/*! 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 30 The figure below shows 2 tangent... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 30

The figure below shows 2 tangent circles such that the 10 -centimeter diameter of the smaller circle is equal to the radius of the larger circle. What is the area, in square centimeters, of the shaded region? \ \(\mathbf{F} .10\) G. 75 H. \(\quad 5 \pi\) J. 10\(\pi\) K. 75\(\pi\)

Short Answer

Expert verified
A. 25蟺 B. 50蟺 C. 75蟺 D. 100蟺 E. 125蟺 Answer: C. 75蟺 Explanation: The shaded area is the difference between the areas of the two circles. We found that the area of the smaller circle is 25蟺 square centimeters, and the area of the larger circle is 100蟺 square centimeters. To find the shaded area, we subtracted the area of the smaller circle from the area of the larger circle (100蟺 - 25蟺), which resulted in 75蟺 square centimeters.

Step by step solution

01

Find the radius of both circles.

As given, the diameter of the smaller circle is 10 cm. To find the radius, we divide the diameter by 2. So, the radius of the smaller circle is 5 cm. Since the diameter of the smaller circle is equal to the radius of the larger circle, the radius of the larger circle is 10 cm.
02

Calculate the area of both circles.

To find the area of a circle, we use the formula, Area = 蟺 * (radius^2). For the smaller circle, the area is 蟺 * (5^2) = 25蟺 square centimeters. For the larger circle, the area is 蟺 * (10^2) = 100蟺 square centimeters.
03

Subtract the area of the smaller circle from the area of the larger circle to find the shaded area.

We need to find the difference between the areas of the two circles to get the area of the shaded region. So, Shaded Area = Area of larger circle - Area of smaller circle = (100蟺 - 25蟺) square centimeters = 75蟺 square centimeters. Therefore, the area of the shaded region is 75蟺 square centimeters (Answer choice K).

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Key Concepts

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

Geometry
Geometry is a branch of mathematics that deals with shapes, sizes, and the properties of space. When solving geometry problems, understanding the basic shapes and their attributes is crucial. One fundamental shape is the circle, which is perfectly round and is defined by its center and its radius. The radius is a straight line from the center to any point on the edge of the circle. A key aspect of solving problems involving circles, like the given exercise, is knowing the relationship between a circle's radius and its diameter鈥攖he diameter is twice the length of the radius.

In our exercise, the interplay between the two circles and their shared dimensions provide vital clues to solving for the shaded area. By recognizing that circles can have different sizes yet share a common diameter and radius, and that the area of a circle increases with the square of its radius, students can unlock the answers to complex geometry problems.
Circle properties
Circles have a set of unique properties that are essential to understanding and solving problems related to them. One of the most important properties is the formula used to calculate a circle's area: \( A = \pi \times r^2 \), where \( A \) is the area and \( r \) is the radius. This formula tells us that the area of a circle is directly proportional to the square of its radius, a concept which we used in the step-by-step solution to calculate the areas of the two tangent circles.

Another vital property is that when two circles are tangent to one another, they do not overlap, and the distance between their centers is equal to the sum of their radii. These relationships allow students to visually and mathematically recognize how to compute the areas necessary to find the shaded region as shown in the exercise.
ACT Math preparation
Preparing for the ACT Math section involves mastering various mathematics topics, including algebra, geometry, and trigonometry. For geometry, specifically, it's important to practice problems that involve circle properties, among other shapes, to be ready for the variety of questions that may appear on the exam. The given exercise is an excellent example of a typical ACT geometry question. Students can prepare by familiarizing themselves with the formulas for calculating a circle's area, understanding the relationship between a circle's radius and diameter, and practicing problems that involve subtracting areas to find the area of shaded regions, just like we did in the step-by-step solution.

Practice also includes improving your skills in visualizing geometric arrangements and manipulating equations to arrive at the correct answers. Timed quizzes and exercises can be beneficial in helping students get used to the pace of the ACT, ensuring that they can solve problems quickly and efficiently under exam conditions.

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