/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 33 How much greater is the light-co... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 33

How much greater is the light-collecting area of a 6 -meter telescope than that of a 3 -meter telescope? (a) two times (b) four times (c) six times

Short Answer

Expert verified
(b) four times

Step by step solution

01

Understanding the Formula for Area of a Circle

The light-collecting area of a telescope is determined by the area of its aperture, which is circular. The formula to calculate the area of a circle is \(A = \pi r^2\) where \(r\) is the radius of the circle.
02

Calculate Area for the 6-Meter Telescope

The diameter of the 6-meter telescope is 6 meters, so its radius is 3 meters (half of the diameter). Plugging this into the area formula gives us an area of \(A = \pi (3)^2 = 9\pi\).
03

Calculate Area for the 3-Meter Telescope

The diameter of the 3-meter telescope is 3 meters, so its radius is 1.5 meters. Using the area formula, we find \(A = \pi (1.5)^2 = 2.25\pi\).
04

Compare the Areas

To find how many times greater the area of the 6-meter telescope is compared to the 3-meter telescope, divide the area of the 6-meter telescope by the area of the 3-meter telescope: \(\frac{9\pi}{2.25\pi} = 4\).
05

Determine the Correct Answer

Based on the result from Step 4, the light-collecting area of a 6-meter telescope is four times greater than that of a 3-meter telescope.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Circle Area Calculation
One important aspect when dealing with telescopes is understanding how to calculate the area of a circle. Telescopes collect light through their aperture, which is a circular opening. To find out how much light they can gather, we need to know the area of that circle. Mathematically, the area (\(A\)) of a circle can be calculated using the formula: \[A = \pi r^2\]Here, \(\pi\) is a constant (approximately 3.14159), and \(r\) is the radius of the circle.
  • First, know the diameter of the circle, which is the distance across its widest point.
  • The radius is half of the diameter.
  • Plug the radius into the formula to find the area.
Knowing how to do this calculation is crucial for analyzing telescopes or any circular object's properties.
Telescope Aperture
The aperture of a telescope is essentially the opening through which light enters. In simple terms, the bigger the aperture, the more light the telescope can collect. This means that a telescope with a larger aperture can gather more light from faint objects, making it possible to see them better.
  • The aperture's size directly affects the telescope's resolution 鈥� larger apertures allow a clearer view of distant objects.
  • For circular apertures, the term 'light-collecting area' refers to the area of the aperture.
  • Larger apertures are often a priority for astronomers because they enhance visibility.
A telescope's aperture is often expressed in meters, just like when comparing the 6-meter and 3-meter telescopes.
Math Problem Solving
Solving math problems, such as comparing telescope light-collecting areas, involves a step-by-step approach. This ensures that all necessary calculations are done correctly.For our example:
  • Determine the radii using the diameters. For a 6-meter telescope, the radius is 3 meters. For a 3-meter one, the radius is 1.5 meters.
  • Calculate the area for each using \(A = \pi r^2\).
  • Find the quotient of the areas to see how many times greater one is than the other.
For the telescopes:
  • The 6-meter telescope: \(A = \pi (3)^2 = 9\pi\).
  • The 3-meter telescope: \(A = \pi (1.5)^2 = 2.25\pi\).
  • Compare the two areas: \(\frac{9\pi}{2.25\pi} = 4\).
Math not only helps in solving such problems but also in understanding how different parameters affect the outcome.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Define wavelength, frequency, and speed for light waves. If light has a long wavelength, what can you say about its frequency? Explain.

The Changing Limitations of Science. In \(1835,\) French philosopher Auguste Comte stated that science would never allow us to learn the composition of stars. Although spectral lines had been seen in the Sun's spectrum at that time, it wasn't until the mid- 19 th century that scientists recognized (primarily through the work of Foucault and Kirchhoff) that spectral lines give clear information about chemical composition. Why might our present knowledge have seemed unattainable in \(1835 ?\) Discuss how new discoveries can change the apparent limitations of science. Today, other questions seem beyond the reach of science, such as the question of how life began on Earth. Do you think such questions will ever be answerable through science? Defend your opinion.

A spectral line that appears at a wavelength of \(321 \mathrm{nm}\) in the laboratory appears at a wavelength of \(328 \mathrm{nm}\) in the spectrum of a distant object. We say that the object's spectrum is (a) redshifted. (b) blueshifted. (c) whiteshifted.

Image Resolution. What happens if you take a photograph from a newspaper, magazine, or book and blow it up to a larger size? Can you see more detail than you could before? Explain clearly. and relate your answer to the concepts of magnification and angular resolution in astronomical observations.

Space Observatory. Visit the website of a major space observatory, either existing or under development. Write a short report about the observatory, including its purpose, its orbit, and how it operates.
See all solutions

Recommended explanations on Physics Textbooks

Electromagnetism

Read Explanation

Fluids

Read Explanation

Electrostatics

Read Explanation

Engineering Physics

Read Explanation

Particle Model of Matter

Read Explanation

Astrophysics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.