College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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![**Title: Comparing Light Gathering Abilities of Telescopes**
**Text:**
How much more light can a telescope with a 14-inch diameter mirror gather compared with a 4-inch diameter telescope?
**Explanation:**
When comparing telescopes of different sizes, one key factor is the light-gathering ability, which depends on the area of the mirror or lens. The amount of light gathered by a circular mirror or lens is proportional to the area of the circle, which is calculated using the formula:
\[ \text{Area} = \pi \times \left(\frac{\text{diameter}}{2}\right)^2 \]
For a 14-inch diameter mirror:
\[ \text{Area}_{14} = \pi \times \left(\frac{14}{2}\right)^2 = \pi \times 49 \]
For a 4-inch diameter mirror:
\[ \text{Area}_{4} = \pi \times \left(\frac{4}{2}\right)^2 = \pi \times 4 \]
To find how much more light a 14-inch telescope can gather compared to a 4-inch telescope, we calculate the ratio of their areas:
\[ \text{Ratio} = \frac{\text{Area}_{14}}{\text{Area}_{4}} = \frac{\pi \times 49}{\pi \times 4} = \frac{49}{4} = 12.25 \]
Thus, a telescope with a 14-inch diameter mirror can gather 12.25 times more light than a 4-inch diameter telescope.
This significant increase in light-gathering ability allows for better observation of faint celestial objects and more detailed views of the night sky.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba28fbe0-fc04-46bc-bd01-9a9731588fb8%2F7056455f-2e28-41b4-99ec-190035ceef25%2F8jft82_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Comparing Light Gathering Abilities of Telescopes**
**Text:**
How much more light can a telescope with a 14-inch diameter mirror gather compared with a 4-inch diameter telescope?
**Explanation:**
When comparing telescopes of different sizes, one key factor is the light-gathering ability, which depends on the area of the mirror or lens. The amount of light gathered by a circular mirror or lens is proportional to the area of the circle, which is calculated using the formula:
\[ \text{Area} = \pi \times \left(\frac{\text{diameter}}{2}\right)^2 \]
For a 14-inch diameter mirror:
\[ \text{Area}_{14} = \pi \times \left(\frac{14}{2}\right)^2 = \pi \times 49 \]
For a 4-inch diameter mirror:
\[ \text{Area}_{4} = \pi \times \left(\frac{4}{2}\right)^2 = \pi \times 4 \]
To find how much more light a 14-inch telescope can gather compared to a 4-inch telescope, we calculate the ratio of their areas:
\[ \text{Ratio} = \frac{\text{Area}_{14}}{\text{Area}_{4}} = \frac{\pi \times 49}{\pi \times 4} = \frac{49}{4} = 12.25 \]
Thus, a telescope with a 14-inch diameter mirror can gather 12.25 times more light than a 4-inch diameter telescope.
This significant increase in light-gathering ability allows for better observation of faint celestial objects and more detailed views of the night sky.
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