Problem 60 - Technical Mathematics with Calculus

Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Applications Involving Powers The volume of a 59.4 -cm-radius sphere (Fig. \(1-7\) ) is \(\frac{4}{3} \pi(59.4)^{3} \mathrm{cm}^{3} .\) Find this volume.

Step-by-Step Solution

Answer

The volume of the 59.4-cm-radius sphere is 880,000 \( \mathrm{cm}^{3} \) to three significant digits.

Step 1: Understanding the Formula for the Volume of a Sphere

The volume of a sphere is given by the formula \( V = \frac{4}{3}\pi r^{3} \), where \( V \) is the volume and \( r \) is the radius of the sphere. In this problem, we are given a sphere with a radius of 59.4 cm, and we'll use the formula to find its volume.

Step 2: Plugging the Radius into the Formula

Substitute the given radius (59.4 cm) into the volume formula: \( V = \frac{4}{3}\pi (59.4 \, \mathrm{cm})^{3} \).

Step 3: Calculating the Volume

Calculate the volume by raising the radius to the third power and multiplying by \( \frac{4}{3}\pi \): \( V = \frac{4}{3}\pi (59.4 \, \mathrm{cm})^{3} = \frac{4}{3}\pi (59.4)^{3} \, \mathrm{cm}^{3} \).

Step 4: Retaining the Proper Number of Significant Digits

Since the radius is given as 59.4, which has three significant digits, the final answer should also be reported with three significant digits. Perform the calculation and round the result accordingly.

Step 5: Reporting the Final Answer

After calculation and rounding, report the volume of the sphere in cubic centimeters with the proper number of significant digits.

Key Concepts

Significant Digits

In mathematical calculations, especially those involving measurements, significant digits play a crucial role in expressing precision. Significant digits, or significant figures, are the digits in a number that carry meaningful contributions to its measurement precision. This includes all nonzero digits, any zeros between them, and any trailing zeros in the decimal portion.

For instance, in the given radius of a sphere, 59.4 cm, there are three significant figures — 5, 9, and 4. The presence of the decimal point indicates that the measurement is precise up to the tenths place. When performing calculations, it is essential to maintain this level of precision to ensure that the final answer reflects the accuracy of the initial measurements. Thus, when we compute the volume of a sphere using this radius, our result must also be rounded to three significant digits to mirror the precision of the given data.

Geometric Formulas

Geometric formulas are vital tools for solving various problems in mathematics and its applications. These formulas often relate different geometric figures' properties, such as their sides, angles, areas, volumes, and more. In our textbook exercise, we are utilizing one of the fundamental geometric formulas which define the volume of a sphere:

\[ V = \frac{4}{3}\pi r^{3} \]

The formula incorporates \(\pi\), a mathematical constant approximately equal to 3.14159, and \(r\), the radius of the sphere. The radius is a straight line from the center of the sphere to any point on its surface. The exponent 3 on the radius signifies that we are dealing with a three-dimensional object, and volume is always expressed in cubic units. This formula is derived from integral calculus and represents the space enclosed by the spherical surface.

Mathematical Calculations

These calculations are structured procedures that involve arithmetic operations, and their complexity can range from simple addition to the application of advanced algebraic formulas. For the sphere's volume calculation, we raise the radius to the third power (cubed), multiply by \(\pi\), and then by \(\frac{4}{3}\). Each step in the calculation reflects the principles of arithmetic and algebra—multiplication, exponentiation, and the application of constants. Careful attention is needed for order of operations to ensure accuracy.

The process of such calculations often involves performing operations with significant figures and rounding off to maintain the proper level of precision. To correctly apply mathematical calculations, one must understand these basics, along with the specific formulas relevant to the problem at hand. A clear systematic approach, as demonstrated in the sphere volume problem, helps in obtaining the correct solution.