Question

Use the following information. The surface area of a cylinder equals the lateral surface area \((2 \pi r \cdot h)\) plus the area of the two bases \(\left(2 \cdot \pi r^{2}\right)\). Evaluate the expression when \(h=10.5\) centimeters and \(r=2.5\) centimeters. Use 3.14 as an approximation for \(\pi .\)

Short answer

The total surface area of the cylinder is 204.55 \(cm^{2}\).

Step-by-step solution

Substitution of given values

Plugging in the given values into the expressions we have: \[ Lateral \, Surface \, Area= 2 \pi r h = 2 \cdot 3.14 \cdot 2.5 cm \cdot 10.5 cm \] \[ Area \, of\, two \, bases = 2 \pi r^{2} = 2 \cdot 3.14 \cdot (2.5 cm)^{2} \]

Calculation of the Lateral Surface Area

For the lateral area, calculate \(2 \cdot 3.14 \cdot 2.5 cm \cdot 10.5 cm = 165.3\, cm^2\)

Calculation of the Area of the two bases

For the area of the two bases, calculate \(2 \cdot 3.14 \cdot (2.5 cm)^{2} = 39.25 cm^2 \)

Sum of the areas

The final step is to add the lateral surface area and the area of the two bases to get the total surface area. The calculation is \(165.3 cm^2 + 39.25 cm^2 = 204.55 cm^2 \)

Key concepts

Lateral Surface Area Calculation

Understanding the lateral surface area of a cylinder is crucial for problems involving curved surfaces. In its most basic form, the formula for the lateral surface area of a cylinder is given by \( 2 \pi r h \), where \( r \) is the radius of the base circle of the cylinder, and \( h \) is the height of the cylinder. This formula calculates the area of the rectangle that would be obtained if we were to 'unroll' the curved surface of the cylinder.

To calculate this, we need to multiply the perimeter of the base circle, which is \( 2 \pi r \) (also known as the circumference), by the height of the cylinder. This gives us the total area of the side surface that wraps around the cylinder. For example, when provided with the radius and the height—even when they are in different units—you should first ensure that you are working in the same units for both measurements before applying the formula.

Area of Cylinder Bases

When dealing with the geometry of a cylinder, it is important to remember that the cylinder has two congruent circular bases. The area of one base can be found using the formula \( \pi r^2 \), where \( r \) represents the radius. Since there are two bases on a cylinder, we simply multiply the area of one base by two to get the total area of both bases, hence the formula \( 2 \pi r^2 \).

It is easy to visualize this by picturing two identical circles at the top and bottom of the cylinder. Determining the area of these bases is just like finding the area of a circle, only that you do it twice and sum them up. This is a simple but powerful concept that often serves as the foundation for more complex geometrical problems. Remember: whenever you’re working with cylinders, you have to account for both bases in your calculations.

Substituting Values in Algebra

An essential skill in algebra is substituting values into an expression or equation. This step is precisely where many mathematical problems begin. Once you have the formula, you will often be given specific values to plug into that formula. To do this correctly, you should substitute these values carefully and perform the operations as indicated.

Following our cylinder example, once the values for \( h \) and \( r \) are provided, they are substituted into the formulas for the lateral surface area and the area of the bases. It’s important to substitute these values consistently to maintain the accuracy of the calculation. Paying attention to detail when substituting helps avoid simple errors that can lead to incorrect answers. Remember to also keep track of the units you're working with, as these can affect the outcome and are frequently a source of mistakes.