Problem 47 - College Algebra with Modeling and Visualization

Question

Find the volume and surface area of the sphere satisfying the given condition, where \(r\) is the radius and dis the diameter. Approximate values to the nearest tenth. \(r=3\) feet

Step-by-Step Solution

Answer

The volume is approximately 113.0 cubic feet, and the surface area is approximately 113.0 square feet.

Step 1: Understand the relationship between radius and diameter

The radius of the sphere is given as \( r = 3 \) feet. The diameter \( d \) can be calculated using the formula \( d = 2r \). In this case, \( d = 2 \times 3 = 6 \) feet.

Step 2: Calculate the volume of the sphere

The volume \( V \) of a sphere can be found using the formula \( V = \frac{4}{3} \pi r^3 \). Substituting \( r = 3 \) into the formula gives:\[V = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi \times 27 = 36 \pi.\]Approximating \( \pi \) as 3.14, the volume is \( V \approx 36 \times 3.14 = 113.0 \) cubic feet.

Step 3: Calculate the surface area of the sphere

The surface area \( A \) of a sphere is calculated using \( A = 4 \pi r^2 \). With \( r = 3 \), the formula becomes:\[A = 4 \pi (3)^2 = 4 \pi \times 9 = 36 \pi.\]Using \( \pi \approx 3.14 \), the surface area is \( A \approx 36 \times 3.14 = 113.0 \) square feet.

Key Concepts

Volume of a Sphere

When you think of the volume of a sphere, you are considering the space inside the sphere. Imagine filling up a ball with air—that's the concept of volume! The formula to find the volume of a sphere is \[ V = \frac{4}{3} \pi r^3 \],where

• \( V \) represents volume,
• \( \pi \) is a constant approximately equal to 3.14, and
• \( r \) is the radius of the sphere.

To compute this, you take the radius of the sphere to the power of three, multiply by \( \pi \), and then multiply by \( \frac{4}{3} \). This might seem complex, but one thing to remember is that the radius plays a central role because it's elevated to the third power. This is why even small changes in the radius can have a big impact on the volume. For example, if a sphere's radius is 3 feet, simply substitute 3 into the radius slot, and follow through with the calculations.

Surface Area of a Sphere

Like finding the amount of wrapping paper you would need to cover a basketball, the surface area of a sphere measures the area covering the outer part of the sphere. The formula for the surface area is \[ A = 4 \pi r^2 \],with

• \( A \) denoting the area,
• \( \pi \) again being 3.14, and
• \( r \) still the radius of the sphere.

To find the surface area, the key step is squaring the radius, multiplying it by \( \pi \), and lastly, multiplying by 4. This tells you how much surface you have in terms of square units. For example, a sphere with a 3-foot radius would have its calculation as \( 4 \times \pi \times 3^2 \), which when evaluated gives you 113.0 square feet using approximate value for \( \pi \). This calculation helps to visualize the spread of the sphere's surface.

Radius and Diameter Relationship

Understanding the relationship between a sphere's radius and its diameter simplifies calculations and aids in conceptual clarity. It's a straightforward relationship where the diameter (\( d \)) is twice the radius (\( r \)). Essentially, you can represent this as: \[ d = 2r \].This means if you know the radius, you can easily find the diameter by doubling it, and vice versa. Conversely, to find the radius from the diameter, you divide the diameter by two. This relationship is crucial in many calculations involving spheres because it offers flexibility in terms of which measurement you start with when solving problems. For the sphere given in our problem, with a radius of 3 feet, the diameter calculates to \( 3 \times 2 = 6 \) feet. This simple yet fundamental relationship underpins further calculations for both surface area and volume.