Question

Use the Intermediate Value Theorem to show that for all spheres with radii in the interval [1,5] , there is one with a volume of 275 cubic centimeters.

Short answer

According to the Intermediate Value Theorem and since the volume function for a sphere is continuous over the given interval, there indeed exists a radius between 1 and 5 that results in a sphere volume of 275 cubic centimeters.

Step-by-step solution

Identify the Function

Identify the function as the volume of a sphere, which is defined as \( V = \frac{4}{3}πr^3 \), where r is the radius of the sphere.

Calculate Function Values for Given Interval

Substitute the values of the interval into the function, thus \( V(1) = \frac{4}{3}π(1)^3 = \frac{4}{3}π \) and \( V(5) = \frac{4}{3}π(5)^3 = \frac{500}{3}π \).

Apply the Intermediate Value Thereom

Since the volume of a sphere as a function of its radius is continuous, and 275 is between \( \frac{4}{3}π \) and \( \frac{500}{3}π \), by Intermediate Value Theorem, there exists a radius between 1 and 5 that will result in a volume of 275 cubic centimeters.

Key concepts

Volume of a Sphere

Understanding the volume of a sphere is fundamental when dealing with geometric shapes and their properties in three-dimensional space. The volume of a sphere is given by the formula \( V = \frac{4}{3}\pi r^3 \), where \( r \) is the radius of the sphere.

This equation is derived from integral calculus and represents the total amount of space enclosed within the sphere. It's critical for students to memorize this formula as it is often used in various disciplines including physics, engineering, and mathematics.

To work with this formula, students should be comfortable manipulating exponents and understanding the concept of \( \pi \) as the ratio of a circle's circumference to its diameter, approximately 3.14159. When calculating the volume, students might also find it useful to express their answers in terms of \( \pi \) to maintain precision before converting to a decimal if necessary.

Continuity

In mathematical analysis, continuity is a characteristic of a function that intuitively indicates that small changes in the input lead to small changes in the output. For a function to be continuous at a point, the following three conditions must be met:

• The function must be defined at the point.
• The limit of the function as it approaches the point must exist.
• The limit of the function as it approaches the point must be equal to the function's value at that point.

For example, the volume of a sphere function \( V = \frac{4}{3}\pi r^3 \) is continuous because for any given radius \( r \), small changes in \( r \) result in small, predictable changes in the volume \( V \).

The concept of continuity is not only central to theorems in calculus but also crucial for understanding real-world phenomena, where we assume conditions change smoothly rather than abruptly.

Mathematical Theorems

Mathematical theorems are statements that have been proven to be true using logic and prior known theorems. They are essential components of mathematical theory, providing the foundation that allows us to understand and predict the behavior of mathematical systems. One such theorem is the Intermediate Value Theorem (IVT), which itself relies on the concept of continuity.

The IVT states that for any continuous function \( f \), over an interval \( [a, b] \) and any number \( N \) between \( f(a) \) and \( f(b) \) there exists at least one point \( c \) in the interval such that \( f(c) = N \).

For the example at hand, the IVT assures us that because the volume of a sphere changes continuously as its radius changes, then for any volume between the volumes at radii 1 and 5, there must be some radius within that range for which the sphere will have that specific volume. This logical consequence is what underpins the use of the IVT in the stated exercise and a multitude of other applications in mathematics.