Area Related To Circles

Explanation
The area of a sector can be calculated by multiplying the central angle (in radians) by the square of the radius of the circle and then dividing by 2. In this case, the radius is given as 21 cm and the angle is 150°. To convert the angle to radians, we use the formula: radians = (π/180) * degrees. So, the angle in radians is (π/180) * 150 = 5π/6. Plugging in the values, we get: (5π/6 * 21^2) / 2 = 577.5 cm2.

Explanation
The formula for the area of a sector of a circle is given by (θ/360) * π * r^2. In this case, the correct answer is 2θπr^2/720°, which is equivalent to (θ/360) * π * r^2. This formula takes into account the central angle (θ), the radius (r), and the fact that the angle is measured in degrees. By multiplying the central angle by π, dividing it by 360°, and then multiplying it by the square of the radius, we can calculate the area of the sector accurately.

Explanation
The length of the arc is given by the formula L = rθ, where L is the length of the arc, r is the radius of the circle, and θ is the central angle in radians. In this case, we are given that L = 5π cm. The area of the sector is given by the formula A = (1/2)r²θ, where A is the area of the sector. We are given that A = 20π cm². By substituting the given values into the formulas, we can solve for the radius. Using L = rθ, we can solve for θ: 5π = rθ. Rearranging the equation, we get θ = 5π/r. Substituting this into the formula for the area, we get A = (1/2)r²(5π/r). Simplifying, we get A = 5πr/2. Equating this to 20π, we get 5πr/2 = 20π. Simplifying, we get r = 8 cm. Therefore, the radius of the circle is 8 cm.

Explanation
The area of a major segment of a circle is given by the formula A = (θ/360)πr² - 0.5r²sinθ, where θ is the central angle of the segment and r is the radius of the circle. In this case, the central angle is 90 degrees since the chord AB makes a right angle at the center of the circle. The radius is given as 10 cm. Plugging in these values, we get A = (90/360)π(10)² - 0.5(10)²sin(90) = 0.25π(100) - 0.5(100)(1) = 25π - 50. Simplifying further, we get A ≈ 78.54 - 50 = 28.54 cm². Therefore, the correct answer is 260.7 cm².

Explanation
The area of the field where the horse can graze can be found by calculating the area of a circle with a radius of 56 meters (which is the length of the string). The formula for the area of a circle is A = πr^2, where A is the area and r is the radius. Plugging in the values, we get A = π(56^2) = 9856 m^2. Therefore, the correct answer is 9856 m^2.

Explanation
The ratio of the circumferences of two circles is directly proportional to the ratio of their radii. Since the ratio of the circumferences is 2:3, the ratio of their radii is also 2:3. The area of a circle is directly proportional to the square of its radius. Therefore, the ratio of the areas of the two circles is (2^2):(3^2), which simplifies to 4:9.

Explanation
The area described by the minute hand on the face of the clock in 5 minutes can be calculated using the formula for the area of a sector of a circle. The formula is (θ/360) * π * r^2, where θ is the angle of the sector in degrees, π is a mathematical constant approximately equal to 3.14, and r is the radius of the circle. In this case, the angle of the sector is 5 minutes out of 60 minutes, which is equal to 5/60 * 360 degrees. The radius of the clock is given as 21 cm. Plugging in these values into the formula, we get (5/60 * 360/360) * 3.14 * 21^2 = 115.5 cm2.

Explanation
The area enclosed between two concentric circles is determined by subtracting the area of the smaller circle from the area of the larger circle. In this case, the area of the larger circle is given as 770 cm2 and its radius is 21 cm. To find the radius of the smaller circle, we need to subtract its area from the area of the larger circle. The only option that results in a difference of 770 cm2 when subtracted from the area of the larger circle is 14 cm. Therefore, the radius of the inner circle is 14 cm.

Explanation
The perimeter of a semi-circular protector is the sum of the curved part and the straight part. Since the curved part is half of the circumference of a full circle, the curved part is equal to π times the radius. The straight part is equal to the diameter. So, we can set up the equation 2πr + d = 72, where r is the radius and d is the diameter. Solving this equation, we find that the diameter is 28 cm.

Explanation
The formula to find the length of an arc is L = (θ/360) * 2πr, where L is the length of the arc, θ is the angle subtended by the arc at the center, and r is the radius of the circle. Given that the length of the arc is 44 cm and the angle is 72°, we can substitute these values into the formula to find the radius of the circle. Once we have the radius, we can use the formula for the area of a circle, A = πr^2, to calculate the area. After performing the calculations, the area of the circle is found to be 3850 cm2.

Explanation
The area of the circle circumscribing a square can be found by taking the diagonal of the square and multiplying it by π/2. In this case, the square has an area of 64 cm2, so each side of the square is 8 cm. The diagonal of the square can be found using the Pythagorean theorem as √(8^2 + 8^2) = √128 = 11.31 cm. Multiplying this by π/2 gives us an area of approximately 100.57 cm2.

Explanation
The track consists of two straight sections and two semicircular ends. The area of each straight section can be calculated by multiplying its length by the width of the track. So, the total area of the straight sections is 2 * 120 m * 14 m = 3360 m². The area of each semicircular end can be calculated by multiplying half the circumference of the circle (π * radius) by the width of the track. So, the total area of the semicircular ends is 2 * (π * 35 m/2) * 14 m = 7056 m². Therefore, the total area of the track is 3360 m² + 7056 m² = 10416 m². Converting this to cm², the answer is 10416 m² * 10000 cm²/m² = 104160000 cm², which is closest to 7056 cm².

Explanation
The ratio of the areas of two circles is equal to the square of the ratio of their radii. In this case, the ratio of the areas is 16:49, which means the ratio of the radii is 4:7. Since the radius of the bigger circle is 14 cm, we can set up the equation 4/7 = x/14, where x is the radius of the smaller circle. Solving for x, we find that x = 8 cm. Therefore, the radius of the smaller circle is 8 cm.

Explanation
To find the area of a triangle, we can use the formula: Area = (1/2) * base * height. In this case, the base of the triangle is AC, and the height is the perpendicular distance from point B to line AC. Since angle ACB is 120 degrees, we can use the formula for the area of an equilateral triangle to find the height. The height will be (sqrt(3)/2) * 18 cm. Plugging in the values into the formula, we get Area = (1/2) * 18 cm * (sqrt(3)/2) * 18 cm = 140.3 cm2. Therefore, the correct answer is 140.3 cm2.

Explanation
The area of a sector can be found by using the formula A = (θ/360) * π * r^2, where θ is the angle of the sector and r is the radius. In this case, the angle is given as 120 degrees and the radius is given as 18 cm. Plugging these values into the formula, we get A = (120/360) * π * (18)^2 = (1/3) * π * 324 = 339.43 cm^2. Therefore, the correct answer is 339.43 cm^2.