This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here's how to find the area of the shaded portion: The figure shows an annulus (a ring shape) formed by two concentric circles. The annulus is divided into 8 equal sectors, with 4 of them shaded. Step 1: Identify the radii of the circles. The inner radius is r_1 = 4 cm. The outer radius is r_2 = 7 cm. Step 2: Calculate the area of the annulus. The area of the annulus is the area of the outer circle minus the area of the inner circle. The formula for the area of a circle is A = r^2. A_annulus = r_2^2 - r_1^2 A_annulus = (r_2^2 - r_1^2) Substitute the given radii: A_annulus = ((7 cm)^2 - (4 cm)^2) A_annulus = (49 cm^2 - 16 cm^2) A_annulus = 33 cm^2 Step 3: Determine the fraction of the annulus that is shaded. The annulus is divided into 8 equal sectors, and 4 of these sectors are shaded. The shaded fraction is (4)/(8) = (1)/(2). Step 4: Calculate the area of the shaded portion. Multiply the area of the annulus by the shaded fraction. A_shaded = (1)/(2) × A_annulus A_shaded = (1)/(2) × 33 cm^2 A_shaded = (33)/(2) cm^2 Using the approximation = (22)/(7): A_shaded = (33)/(2) × (22)/(7) cm^2 A_shaded = 33 × (11)/(7) cm^2 A_shaded = (363)/(7) cm^2 The area of the shaded portion is (363)/(7) cm^2. 4 done, 1 left today. You're making progress.