The length of an arc of a circle is given by the formula L = (theta)/(360^) x 2pi r. Calculate the angle theta if the arc length is 1 (19)/(36) cm and the radius is 3.5 cm. Use pi = (22)/(7).

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Asked on April 9, 2026Mathematics

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.

Accepted Answer · 2026-04-09T09:48:54.686Z

Here are the solutions to the questions. Question 31: The length of an arc of a circle is given by the formula L = ()/(360^) × 2 r. Step 1: Convert the given arc length and radius to fractions. L = 1 (19)/(36) cm = (36 × 1 + 19)/(36) cm = (55)/(36) cm r = 3.5 cm = (7)/(2) cm We are given = (22)/(7). Step 2: Substitute the values into the arc length formula. (55)/(36) = ()/(360^) × 2 × (22)/(7) × (7)/(2) Step 3: Simplify the right side of the equation. (55)/(36) = ()/(360^) × (2 × (22)/(7) × (7)/(2)) (55)/(36) = ()/(360^) × 22 Step 4: Solve for . = (55)/(36) × (360^)/(22) = (55 × 360^)/(36 × 22) = (55 × 10^)/(22) = (5 × 10^)/(2) = 5 × 5^ = 25^ The final answer is C. 25^. Question 32: In the diagram, PU SR and PS TR. This means that PTRS is a parallelogram. We are given that the area of SUR = 24 cm^2 and SR = 8 cm. Step 1: Find the height of SUR. Let h be the perpendicular height from U to the base SR. The area of a triangle is given by (1)/(2) × base × height. 24 cm^2 = (1)/(2) × SR × h 24 cm^2 = (1)/(2) × 8 cm × h 24 cm^2 = 4 cm × h h = 24 cm^24 cm = 6 cm Step 2: Calculate the area of parallelogram PTRS. Since PU SR, the perpendicular distance between the parallel lines PU and SR is h. The parallelogram PTRS has base SR = 8 cm and height h = 6 cm. The area of a parallelogram is given by base × height. Area of PTRS = SR × h Area of PTRS = 8 cm × 6 cm Area of PTRS = 48 cm^2 The final answer is B. 48 cm^2. Question 33: In the diagram, PQR is a circle with center O. OP, OQ, and OR are radii of the circle. We are given OPQ = 48^. Step 1: Find POQ. Since OP and OQ are radii, OPQ is an isosceles triangle with OP = OQ. Therefore, the base angles are equal: OQP = OPQ = 48^. The sum of angles in a triangle is 180^. POQ = 180^ - ( OPQ + OQP) POQ = 180^ - (48^ + 48^) POQ = 180^ - 96^ POQ = 84^ Step 2: Find the value of m^, which is ROQ. From the diagram, R, O, P lie on a straight line, which means RP is a diameter. Angles on a straight line sum to 180^. ROQ + POQ = 180^ m^ + 84^ =

Accepted Answer · 2026-04-09T09:48:54.686Z

Here are the solutions to the questions. Question 31: The length of an arc of a circle is given by the formula L = ()/(360^) × 2 r. Step 1: Convert the given arc length and radius to fractions. L = 1 (19)/(36) cm = (36 × 1 + 19)/(36) cm = (55)/(36) cm r = 3.5 cm = (7)/(2) cm We are given = (22)/(7). Step 2: Substitute the values into the arc length formula. (55)/(36) = ()/(360^) × 2 × (22)/(7) × (7)/(2) Step 3: Simplify the right side of the equation. (55)/(36) = ()/(360^) × (2 × (22)/(7) × (7)/(2)) (55)/(36) = ()/(360^) × 22 Step 4: Solve for . = (55)/(36) × (360^)/(22) = (55 × 360^)/(36 × 22) = (55 × 10^)/(22) = (5 × 10^)/(2) = 5 × 5^ = 25^ The final answer is C. 25^. Question 32: In the diagram, PU SR and PS TR. This means that PTRS is a parallelogram. We are given that the area of SUR = 24 cm^2 and SR = 8 cm. Step 1: Find the height of SUR. Let h be the perpendicular height from U to the base SR. The area of a triangle is given by (1)/(2) × base × height. 24 cm^2 = (1)/(2) × SR × h 24 cm^2 = (1)/(2) × 8 cm × h 24 cm^2 = 4 cm × h h = 24 cm^24 cm = 6 cm Step 2: Calculate the area of parallelogram PTRS. Since PU SR, the perpendicular distance between the parallel lines PU and SR is h. The parallelogram PTRS has base SR = 8 cm and height h = 6 cm. The area of a parallelogram is given by base × height. Area of PTRS = SR × h Area of PTRS = 8 cm × 6 cm Area of PTRS = 48 cm^2 The final answer is B. 48 cm^2. Question 33: In the diagram, PQR is a circle with center O. OP, OQ, and OR are radii of the circle. We are given OPQ = 48^. Step 1: Find POQ. Since OP and OQ are radii, OPQ is an isosceles triangle with OP = OQ. Therefore, the base angles are equal: OQP = OPQ = 48^. The sum of angles in a triangle is 180^. POQ = 180^ - ( OPQ + OQP) POQ = 180^ - (48^ + 48^) POQ = 180^ - 96^ POQ = 84^ Step 2: Find the value of m^, which is ROQ. From the diagram, R, O, P lie on a straight line, which means RP is a diameter. Angles on a straight line sum to 180^. ROQ + POQ = 180^ m^ + 84^ =

The length of an arc of a circle is given by the formula L = (theta)/(360^) x 2pi r. Calculate the angle theta if the arc length is 1 (19)/(36) cm and the radius is 3.5 cm. Use pi = (22)/(7).

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The length of an arc of a circle is given by the formula L = (theta)/(360^) x 2pi r. Calculate the angle theta if the arc length is 1 (19)/(36) cm and the radius is 3.5 cm. Use pi = (22)/(7).

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