Question

A science class compares the relative strengths of two telescopic lenses. Lens \(X\) produces a magnification of 3 \(\times 10^5\), and Lens \(Y\) produces a magnification of \(6 \times 10^2\). Which of the following statements accurately describes the relationship between the two lenses? A. Lens \(X\) produces a magnification \(200 \%\) that of lens \(Y\). B. Lens \(X\) is 200 times as strong as Lens \(Y\). C. Lens \(X\) produces a magnification \(500 \%\) that of lens \(Y\). D. Lens \(X\) is 500 times as strong as Lens \(Y\).

Short answer

Lens X is 500 times as strong as Lens Y.

Step-by-step solution

Write down given magnifications

Lens X magnification: \(3 \times 10^5\) Lens Y magnification: \(6 \times 10^2\)

Calculate the ratio of Lens X to Lens Y

Divide the magnification of Lens X by the magnification of Lens Y to find the relationship between the two lenses: \[\frac{3 \times 10^5}{6 \times 10^2}\]

Simplify the expression

Cancel out the common factors and simplify the expression: \[\frac{3 \times 10^5}{6 \times 10^2} = \frac{3}{6} \times \frac{10^5}{10^2}\] The fraction can be simplified to: \[\frac{1}{2} \times 10^3\] Now, we multiply by 1000 (since \(10^3 = 1000\)): \[500\]

Compare with the given statements

Now we check which statement matches with our result: A. Lens X produces a magnification 200% that of lens Y. (Incorrect) B. Lens X is 200 times as strong as Lens Y. (Incorrect) C. Lens X produces a magnification 500% that of lens Y. (Incorrect) D. Lens X is 500 times as strong as Lens Y. (Correct) The correct statement describing the relationship between Lens X and Lens Y is: \(D. \)Lens X is 500 times as strong as Lens Y.

Key concepts

Telescopic Lenses Comparison

Understanding how telescopic lenses compare is vital for astronomers, hobbyists, and students alike. When comparing the strength of lenses, such as in the exercise provided, it's important to consider the magnification power each lens offers. Magnification is a measure of how much larger a lens can make objects appear than they do to the naked eye.

For instance, a lens with a magnification of 3x can make an object appear three times closer than its actual distance. When comparing two lenses, such as Lens X and Lens Y with magnification powers of 3x10^5 and 6x10^2, respectively, you'll want to determine how many times more powerful one lens is relative to the other. In our case, we determined that Lens X is indeed 500 times more powerful than Lens Y, a significant difference that would be clearly noticeable in practical use.
This knowledge is applicable whenever you're faced with a scenario requiring a direct comparison of the capabilities of different lenses whether for studying celestial bodies or for photography.

Scientific Notation

Scientific notation is a method of expressing numbers that are too large or too small to be conveniently written in decimal form. It is often used by scientists, mathematicians, and engineers to make computations more manageable and to easily communicate very precise measurements. Scientific notation is expressed as the product of a number between 1 and 10 and a power of 10.

For example, in the magnifications cited for telescopic lenses exercise, Lens X has a magnification written as 3×10^5. This means that the lens can magnify an image 300,000 times. In contrast, Lens Y has a lesser magnification of 6×10^2, or 600 times. The powers of 10 notably simplify multiplication or division, as seen when calculating ratios, because you can add or subtract the exponents accordingly. Understanding how to interpret and manipulate scientific notation is crucial for handling data efficiently and avoiding errors in calculations.

Ratio and Proportion

The concept of ratio and proportion is foundational in mathematics, and it is extensively used in practical applications such as recipe adjustments, map reading, and in our example, comparing lens magnifications. A ratio is a quantitative relationship between two numbers showing the number of times one value contains or is contained within the other. Proportion, on the other hand, states that two ratios are equal.

In this exercise, we calculated the ratio of the magnification of Lens X to Lens Y by dividing the former by the latter. Simplification of the resulting fraction and the multiplication by a power of 10 gave us the exact number by which the magnification of one lens exceeds the other. This straightforward computation demystifies what might initially appear as a complex problem and equips students with the skill to compare various elements quantitatively, a skill that can be transferred to endless real-world situations.