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Chapter 9: Problem 282

Find a the volume and b the surface area of the sphere with the given radius. Round answers to the nearest hundredth. 2.1 yards

Short Answer

Expert verified
Volume: 38.84 cubic yards; Surface Area: 55.42 square yards.

Step by step solution

01

Write the Formula for the Volume of a Sphere

The formula for the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius.
02

Plug in the Radius for Volume

Given the radius (r) is 2.1 yards, substitute this value into the formula: \( V = \frac{4}{3} \pi (2.1)^3 \).
03

Calculate the Volume

First calculate \( (2.1)^3 \): \[ (2.1)^3 = 2.1 \times 2.1 \times 2.1 = 9.261 \] Then multiply by 4/3 and \( \pi \): \[ V = \frac{4}{3} \pi \times 9.261 \approx 38.84 \text{ cubic yards} \] (Rounded to the nearest hundredth).
04

Write the Formula for the Surface Area of a Sphere

The formula for the surface area of a sphere is \( A = 4 \pi r^2 \), where \( r \) is the radius.
05

Plug in the Radius for Surface Area

Given the radius (r) is 2.1 yards, substitute this value into the formula: \[ A = 4 \pi (2.1)^2 \].
06

Calculate the Surface Area

First calculate \( (2.1)^2 \): \[ (2.1)^2 = 2.1 \times 2.1 = 4.41 \] Then multiply by 4 and \( \pi \): \[ A = 4 \pi \times 4.41 \approx 55.42 \text{ square yards} \] (Rounded to the nearest hundredth).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Volume Calculation
Understanding how to calculate the volume of a sphere is essential in geometry. The formula used for this is \( V = \frac{4}{3} \pi r^3 \). This formula stems from integral calculus, but the main idea is to measure how much space a spherical object occupies.

Here's a step-by-step guide on calculating the volume:
  • First, identify the radius (r) of the sphere. In our case, the radius is 2.1 yards.
  • Next, use the volume formula: \( V = \frac{4}{3} \pi (2.1)^3 \).
  • Simplify the expression by calculating the cube of the radius: \( (2.1)^3 = 9.261 \).
  • Then, multiply by \( \frac{4}{3} \) and \( \pi \): \( \frac{4}{3} \pi \times 9.261 \).
  • This gives approximately 38.84 cubic yards, as rounded to the nearest hundredth.
Breaking it down this way makes it easier to understand each part of the calculation.
Surface Area Calculation
Next, let's talk about calculating the surface area of a sphere. The formula for surface area is \( A = 4 \pi r^2 \). This formula helps determine how much area covers the outer surface of the sphere.

Here are the steps to calculate it:
  • Start with the radius of the sphere, which is again 2.1 yards.
  • Use the surface area formula: \( A = 4 \pi (2.1)^2 \).
  • Calculate the square of the radius: \( (2.1)^2 = 4.41 \).
  • Multiply the result by 4 and then by \( \pi \): \( 4 \pi \times 4.41 \).
  • The final result is approximately 55.42 square yards, rounded to the nearest hundredth.
This step-by-step method helps you keep track of each calculation.
Sphere Geometry
To fully grasp the previous calculations, it's important to understand the basics of sphere geometry. A sphere is a perfectly round three-dimensional shape, where every point on its surface is equidistant from its center.
  • The radius (r) is the distance from the center of the sphere to any point on its surface.
  • The diameter is twice the radius, giving the longest distance across the sphere.
  • Volume refers to the space inside the sphere.
  • Surface area is the total area that the surface of the sphere occupies.
Understanding these basic aspects helps in visualizing and solving problems relating to spheres. Additionally, knowing these elements allows you to quickly identify which formulas to use when calculating different properties of a sphere.

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Most popular questions from this chapter

In the following exercises, solve using the problem-solving strategy for word problems. Remember to write a complete sentence to answer each question. There are 9 saxophone players in the band. The number of saxophone players is one less than twice the number of tuba players. Find the number of tuba players.

In the following exercises, find the a volume b surface area of the cylinder a cylinder with radius 2 yards and height 6 yards

A traffic cone has height 75 centimeters. The radius of the base is 20 centimeters. Find the volume of the cone. Round to the nearest tenth.

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In the following exercises, find the a volume b surface area of the sphere. a baseball with radius 1.45 inches
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