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Chapter 8: Problem 2

Find the surface area of each sphere. \(r=30\) in.

Short Answer

Expert verified
The surface area of the sphere is approximately \(11304\) square inches.

Step by step solution

01

- Recall the formula for the surface area of a sphere

The formula to calculate the surface area of a sphere is given by \[ A = 4 \pi r^2 \] where \(A\) is the surface area and \(r\) is the radius of the sphere.
02

- Substitute the value of the radius into the formula

Given \(r = 30\) inches, substitute this value into the formula: \[ A = 4 \pi (30)^2 \]
03

- Calculate the square of the radius

Calculate \(30^2\): \[ 30^2 = 900 \]
04

- Multiply by 4蟺

Multiply \(900\) by \(4 \pi\): \[ A = 4 \pi \times 900 \] \[ A = 3600\pi \]
05

- Provide the final answer in approximate terms

To express the surface area in approximate terms, use \( \pi \approx 3.14 \): \[ A \approx 3600 \times 3.14 \] \[ A \approx 11304 \text{ square inches} \]

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Key Concepts

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

sphere surface area formula
To find the surface area of a sphere, we use a specific formula. This formula is: \[A = 4 \pi r^2\]. In this equation, \(A\) represents the surface area, \(r\) represents the radius, and \(\pi\) is a constant approximately equal to 3.14.
Understanding this formula is essential in geometry, particularly for problems involving spherical objects.
Let's break down why this formula works. The term \[4 \pi r^2\] comes from integrating the surface area of a sphere.
This formula provides a direct relationship between the radius of the sphere and its surface area.
It tells us that if we know the radius, we can find the surface area easily.
  • \(A\): Surface area of the sphere
  • \(r\): Radius of the sphere
  • \(\pi\): Mathematical constant (~3.14)
geometry calculations
Geometry calculations involve using formulas and mathematical principles to measure shapes and spaces.
In this exercise, we calculated the surface area of a sphere by substituting the given values into the formula.
Here鈥檚 a simple step-by-step approach:
  • Identify the formula: \[A = 4 \pi r^2\]
  • Substitute the given radius into the formula: Given \(r = 30\) inches, substitute it to get \[A = 4 \pi (30)^2\]
  • Calculate the square of the radius: \[30^2 = 900\]
  • Multiply the result by 4 and \(\pi\): \[A = 4 \pi \times 900 = 3600 \pi\]

Finally, if necessary, approximate the value using \(\pi \approx 3.14\): \[A \approx 3600 \times 3.14 \approx 11304 \text{ square inches}\]
This breakdown helps you see each step clearly and understand the logical progression from identifying the formula to reaching the final answer.
substitution in formulas
Substituting values into a formula is a key skill in solving geometry problems. It involves replacing the variables in the formula with the given numerical values. Simply follow these steps:
  • Identify the known variables and their values
  • Replace the variables in the formula with the known values
  • Perform the necessary arithmetic operations

For example, in our exercise:
  • We identified the radius \( r = 30 \text{ inches}\) and the formula \[A = 4 \pi r^2\]
  • We substituted \(30\) into the formula: \[A = 4 \pi (30)^2\]
  • We then calculated \[30^2 = 900\]

This process helps make complex calculations more manageable by breaking them down into simpler arithmetic steps.

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Most popular questions from this chapter

find the surface area of each cone. Round to nearest tenth. $$r=7 \mathrm{cm}, \ell=17 \mathrm{cm}$$

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