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.
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Here are the solutions to the problems from the image: 18. (d) Adam earned 36 515 in 2019. This was an increase of 9% on his earnings in 2018. Work out his earnings in 2018. Step 1: Understand the relationship between 2018 and 2019 earnings. If Adam's earnings increased by 9% in 2019, his 2019 earnings represent 100% + 9% = 109% of his 2018 earnings. Let E_2018 be his earnings in 2018. E_2019 = E_2018 × (1 + 0.09) E_2019 = E_2018 × 1.09 Step 2: Calculate his earnings in 2018. We are given E_2019 = \36\,515. \36\,515 = E_2018 × 1.09 E_2018 = (\36\,515)/(1.09) E_2018 = \33\,500 Adam's earnings in 2018 were \33\,500. 19. The diagram shows a sector of a circle with centre O, radius 8 cm and sector angle 165°. (a) Calculate the total perimeter of the sector. Step 1: Calculate the length of the arc. The formula for arc length L is L = ()/(360^) × 2 r. Given = 165^ and r = 8 cm. L = (165)/(360) × 2 × × 8 L = (165)/(360) × 16 L = (11)/(24) × 16 L = (11 × 2)/(3) × L = (22)/(3) cm L ≈ 23.038 cm Step 2: Calculate the total perimeter of the sector. The perimeter of the sector is the sum of the arc length and two radii. Perimeter P = L + 2r P = (22)/(3) + 2(8) P = (22)/(3) + 16 P ≈ 23.038 + 16 P ≈ 39.038 cm Rounding to one decimal place: P ≈ 39.0 cm The total perimeter of the sector is 39.0 cm. (b) The surface area of a sphere is the same as the area of the sector. Calculate the radius of the sphere. [The surface area, A, of a sphere with radius r is A = 4 r^2] Step 1: Calculate the area of the sector. The formula for the area of a sector A_sector is A_sector = ()/(360^) × r^2. Given = 165^ and r = 8 cm. A_sector = (165)/(360) × × 8^2 A_sector = (165)/(360) × 64 A_sector = (11)/(24) × 64 A_sector = (11 × 8)/(3) × A_sector = (88)/(3) cm^2 A_sector ≈ 92.153 cm^2 Step 2: Set the area of the sector equal to the surface area of the sphere. Let r_s be the radius of the sphere. The surface area of the sphere is A_sphere = 4 r_s^2. A_sphere = A_sector 4 r_s^2 = (88)/(3) Step 3: Solve for the radius of the sphere, r_s. Divide both sides by : 4r_s^2 = (88)/(3) Divide both sides by 4: r_s^2 = (88)/(3 × 4) r_s^2 = (88)/(12) r_s^2 = (22)/(3) r_s = sqrt((22)/(3)) r_s ≈ sqrt(7.33333) r_s ≈ 2.7080 cm Rounding to two decimal places (or 3 significant figures): r_s ≈ 2.71 cm The radius of the sphere is 2.71 cm. Got more? Send 'em.