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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36. Step 1: Convert each fraction to a decimal to compare them. (5)/(11) ≈ 0.4545 (3)/(7) ≈ 0.4286 (6)/(13) ≈ 0.4615 (2)/(5) = 0.4 (4)/(9) ≈ 0.4444 Step 2: Arrange the decimals in ascending order. 0.4, 0.4286, 0.4444, 0.4545, 0.4615 Step 3: Write the original fractions in ascending order. (2)/(5), (3)/(7), (4)/(9), (5)/(11), (6)/(13) 37. Step 1: Identify the type of triangle. The markings indicate it is an isosceles triangle, meaning two sides are equal, and the base angles opposite those sides are also equal. Step 2: Calculate the base angles of the isosceles triangle. The sum of angles in a triangle is 180^. Let the base angles be x. 48^ + x + x = 180^ 48^ + 2x = 180^ 2x = 180^ - 48^ 2x = 132^ x = (132^)/(2) x = 66^ Step 3: Determine the value of angle t^. Angle t^ and the base angle x form a linear pair on a straight line, so their sum is 180^. t^ + x = 180^ t^ + 66^ = 180^ t^ = 180^ - 66^ t^ = 114^ The value of the angle marked t^ is 114^. 38. Step 1: Use the area of a right-angled triangle formula to find the height. Area A = (1)/(2) × base × height Given A = 96 m^2 and base b = 12 m. 96 m^2 = (1)/(2) × 12 m × h 96 m^2 = 6 m × h h = 96 m^26 m h = 16 m Step 2: Use the Pythagorean theorem to find the hypotenuse. Let the hypotenuse be c. c^2 = b^2 + h^2 c^2 = (12 m)^2 + (16 m)^2 c^2 = 144 m^2 + 256 m^2 c^2 = 400 m^2 c = sqrt(400 m)^2 c = 20 m Step 3: Calculate the perimeter of the triangle. Perimeter P = base + height + hypotenuse P = 12 m + 16 m + 20 m P = 48 m The perimeter of the triangle is 48 m. 39. Step 1: Identify the shape. The figure shows a semicircle with a right-angled triangle inscribed within it. The shaded region is the area of the semicircle minus the area of the triangle. The sides 42 cm and 56 cm are the legs of the right-angled triangle, and its hypotenuse is the diameter of the semicircle. Step 2: Calculate the diameter of the semicircle using the Pythagorean theorem. d^2 = (42 cm)^2 + (56 cm)^2 d^2 = 1764 cm^2 + 3136 cm^2 d^2 = 4900 cm^2 d = sqrt(4900 cm)^2 d = 70 cm Step 3: Calculate the radius of the semicircle. r = (d)/(2) = 70 cm2 = 35 cm Step 4: Calculate the area of the semicircle. Use = (22)/(7). A_semicircle = (1)/(2) r^2 A_semicircle = (1)/(2) × (22)/(7) × (35 cm)^2 A_semicircle = (1)/(2) × (22)/(7) × 1225 cm^2 A_semicircle = 11 × (1225)/(7) cm^2 A_semicircle = 11 × 175 cm^2 A_semicircle = 1925 cm^2 Step 5: Calculate the area of the triangle. A_triangle = (1)/(2) × base × height A_triangle = (1)/(2) × 42 cm × 56 cm A_triangle = 21 cm × 56 cm A_triangle = 1176 cm^2 Step 6: Calculate the area of the shaded region. A_shaded = A_semicircle - A_triangle A_shaded = 1925 cm^2 - 1176 cm^2 A_shaded = 749 cm^2 The area of the shaded region is 749 cm^2. 40. Step 1: List the number of slips of each color. Number of blue slips = 3 Number of red slips = 4 Number of green slips = 2 Number of yellow slips = 6 Step 2: Calculate the total number of slips. Total slips = 3 + 4 + 2 + 6 = 15 Step 3: Determine the number of red slips. Number of red slips = 4 Step 4: Calculate the probability of drawing a red slip. P(red) = Number of red slipsTotal number of slips P(red) = (4)/(15) The probability of it being a red slip is (4)/(15). That's 2 down. 3 left today — send the next one.