Question

The surface area of the can, Fig. \(2-5,\) obtained by adding the area of the curved side to the area of each end, is \(\pi r^{2}+2 \pi r h+\pi r^{2} .\) Simplify this expression.

Short answer

\(2\pi r^{2} + 2 \pi r h\).

Step-by-step solution

Identify Like Terms

Review the original expression \(\pi r^{2}+2 \pi r h+\pi r^{2}\) and identify terms that are similar. In this expression, the like terms are the ones that have \(\pi r^{2}\) because they both represent the area of the circular ends of the can.

Combine Like Terms

Combine the like terms by adding their coefficients. Since we have two \(\pi r^{2}\) terms, we add them together to get \(2\pi r^{2}\).

Write the Simplified Expression

Include the term representing the curved surface area which has not changed, and write the fully simplified expression: \(2\pi r^{2} + 2 \pi r h\).

Key concepts

Like Terms in Algebra

Understanding 'like terms' in algebra is essential for simplifying algebraic expressions efficiently. But what exactly are like terms? Simply put, like terms are terms within an algebraic expression that have the same variables raised to the same power, although their coefficients (the numerical part of the terms) can be different. For instance, in the expression given in our exercise, \(\pi r^{2}\) and \(\pi r^{2}\) are like terms because they both contain the variable \(r\) raised to the second power, and they're both multiplied by \(\pi\).

When simplifying an expression, combining like terms is a crucial step. This involves adding or subtracting the coefficients of these terms. In the context of our problem, we added together the coefficients of the \(\pi r^{2}\) terms. This process is just like saying if you have two apples and a friend gives you two more apples, you now have four apples—or in algebraic terms, \(\pi r^{2} + \pi r^{2} = 2\pi r^{2}\). Like terms make our expressions neat and easier to work with, stripping down complicated problems to simpler, solvable equations.

Surface Area of a Cylinder

The surface area of a cylinder is an important concept in geometry that has practical applications in fields such as engineering and manufacturing. A cylinder has three parts to consider when calculating its surface area—the two circular ends and the curved side. The formula consists of two parts: the areas of the circles (which there are two of) and the area of the curved side.

The formula for the area of a circle is \(\pi r^{2}\), where \(r\) stands for the radius. Since we have two circular ends on a cylinder, we multiply this area by two. This is where the \(2\pi r^{2}\) in our exercise comes from. Next, to find the area of the curved side, imagine 'unrolling' it into a rectangle. The width of this rectangle would be the height \(h\) of the cylinder, and the length would be the circumference of the circle, which is \(2\pi r\). Therefore, the area of the curved surface equals \(2\pi r h\).

Now, following our exercise, we take the sum of the area of the circles and the curved side to express the total surface area of the cylinder as \(2\pi r^{2} + 2\pi r h\). Isn't it fascinating how we can break down a three-dimensional object into simpler components to understand and calculate its properties?

Algebraic Expression Simplification

Algebraic expression simplification is like tidying up a room by putting things in order and simplifying the space. In algebra, simplifying an expression means to reduce it to its simplest form, making it easier to understand and work with. This usually involves combining like terms, as we've seen, but can also include other operations like factoring, canceling out terms, or using the distributive property.

Imagine an algebraic expression as a long sentence. To make it simpler, you would remove redundant words, combine similar ideas, and strip it down to its essential meaning. Just as you would say 'six apples' instead of 'three apples plus three apples,' algebraic simplification combines all like terms and performs available operations to shorten and present the expression most cleanly and precisely.

In our textbook problem, simplifying the expression for the surface area of a cylinder involved identifying and combining like terms, which is often the critical first step in simplification. Remember, by simplifying an expression, you are not changing the expression's value—just making it easier to read and possibly to further manipulate it mathematically.