Problem 31 - Algebra Form and Function

Question

If the side length of a cube is increased by \(10 \%,\) what happens to its surface area?

Step-by-Step Solution

Answer

Answer: The surface area increases by 21%.

Step 1: 1. Understand the problem

We are given that the side length of a cube is increased by \(10 \%\). We need to determine the new surface area and compare it with the initial surface area.

Step 2: 2. Calculate the new side length

Let the initial side length of the cube be \(s\). If we increase it by \(10 \%,\) the new side length will be \(1.10s\). This is because a \(10 \%\) increase is equivalent to multiplying by \(1 + \frac{10}{100} = 1.1\).

Step 3: 3. Compute the initial surface area

The surface area of a cube with side length \(s\) is given by the formula \(6s^2\). Substitute the initial side length \(s\) into the formula: Initial Surface Area \(=6s^2\)

Step 4: 4. Compute the new surface area

Now that we have the new side length \(1.10s\), we can find the new surface area. Use the same surface area formula as before, substituting the new side length \(1.10s\): New Surface Area \(= 6(1.10s)^2\)

Step 5: 5. Simplify the new surface area

Next, we want to simplify the expression to better compare the initial surface area and the new one: New Surface Area \(= 6(1.21s^2) = 7.26s^2\)

Step 6: 6: Compare the surface areas

Now, we can compare the initial surface area (\(6s^2\)) to the new surface area (\(7.26s^2\)). We can clearly see that the new surface area is more significant. In particular, the surface area has increased by a factor of \(\frac{7.26s^2}{6s^2}=1.21\). In percentage terms, the surface area has increased by \(21\%\) as a result of increasing the side length by \(10 \%\).

Key Concepts

Percentage Increase

When we talk about a percentage increase, we are referring to how much something grows in terms of percent. To find a percentage increase, you add the percentage to 100% and multiply it by the original value. For example, if you increase the length of something by 10%, you multiply the original length by 1 + \( \frac{10}{100} \), which simplifies to 1.1. This means the new length is 110% of the original length, representing a 10% growth.

If a side length of a cube is increased by 10%, it implies multiplying the original side length by 1.1. However, when it comes to cube-related problems, we don't just stop here. We often look at how this increase affects other properties like volume or in this case, surface area. By understanding and calculating percentage increases accurately, you can understand how small changes in one dimension affect the entire shape in significant ways.

In this exercise, you discover that a 10% increase in side length results in a 21% increase in surface area, which is more substantial than it might initially appear.

Cube Geometry

The geometry of a cube revolves around its perfect symmetry and simple structure. A cube is a three-dimensional shape with six equal square faces. This means that if you know one side length of the cube, you can calculate its entire geometry.

Few essential properties of the cube include:

• All sides are equal, which means it is very straightforward to calculate areas and volumes.
• The surface area of a cube is found using the formula \( 6s^2 \), where \( s \) is the side length. This formula is derived from the fact that a cube has six identical square faces, each with an area of \( s^2 \).

Understanding these properties can help solve numerous geometrical problems that involve cubes. For this exercise, the critical concept is that any change in the side length proportionally affects these properties. Increasing the side length by 10% changes the dimensions of the cube and has a larger impact on the surface area than one might expect from the percentage increase alone.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They allow us to represent real-world situations in a mathematical form, making complex calculations more straightforward. For instance, in problems related to cubes, algebraic expressions are key to finding dimensions and areas.

In this scenario, algebraic expressions are illustrated by the calculation of the surface area. We start with the expression for the surface area of a cube, \( 6s^2 \), and then substitute the expression for the increased side length, \( 1.10s \). This substitution step is crucial because it turns a relatively simple formula into a new expression that includes the factor accounting for the increase:

\[ \text{New Surface Area} = 6(1.10s)^2 = 6 \times 1.21s^2 = 7.26s^2 \]
This expression now shows the new surface area. Algebra teaches us that the manipulation of these expressions is vital to understanding how different changes affect the equation's outcome. Practices like expanding, simplifying, and substituting help in comprehending both straightforward and intricate problems efficiently.