Calculating Sphere Volume Expression For Radius 19 Units

by THE IDEN 57 views

Understanding the formulas for calculating geometric properties is a fundamental aspect of mathematics. When it comes to three-dimensional shapes, the sphere is a particularly interesting one. A sphere is a perfectly round geometrical object in three-dimensional space, much like a ball. Its volume, which is the amount of space it occupies, is a critical property that has numerous applications in various fields such as physics, engineering, and even art. To calculate the volume of a sphere, we use a specific formula that involves the radius of the sphere. In this article, we will explore this formula and apply it to a specific problem: determining the volume of a sphere with a radius of 19 units. This exercise will not only reinforce our understanding of the formula but also enhance our ability to apply mathematical concepts to practical problems. Before diving into the solution, let's first discuss the formula for the volume of a sphere and its components. The formula is derived from integral calculus, but for practical purposes, we can use the readily available formula. This discussion will set the stage for a detailed explanation of how to arrive at the correct answer and why the other options are incorrect. Understanding the underlying principles and the correct application of the formula is key to mastering this concept.

The volume of a sphere is calculated using a well-established formula that relates the volume to the radius of the sphere. This formula is expressed as:

V=43Ï€r3V = \frac{4}{3} \pi r^3

Where:

  • V represents the volume of the sphere.
  • Ï€ (pi) is a mathematical constant approximately equal to 3.14159.
  • r is the radius of the sphere, which is the distance from the center of the sphere to any point on its surface.

This formula tells us that the volume of a sphere is directly proportional to the cube of its radius. This means that if you double the radius of a sphere, its volume will increase by a factor of eight (2^3). The π (pi) in the formula is a constant that relates a circle's circumference to its diameter, and it appears in many formulas involving circles and spheres. The fraction 4/3 is a result of the mathematical derivation of the volume formula using integral calculus. The formula is a cornerstone in geometry and is used extensively in various scientific and engineering applications. For example, it is used to calculate the volume of spherical objects in astronomy, the capacity of spherical tanks in chemical engineering, and the size of ball bearings in mechanical engineering. Understanding this formula is crucial for anyone working with three-dimensional geometry and its applications. The ability to correctly apply this formula is not only important for solving mathematical problems but also for understanding the physical world around us. Now that we have a clear understanding of the formula, let's apply it to the problem at hand.

The problem asks us to find the expression that gives the volume of a sphere with a radius of 19 units. To solve this, we need to substitute the given radius (r = 19) into the formula for the volume of a sphere:

V=43Ï€r3V = \frac{4}{3} \pi r^3

Substituting r = 19 into the formula, we get:

V=43Ï€(19)3V = \frac{4}{3} \pi (19)^3

This expression represents the volume of a sphere with a radius of 19 units. Now, let's compare this expression with the given options to identify the correct answer. We are looking for the option that matches our calculated expression. By substituting the value of the radius into the formula, we have a concrete expression that we can directly compare with the given options. This step is crucial to ensure that we are selecting the correct answer. It also helps us understand why the other options are incorrect. The process of substitution is a fundamental skill in algebra and is used extensively in solving mathematical problems. It allows us to take a general formula and apply it to a specific case. In this case, we are applying the general formula for the volume of a sphere to a specific sphere with a radius of 19 units. This process of applying general formulas to specific cases is a key part of mathematical problem-solving. Now that we have substituted the value and obtained the expression, we can confidently compare it with the given options and select the correct answer.

We have the expression for the volume of the sphere with a radius of 19 units:

V=43Ï€(19)3V = \frac{4}{3} \pi (19)^3

Now, let's analyze the given options:

  • A. $\frac{4}{3} \pi(19^2)$ This option has 19 raised to the power of 2 (19^2), which represents the square of the radius. The formula for the volume of a sphere requires the radius to be cubed (r^3), not squared. Therefore, this option is incorrect. It is a common mistake to confuse the formulas for the surface area and volume of a sphere. The surface area formula involves the square of the radius, while the volume formula involves the cube of the radius. This option highlights the importance of carefully remembering the correct formula.

  • B. $4 \pi(19^3)$ This option has the correct power of 19 (19^3), but it is missing the fraction $ rac{4}{3}$ in front of $\pi$. The correct formula includes the fraction $ rac{4}{3}$, so this option is also incorrect. This option demonstrates the importance of paying attention to all parts of the formula, including the constants and fractions. Even if the radius is raised to the correct power, missing a constant factor will result in an incorrect answer.

  • C. $\frac{4}{3} \pi(19^3)$ This option matches our calculated expression exactly. It has the correct fraction $ rac{4}{3}$, the constant $\pi$, and the radius raised to the power of 3 (19^3). Therefore, this is the correct answer.

  • D. $4 \pi(19^2)$ This option is incorrect for two reasons. First, it is missing the fraction $\frac{4}{3}$. Second, it has 19 raised to the power of 2 (19^2) instead of 3 (19^3). This option combines the mistakes from options A and B, making it clearly incorrect. This option serves as a good example of how multiple errors can lead to a wrong answer.

By carefully analyzing each option and comparing it with the correct expression, we can confidently identify the correct answer and understand why the other options are incorrect. This process of elimination and comparison is a valuable problem-solving strategy in mathematics.

In conclusion, the expression that gives the volume of a sphere with a radius of 19 units is:

43Ï€(193)\frac{4}{3} \pi(19^3)

This corresponds to option C. We arrived at this answer by correctly applying the formula for the volume of a sphere, which is V = (4/3)Ï€r^3. We substituted the given radius, r = 19, into the formula and obtained the expression (4/3)Ï€(19^3). By comparing this expression with the given options, we were able to identify the correct answer. The other options were incorrect because they either had the radius raised to the wrong power (19^2 instead of 19^3) or were missing the fraction (4/3) in the formula. This exercise demonstrates the importance of understanding and correctly applying mathematical formulas. It also highlights the need to pay attention to detail and carefully analyze each option when solving problems. The formula for the volume of a sphere is a fundamental concept in geometry, and mastering it is essential for success in mathematics and related fields. Understanding the formula and its application allows us to solve a wide range of problems involving spherical objects. This problem-solving approach can be applied to various other mathematical problems as well. By carefully understanding the problem, identifying the relevant formulas, and applying them correctly, we can arrive at the correct solution.