Problem 125 - Intermediate Algebra

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 \(\mathrm{m}\). 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.) (IMAGE CANNOT COPY)

Step-by-Step Solution

Answer

The radius of the zorb is approximately 1.6 meters.

Step 1: Identify the Formula

We need to find the radius of a sphere using the given surface area formula. The formula provided is:\[r = \sqrt{\frac{A}{4\pi}}\]where \( A \) is the surface area of the sphere.

Step 2: Substitute the Given Surface Area

The problem states that the surface area \( A \) of the zorb is 32.17 square meters. Substitute \( A = 32.17 \) into the formula:\[r = \sqrt{\frac{32.17}{4\pi}}\]

Step 3: Calculate \( 4\pi \)

Calculate the value of \( 4\pi \) using the approximation \( \pi \approx 3.14159 \):\[4\pi = 4 \times 3.14159 = 12.56636\]

Step 4: Divide Surface Area by \( 4\pi \)

Divide the surface area (32.17) by \( 4\pi \) (12.56636):\[\frac{32.17}{12.56636} \approx 2.560181\]

Step 5: Calculate the Square Root

Find the square root of the result from the previous step:\[r = \sqrt{2.560181} \approx 1.6\]

Step 6: Round the Answer

Round the result to the nearest tenth as requested by the problem:\( r \approx 1.6 \) meters.

Key Concepts

Surface Area of a Sphere

The surface area of a sphere is a mathematical concept that refers to the total area that the surface of a sphere occupies. To calculate this, you use the formula: \[ A = 4 \pi r^2 \]where \( A \) is the surface area, and \( r \) is the radius of the sphere. This formula shows that the surface area depends on the square of the radius. Hence, even small changes in the radius significantly impact the surface area.
The "4\(\pi\)" in the formula essentially represents the proportionality constant which derives from the integration over the surface of a sphere.Understanding this is crucial because in various applications, such as engineering or physics, the precise surface area can determine factors like material costs or surface resistance.

Mathematical Formula Substitution

Mathematical formula substitution is an essential concept in solving equations where a specific value is inserted into an existing formula to get a solution. In our problem, the task is to find the radius of a sphere given its surface area. The formula is: \[ r = \sqrt{\frac{A}{4\pi}} \]Here, substitution involves replacing \( A \) with the known surface area of the zorb, which is 32.17 square meters. By substituting, the formula becomes: \[ r = \sqrt{\frac{32.17}{4\pi}} \]This step simplifies solving the equation, as it sets up the mathematical expression needed to isolate and calculate the desired variable, which is the radius.

Square Root Calculation

Calculating the square root is a mathematical operation that is crucial in determining the radius once the equation has been set up. In this context, after substituting the surface area into our formula, we have:\[ r = \sqrt{2.560181} \]Finding the square root of this number means identifying a value that, when multiplied by itself, gives approximately 2.560181. In simpler terms, if \( x^2 = 2.560181 \), then \( x \) is what we need to find. For many practical problems, you'll either use a calculator or tables to find square roots. Here, we approximate the square root to the nearest tenth, which is actually 1.6. Understanding how to accurately perform this step is vital in solving problems involving quadratic equations or geometry.

Rounding to Nearest Tenth

Rounding numbers to the nearest tenth involves finding the closest number with one decimal place. This is especially important in many practical scenarios, such as engineering or manufacturing, where exact precision is often not feasible. In our example, after calculating the square root, we obtain the number 1.599984. To round this to the nearest tenth, we look at the digit in the tenths place (1.5) and the digit immediately following it (1.59). - If the next digit is 5 or more, we round up. - If it's less than 5, we round down. Therefore, 1.599984 rounds up to 1.6. Rounding helps in simplifying numbers while retaining a reasonable accuracy, making them easier to communicate and utilize in further calculations.