$
Divide both sides by 88:
$h = \frac{1760}{88}$
$h = 20$ cm
So, the height of the cylinder is 20 cm.
Now that we have the radius ($r=7$ cm) and the height ($h=20$ cm), we can calculate the volume of the cylinder using the volume formula:
Volume = $\pi r^2h$
Substitute the values:
Volume = $\frac{22}{7} \times (7)^2 \times 20$
Volume = $\frac{22}{7} \times 49 \times 20$
Volume = $22 \times 7 \times 20$
Volume = $154 \times 20$
Volume = $3080$ cm3
| Step | Description | Calculation |
|---|---|---|
| 1 | Equation for sum of CSA and TSA | $4\pi rh + 2\pi r^2 = 2068$ |
| 2 | Substitute r=7, $\pi$=22/7 | $4 \times \frac{22}{7} \times 7 \times h + 2 \times \frac{22}{7} \times 7^2 = 2068$ |
| 3 | Simplify and solve for h | $88h + 308 = 2068 \implies 88h = 1760 \implies h = 20$ cm |
| 4 | Calculate Volume | Volume = $\pi r^2 h = \frac{22}{7} \times 7^2 \times 20 = 3080$ cm$^3$ |
The volume of the cylinder is 3080 cm3.
Revision Table: Cylinder Calculations
| Concept | Formula | Variables |
|---|---|---|
| Curved Surface Area (CSA) | $2\pi rh$ | r = radius, h = height |
| Total Surface Area (TSA) (Solid) | $2\pi rh + 2\pi r^2$ | r = radius, h = height |
| Volume | $\pi r^2h$ | r = radius, h = height |
Additional Information: Understanding Cylinder Geometry
A solid cylinder is a 3D shape with two parallel circular bases and a curved surface connecting them. Imagine a can of soup; that's a good example of a solid cylinder.
- The radius (r) is the distance from the center of the circular base to its edge.
- The height (h) is the perpendicular distance between the two circular bases.
- The curved surface area is the area of the side part that wraps around the cylinder. If you unroll it, it forms a rectangle with width equal to the height and length equal to the circumference of the base ($2\pi r$).
- The total surface area of a solid cylinder includes the curved surface area plus the area of the two circular bases. Each base is a circle with area $\pi r^2$.
- The volume is the amount of space inside the cylinder. It's calculated by multiplying the area of the base ($\pi r^2$) by the height (h).
Problems involving cylinders often require using these formulas to find a missing dimension (like height or radius) or calculating an area or volume based on given dimensions or relationships between areas.
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