Question

What is the size of the non-shaded area? (1) The square has sides 10 cm long (2) The semicircle has a radius of 5 cm A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

Short answer

C. 1 and 2 together (need both)

Step-by-step solution

Understand the problem

We need to determine the size of the non-shaded area given a square with sides of 10 cm and a semicircle with a radius of 5 cm.

Calculate the area of the square

The area of a square is given by the formula: \[ \text{Area}_{\text{square}} = \text{side}^2 \] Given the side length of the square is 10 cm, the area is: \[ \text{Area}_{\text{square}} = 10^2 = 100 \text{ cm}^2 \]

Calculate the area of the semicircle

The area of a semicircle is half the area of a full circle. The area of a full circle is given by the formula: \[ \text{Area}_{\text{circle}} = \pi r^2 \] Given the radius of the semicircle is 5 cm, the area of the circle is: \[ \text{Area}_{\text{circle}} = \pi (5^2) = 25\pi \text{ cm}^2 \] Thus the area of the semicircle is: \[ \text{Area}_{\text{semicircle}} = \frac{1}{2} \times 25\pi = 12.5\pi \text{ cm}^2 \]

Calculate the non-shaded area

Subtract the area of the semicircle from the area of the square to find the size of the non-shaded area: \[ \text{Area}_{\text{non-shaded}} = \text{Area}_{\text{square}} - \text{Area}_{\text{semicircle}} \] Substituting the values: \[ \text{Area}_{\text{non-shaded}} = 100 - 12.5\pi \] \[ \text{Area}_{\text{non-shaded}} \approx 100 - 39.27 \] \[ \text{Area}_{\text{non-shaded}} \approx 60.73 \text{ cm}^2 \]

Analyze the given choices

From the calculations, both the information about the square's side and the semicircle's radius were necessary to determine the non-shaded area. Therefore, the answer is: C. 1 and 2 together (need both)

Key concepts

Geometric Area Calculation

Let's start by exploring the basics of geometric area calculation. In the given problem, we have two geometric shapes: a square and a semicircle. Calculating the area of these shapes individually is the first step. For a square, the area formula is \( \text{side}^2 \). Given that each side of the square is 10 cm, the area comes out to be 100 square centimeters. For a semicircle, the area formula is half the area of a full circle: \( \frac{1}{2} \times \text{Area}_{\text{circle}} \). The full circle's area, given a radius of 5 cm, is \( \text{Area}_{\text{circle}} = \pi r^2 \). Thus, the semicircle's area is \( 12.5\pi \). By subtracting this from the square's area, we derive the non-shaded area.

GMAT Test Preparation

Preparing for GMAT can be daunting, but understanding problems like this one can enhance your quantitative reasoning skills. Focus on breaking down the problem into manageable steps. Recognize that geometry questions may require knowledge of basic formulas and the ability to apply them. For instance, the problem here required the use of area formulas for a square and a circle. Ensure you have these formulas memorized: Area of a square is \( \text{side}^2 \) and Area of a circle is \( \pi r^2 \). Practice solving similar problems within a set time to mimic actual test conditions. Keep practicing different types of geometry problems to build confidence.

Quantitative Reasoning

Quantitative reasoning involves using mathematical concepts to solve problems. The exercise is a perfect example. We start by interpreting the problem, recognize the shapes involved, and then decide which formulas apply. Key to quantitative reasoning is understanding the relationships between different elements of the problem. To solve, we first calculate the area of the square, \(10^2 = 100 \text{cm}^\text{2} \). Then, we calculate the semicircle's area, remembering that it’s half a circle’s area: \( \frac{1}{2} \times \pi r^2 = 12.5 \pi \). These calculations, followed by subtracting the semicircle's area from the square's area, demonstrate the logical steps required. Always recheck your work to catch any mistakes and ensure your final answer is correct.