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Chapter 13: Problem 19

Find the surface area of each sphere or hemisphere. Round to the nearest tenth. hemisphere: The circumference of a great circle is 40.8 inches.

Short Answer

Expert verified
The surface area of the hemisphere is approximately 397.6 square inches.

Step by step solution

01

Understanding the Problem

We are given a hemisphere, and we need to find its surface area. The only information we have is the circumference of the great circle, which is 40.8 inches. We will use this to find the radius first.
02

Find the Radius

The formula for the circumference of a circle is \( C = 2\pi r \). Given that the circumference \( C \) is 40.8 inches, we can solve for the radius \( r \). \[ r = \frac{C}{2\pi} = \frac{40.8}{2\pi} \approx 6.5 \text{ inches} \] (rounded to the nearest tenth)
03

Calculate the Surface Area of the Hemisphere

The formula for the surface area of a hemisphere is \( A = 3\pi r^2 \) because it includes the curved surface area and the base area (a great circle). Using the radius from Step 2, we calculate:\[ A = 3\pi (6.5)^2 \approx 397.6 \text{ square inches} \] (rounded to the nearest tenth).

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Key Concepts

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

Geometry
In geometry, we explore various shapes and their properties. A hemisphere is one such shape, being half of a sphere. It's like slicing an orange in half to see the round part and the flat surface.
This shape comes with its own set of surface area considerations. When we talk about the surface area of a hemisphere, we take into account not only the curved surface but also the flat circular base.
The total surface area involved is crucial in many applications, such as calculating the paint needed to cover a dome or designing sports equipment.
  • Curved Surface: This is the external, dome-like area of the hemisphere.
  • Base Surface: The flat circular part of the hemisphere.
Circumference
The circumference of a circle is the distance around it, similar to the perimeter of a polygon. For our task, it's critical to understand this concept because the diameter 鈥� and thus the radius 鈥� derives directly from it.
When calculating the surface area of a hemisphere, knowing the circumference allows us to find the radius using the formula:
  • The formula is:
    \[ C = 2\pi r \]
Once you know the circumference (given as 40.8 inches), you can rearrange the formula to solve for the radius. This relationship is foundational in geometry, especially when working with circles and spherical objects.
Radius Calculation
Calculating the radius is a pivotal step in solving many mathematical problems involving circular shapes. In our exercise, finding the radius is done through rearranging the circumference formula:
\[ r = \frac{C}{2\pi} \]
Given that our circumference is 40.8 inches, we substitute it into the equation to get:
  • \( r = \frac{40.8}{2\pi} \approx 6.5 \text{ inches} \)
The radius is fundamental as it links directly to other calculations, such as finding the surface area or volume of circular and spherical shapes. In our specific problem, it sets the stage for determining the hemisphere's surface area.
Mathematical Problem Solving
Solving mathematical problems, especially in geometry, often involves a series of logical steps. For instance, in finding the surface area of a hemisphere, we:
  1. Identify the given information: the circumference in this case.
  2. Use appropriate formulas: such as for the radius and surface area.
  3. Perform calculations: to find the radius and substitute it into the surface area formula.
  4. Round the answers appropriately: to ensure accurate representation of measurements.
Using a structured approach, problems become less daunting and more manageable. It teaches us not only how to solve them but also underlines the importance of understanding each step along the way. This methodology is what makes geometry not just about numbers and shapes 鈥� but about problem-solving and logical reasoning.

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Most popular questions from this chapter

Find the volume of the empty space in a cylindrical tube of three tennis balls. The diameter of each ball is about 2.5 inches. The cylinder is 2.5 inches in diameter and is 7.5 inches tall.

Find the volume of each sphere or hemisphere. Round to the nearest tenth. A hemisphere has a surface area of \(18.75 \pi\) square meters.

A crane is used to move a crate from the loading dock to a place in a warehouse. To allow the storage company to locate and identify the crates, they assign ordered triples to the corners using positive \(x, y,\) and \(z\) values. Another crate is a cube 4 feet on each side. If the crate is stored 15 feet back, 11 feet to the right, and 25 feet up, find the ordered triples of the vertices describing the new location. Use the translation \((x, y, z) \rightarrow(x-15, y+11, z+25)\).

An oatmeal container is a cardboard cylinder 10 inches tall with a plastic lid that has a diameter of \(2 \frac{1}{2}\) inches. What is the lateral area of the container?

Find the volume of a sphere given the radius or diameter. Round to the nearest tenth. $$r=17.2 \mathrm{m}$$
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