Problem 109 - Algebra A Combined Function

Question

The formula for the radius \(r\) of a sphere with surface area \(A\) is given by \(r=\sqrt{\frac{A}{4 \pi}}\). Calculate the radius of a standard zorb whose outside surface area is 32.17 sq \(m\). Round to the nearest tenth. (A zorb is a large inflated ball within a ball in which a person, strapped inside, may choose to roll down a hill. Source: Zorb, Ltd.)

Step-by-Step Solution

Answer

The radius is approximately 1.6 meters.

Step 1: Understand the Formula

The formula given is used to calculate the radius of a sphere from its surface area: \[ r = \sqrt{\frac{A}{4\pi}} \]where \( A \) represents the surface area and \( \pi \approx 3.14159 \).

Step 2: Substitute the Given Values

We need to substitute the given surface area into the formula. The given surface area \( A \) is 32.17 square meters. Substitute \( A = 32.17 \) into the formula:\[ r = \sqrt{\frac{32.17}{4\pi}} \]

Step 3: Calculate Inside the Square Root

First, calculate the expression inside the square root: \[ \frac{32.17}{4\pi} = \frac{32.17}{4 \times 3.14159} \]Calculating further gives:\[ \frac{32.17}{12.56636} \approx 2.560 \].

Step 4: Compute the Square Root

Now compute the square root of the result from Step 3:\[ \sqrt{2.560} \approx 1.600 \].

Step 5: Round to the Nearest Tenth

Round the final result to the nearest tenth: 1.600 rounded to the nearest tenth is 1.6.

Key Concepts

Radius Calculation

In the realm of geometry, understanding how to calculate the radius of a sphere from its surface area is incredibly useful. The formula for the radius \( r \) of a sphere given its surface area \( A \) is expressed as follows: \[ r = \sqrt{\frac{A}{4\pi}} \] This equation helps us find the radius when the surface area is known. To use the formula, you'll need the surface area value, which in the case of a standard zorb is given as 32.17 square meters. By substituting the surface area into the formula, you set the stage for finding the radius. By understanding this concept, you'll gain the ability to transition from an area measurement to a linear one.

Mathematical Formulas

Mathematical formulas are like recipes for solving problems. In our exercise, the formula \( r = \sqrt{\frac{A}{4\pi}} \) is used. This neatly encapsulates the relationship between surface area and radius for a sphere. Let's break it down:

• \( A \) is the surface area of the sphere.
• \( \pi \) is a constant, approximately 3.14159, which relates the circumference to the diameter of a circle.
• The division by \( 4\pi \) connects surface area to a sphere's radius.

In mathematics, such formulas save time and ensure precision by streamlining the calculation process.

Square Roots

Square roots are essential components in mathematical calculations, especially when it comes to solving for the radius in our sphere equation. When you see a square root symbol \( \sqrt{} \), it asks you to find a number which, when multiplied by itself, gives the original number inside the radical. For example, in our step-by-step solution, we calculate \( \sqrt{2.560} \). This means we are looking for a number which, when multiplied by itself equals approximately 2.560.

• The square root of 2.560 is approximately 1.600.
• Hence, the radius of the sphere (zorb in this example) was found.

Understanding square roots is crucial for interpreting and solving many geometric problems.

Pi Approximation

Approximating \( \pi \) is an important skill in mathematics, specifically when performing calculations involving circles and spheres. The number \( \pi \) is irrational, meaning it has non-repeating, infinite decimal places, but for practical purposes, it is usually approximated as 3.14159.

• When calculating the radius from a surface area, \( 4\pi \) becomes a key part of the denominator in our formula.
• In practical calculations, using 3.14159 for \( \pi \) gives the precision needed for most real-world applications.

The approximation allows you to perform calculations without needing to delve into the infinite decimal nature of \( \pi \), making it accessible and functional.