Surface Area Of A Sphere With Radius 3x Calculation And Explanation

The question at hand involves determining the correct expression for the surface area of a sphere given a specific radius. Understanding the formula for the surface area of a sphere is the first crucial step in solving this problem. This article will delve deep into the concept, the formula, and the step-by-step solution to the given problem, ensuring a comprehensive understanding for anyone looking to grasp this mathematical concept.

Understanding the Surface Area of a Sphere

The surface area of a sphere is a fundamental concept in geometry, representing the total area that the surface of the sphere occupies. Unlike a flat surface, a sphere is a three-dimensional object, and its surface area is calculated using a specific formula that takes into account its radius. The radius is the distance from the center of the sphere to any point on its surface. Grasping this concept is crucial for various applications, from calculating the amount of material needed to construct spherical objects to understanding physical phenomena like the surface tension of liquid droplets.

The Formula for Surface Area

The formula to calculate the surface area (*A*) of a sphere is given by:

${ A = 4 \pi r^2 }$

Where:

• (*A*) represents the surface area of the sphere.
• ${\pi}$ (pi) is a mathematical constant approximately equal to 3.14159.
• (*r*) is the radius of the sphere.

This formula tells us that the surface area of a sphere is directly proportional to the square of its radius. This means that if you double the radius of a sphere, its surface area will increase by a factor of four. The ${4 \pi}$ factor in the formula arises from the geometric properties of the sphere and how its surface curves in three-dimensional space. Understanding this formula is the cornerstone to solving any problem related to the surface area of spheres.

Importance of Understanding Surface Area

Understanding the surface area of a sphere isn't just an academic exercise; it has numerous practical applications in various fields. In engineering, it's crucial for designing spherical tanks, domes, and other structures. For example, calculating the surface area helps determine the amount of material needed, the cost of construction, and the structural integrity of the design. In physics, the concept is vital in understanding phenomena like heat transfer and fluid dynamics. The rate at which an object loses or gains heat, for instance, is directly related to its surface area. Similarly, in chemistry, the surface area plays a critical role in understanding reaction rates and the behavior of molecules. The larger the surface area of a reactant, the faster the reaction can occur. Even in everyday life, understanding surface area can be useful, from estimating the amount of paint needed to cover a spherical object to understanding why larger bubbles are more fragile.

Problem Statement: Sphere with Radius 3x

Now, let's focus on the specific problem presented. We are asked to find the expression that gives the surface area of a sphere with a radius of ${3x}$. This problem tests our ability to apply the surface area formula correctly, especially when the radius is given as an algebraic expression rather than a numerical value. This requires substituting ${3x}$ for ${r}$ in the surface area formula and simplifying the expression. The key here is to remember the order of operations and to correctly square the term ${3x}$. This type of problem is common in algebra and geometry, as it combines algebraic manipulation with geometric concepts.

Breaking Down the Radius

The radius of the sphere is given as ${3x}$, which means the radius is three times the value of ${x}$. Here, ${x}$ can be any variable, representing a length unit. The expression ${3x}$ emphasizes that the radius is not a fixed number but rather a multiple of a variable. This is a common way to introduce algebraic concepts into geometry problems, allowing for more generalized solutions. For instance, if ${x}$ represents centimeters, then the radius ${3x}$ would be three times that centimeter value. Understanding that ${3x}$ represents the radius is the first step in correctly applying the surface area formula. We must treat ${3x}$ as a single term when substituting it into the formula and remember to square both the constant (3) and the variable (${x}$).

Step-by-Step Solution

To find the surface area of the sphere with radius ${3x}$, we will substitute ${3x}$ for ${r}$ in the surface area formula and simplify the expression step-by-step. This process will illustrate how to correctly apply the formula and handle algebraic terms within a geometric context. Each step is crucial in arriving at the correct answer and avoiding common mistakes.

Step 1: Substitute the Radius into the Formula

We start with the formula for the surface area of a sphere:

${ A = 4 \pi r^2 }$

Now, we substitute ${r}$ with ${3x}$:

${ A = 4 \pi (3x)^2 }$

This substitution is the core of the problem. It directly applies the given information (the radius ${3x}$) to the general formula. The next step involves simplifying this expression, which requires understanding how to handle the squared term.

Step 2: Simplify the Expression

Next, we need to simplify the term ${(3x)^2}$. Remember that squaring a term means multiplying it by itself:

${ (3x)^2 = (3x) \times (3x) }$

Using the rules of exponents and multiplication, we get:

${ (3x)^2 = 3^2 \times x^2 = 9x^2 }$

Now, we substitute this back into the surface area formula:

${ A = 4 \pi (9x^2) }$

Step 3: Final Calculation

Finally, we multiply the constants together:

${ A = 4 \times 9 \pi x^2 }$

${ A = 36 \pi x^2 }$

Thus, the surface area of the sphere with radius ${3x}$ is ${36 \pi x^2}$. This result is a simplified algebraic expression, representing the surface area in terms of the variable ${x}$. This final step completes the solution, providing the answer in the required format.

Analyzing the Options

Now that we have the solution, ${36 \pi x^2}$, let's analyze the given options to identify the correct one. This involves comparing our calculated result with each of the options provided and understanding why the other options are incorrect.

The options given are:

• A. ${9 \pi x^2}$
• B. ${12 \pi x^2}$
• C. ${4 \pi x^2}$
• D. ${36 \pi x^2}$

By comparing our calculated surface area, ${36 \pi x^2}$, with the options, we can clearly see that:

• Option D, ${36 \pi x^2}$, matches our result. Therefore, this is the correct answer.
• Options A, B, and C do not match our calculated surface area. These options likely represent common mistakes in applying the formula or simplifying the expression.

Why Other Options Are Incorrect

Understanding why the other options are incorrect is as important as finding the correct answer. It helps to solidify the understanding of the concept and avoid similar mistakes in the future.

• Option A: ${9 \pi x^2}$ This option is incorrect because it only squares the ${x}$ term and not the 3 in the radius ${3x}$. It represents the surface area if the radius was mistakenly taken as ${x}$ instead of squaring the entire term ${3x}$.
• Option B: ${12 \pi x^2}$ This option might arise from multiplying 4 by 3 but failing to square the 3 in the radius. It doesn't account for the correct application of the squaring operation in the formula.
• Option C: ${4 \pi x^2}$ This option represents the surface area of a sphere with a radius of ${x}$, not ${3x}$. It completely neglects the effect of the 3 in the given radius.

By analyzing these incorrect options, we reinforce the importance of correctly applying the formula and paying attention to each step in the simplification process.

Conclusion

In conclusion, the correct expression for the surface area of a sphere with a radius of ${3x}$ is ${36 \pi x^2}$. This was determined by substituting ${3x}$ into the surface area formula, ${A = 4 \pi r^2}$, and simplifying the expression step-by-step. Understanding the formula, applying it correctly, and simplifying algebraic expressions are key skills demonstrated in this problem. Furthermore, analyzing the incorrect options helps to solidify the understanding and avoid common mistakes. This problem serves as an excellent example of how algebraic concepts and geometric formulas combine to solve mathematical problems. Mastering this concept is crucial for further studies in mathematics and its applications in various fields.

By understanding the concepts, formula, and step-by-step solution, anyone can confidently tackle similar problems involving the surface area of spheres. Remember to always pay attention to the details, apply the formula correctly, and simplify the expressions carefully to arrive at the correct answer.