Question

I need help with this problem. Given: BC = 10 inches AC = √50 inches m∠CBD = 60° m∠CAD = 90° Calculate the exact area of the shaded region. Show all work for full credit. picture shown!

Answer 1

Answer:

Area of the shaded region = 23.33 in²

Step-by-step explanation:

Area of a sector = $\frac{\theta}{360}(\pi r^{2})$

Where θ = Central angle subtended by an arc

r = radius of the circle

Area of the sector BCD = $\frac{60}{360}(\pi) (10^{2})$

                                       = 52.36 in²

Area of equilateral triangle BCD = $\frac{\sqrt{3} }{4}(\text{Side})^2$

                                                      = $\frac{\sqrt{3} }{4}(10)^2$

                                                      = $25\sqrt{3}$ in²

                                                      = 43.30 in²

Area of the shaded portion in ΔBCD = 52.36 - 43.3

                                                             = 9.06 in²

Area of sector CAD = $\frac{90}{360}(\pi)(\sqrt{50})^2$

                                 = 39.27 in²

Area of right triangle CAD = $\frac{1}{2}(\text{Base})(\text{Height})$

                                            = $\frac{1}{2}(\text{AC})(\text{AD})$

                                            = $\frac{1}{2}(\sqrt{50})(\sqrt{50})$

                                            = 25 in²

Area of the shaded part in the ΔACD = 39.27 - 25

                                                                         = 14.27 in²

Area of the shaded part of the figure = 9.06 + 14.27

                                                                = 23.33 in²