Mensuration

Concept: The surface area ( ) of a sphere with radius is given by the formula . The diameter ( ) of a sphere is twice its radius ( ). Step 1: Given information and formula Given: Surface area of the sphere, . Formula: . We will use the approximation . Step 2: Substitute the given surface area into the formula to find the radius ( ) Step 3: Solve for To isolate , multiply both sides by : Let's simplify this. We can see if 616 is divisible by 88. : Try dividing by common factors. 88 is . Is 616 divisible by 11? , so yes. . So, . Now, . Step 4: Solve for Take the square root of both sides: (Since radius must be positive, we take the positive square root). Step 5: Calculate the diameter ( ) The diameter . The diameter of the sphere is . This matches option (1).

Concept: The area of a sector of a circle with radius and central angle (in degrees) is given by the formula: Area of sector = . Step 1: Identify the given values
Radius of the sector, .
Angle of the sector, . Step 2: Substitute the values into the formula for the area of a sector Step 3: Simplify the fraction Step 4: Calculate Step 5: Calculate the area of the sector To simplify : Divide by common factors. Both are divisible by 2: . Divide by 2 again: . Divide by 3: . So, . The area of the sector is . This matches option (3).

Concept: The perimeter of a shape is the total length of its outer boundary. For this composite figure (a rectangle topped with a semi-circle), we need to sum the lengths of all the straight and curved parts that form the outside edge. Step 1: Identify the parts forming the outer boundary The given figure looks like a rectangle with a semi-circle placed on one of its longer sides. The outer boundary consists of:
The bottom straight side of the rectangle.
The two vertical straight sides of the rectangle.
The curved edge of the semi-circle on top. (Note: The straight side of the rectangle that forms the diameter of the semi-circle is {not} part of the outer perimeter of the composite figure, as it's an internal line.) Step 2: Determine the lengths of the straight sides from the diagram From the diagram:
Length of the bottom side of the rectangle = units (let's assume meters or cm, as the unit isn't specified but will be consistent).
Length of each vertical side of the rectangle = units. So, the contribution from these straight sides to the perimeter is units. Step 3: Determine the length of the curved semi-circular part The top side of the rectangle, which serves as the diameter of the semi-circle, has a length of units.
Diameter ( ) of the semi-circle = units.
Radius ( ) of the semi-circle = Diameter / 2 = units. (The "7" marked inside the semi-circle likely indicates this radius). The circumference (length of the curved part) of a full circle is given by the formula . For a semi-circle, the length of its curved part is half of this: . Using the common approximation : Length of the semi-circular arc = units. Step 4: Calculate the total perimeter of the figure The total perimeter is the sum of the lengths of all outer boundary parts: Perimeter = (Length of bottom side) + (Length of left vertical side) + (Length of right vertical side) + (Length of semi-circular arc) Perimeter = units Perimeter = units Perimeter = units. Step 5: Match with the given options The calculated perimeter is 56 units. This corresponds to option (2).

Concept: To cut the largest possible right circular cone from a cube, the base of the cone must be inscribed within one of the faces of the cube, and the height of the cone will be equal to the edge of the cube. Step 1: Visualize the situation Imagine a cube. The largest circular base of a cone that can fit on one face of the cube will have its diameter equal to the side length (edge) of that face. The height of this cone can be at most the edge of the cube if its apex touches the opposite face. Step 2: Relate the cone's dimensions to the cube's edge Let the edge of the cube be . Given, edge of the cube . For the largest cone:
The diameter of the base of the cone will be equal to the edge of the cube, . Diameter of cone's base = .
The radius ( ) of the base of the cone is half of its diameter. .
The height ( ) of the largest cone will be equal to the edge of the cube, . . (Though the height is not needed to find the radius of the base). Step 3: Calculate the radius of the cone Using and : The radius of the largest right circular cone that can be cut out is . This matches option (2).

Concept: The volume ( ) of a cylinder is given by the formula , where is the radius of the base and is the height of the cylinder. Step 1: Define the ratios for the two cylinders Let the first cylinder be Cylinder 1 and the second cylinder be Cylinder 2. Let their radii be and , their heights be and , and their volumes be and . Given: Ratio of radii: This means we can write and for some constant . Ratio of heights: This means we can write and for some constant . Step 2: Write the formulas for the volumes of the two cylinders Volume of Cylinder 1: Volume of Cylinder 2: Step 3: Find the ratio of their volumes We need to find : The terms cancel out: This can be written as: Step 4: Substitute the given ratios into the expression Substitute and : Step 5: Calculate the final ratio So, the ratio of their volumes is . This matches option (2).

Concept: (A) The diagonal ( ) of a cube with side length is given by the formula . (B) The lateral surface area (LSA) of a cube is the sum of the areas of its four vertical faces. Since each face is a square with area , the LSA is . Step 1: Find the side length ( ) of the cube using the given diagonal Given: Diagonal of the cube, . We know that . Substitute the given value of : To solve for , divide both sides by : We can simplify this using the property : So, the side length of the cube is . Step 2: Calculate the lateral surface area (LSA) of the cube The formula for the lateral surface area of a cube is LSA = . Substitute the value of into this formula: Since , we have: Step 3: Compare with the given options The calculated lateral surface area is . This matches option (4).

Concept: The total surface area (TSA) of a cone is the sum of the area of its circular base and its curved surface area (CSA). The formulas are:
Area of base =
Curved Surface Area (CSA) =
Total Surface Area (TSA) = TSA = Step 1: Identify the given dimensions of the cone Let the actual radius of the cone be and its actual slant height be . Given:
Radius of the cone,
Slant height of the cone, Here, and are general variables used in the problem statement and options. Step 2: Substitute these dimensions into the TSA formula The general formula for TSA of a cone is TSA = . Substitute and : Step 3: Simplify the expression First, rearrange the terms inside the second parenthesis to match common order: Now, distribute the term: Factor out from both terms: Step 4: Compare with the given options The simplified expression is . This matches option (2).

Concept: In an isosceles triangle, the altitude (height) from the vertex angle to the base bisects the base and forms two congruent right-angled triangles. We can use the Pythagorean theorem and the formula for the area of a triangle. Area of a triangle = . Step 1: Define variables and set up relationships Let be the height of the isosceles triangle (in cm). Let be the length of each of the equal sides (in cm). Given: "Each of equal sides of an Isosceles triangle is 2 cm greater than its height." So, . Given: Base of the triangle = . Step 2: Use the property of the altitude in an isosceles triangle The altitude to the base of an isosceles triangle divides the base into two equal parts. Length of half the base = . This altitude ( ), half the base (6 cm), and one of the equal sides ( ) form a right-angled triangle. Step 3: Apply the Pythagorean theorem In the right-angled triangle formed: Hypotenuse = One leg = Other leg = 6 cm By Pythagorean theorem: Substitute into this equation: Step 4: Solve for the height ( ) Expand : . So the equation becomes: Subtract from both sides: Subtract 4 from both sides: Divide by 4: So, the height of the triangle is 8 cm. (We can also find the equal side: . Check: . This is correct.) Step 5: Calculate the area of the triangle Area = Base = Height ( ) = Area = Area = Area = This matches option (4).

Concept: In a right-angled triangle, the altitude to the hypotenuse creates similar triangles and specific geometric relationships. The area can also be used. Given: is right-angled at A. AD BC. AB = 1 m = 100 cm. AC = 25 cm. (The value BD = 96.08 cm appears inconsistent with AB and AC if standard right triangle properties are strictly applied, so we will primarily derive AD from AB, AC, and the resulting BC.) Step 1: Calculate the hypotenuse BC In right , by Pythagorean theorem: . . cm. Numerically, cm. Step 2: Calculate altitude AD using area of Area of can be expressed in two ways: 1. (using legs) 2. (using hypotenuse and altitude AD) Equating them: . Substitute values: cm. Step 3: Numerical value of AD cm. Step 4: Conclusion The calculated value cm. This is closest to option (3) 24.02cm. The discrepancy likely arises from the given value of BD being inconsistent with AB and AC, or use of rounded values in the options. Based on the primary lengths AB and AC, 24.02cm is the most plausible intended answer among the choices.

Concept: This problem requires knowing the volume formulas for a cone, a hemisphere, and a cylinder, and applying the given conditions about equal bases and same height. Step 1: Define dimensions based on conditions "Equal bases" means they all have the same radius, let it be . "Same height" means they all have the same height, let it be . For a hemisphere standing on its circular base, its height is equal to its radius. So, if the hemisphere has the same height as the cylinder and cone, then for the hemisphere, its radius must also be . Since all three have equal bases (same radius ) AND same height ( ):
For the hemisphere: radius = height. So, .
Since all bases are equal, the radius of the cone and cylinder is also .
Since all heights are equal, the height of the cone and cylinder is also . The crucial condition is that for a hemisphere, its "height" (when standing on its base) *is* its radius. So, if they all have the same height , then for the hemisphere, its radius must be . And since they all have equal bases, the radius of the cone and cylinder must also be . Thus, for all three shapes: radius , and height . Step 2: Volume formulas
Volume of Cone ( ):
Volume of Hemisphere ( ):
Volume of Cylinder ( ): Step 3: Substitute and into the formulas
Cone:
Hemisphere: Its radius is . If it "stands on its base" and has height , then its radius must be equal to . So,
Cylinder: Step 4: Find the ratio of their volumes Ratio We can cancel out the common factor from all terms (assuming ): To get rid of the fractions, multiply all parts of the ratio by 3: So, the ratio of their volumes is 1 : 2 : 3.

Concept: This problem involves a rectangular solid (cuboid) where the dimensions are in a given ratio and the volume is known. We need to find the total surface area. Formulas for a cuboid with length , width , and height :
Volume ( ) =
Total Surface Area ( ) = Step 1: Express dimensions in terms of a common factor The ratio of length : width : height is 3 : 2 : 1. Let the common factor be . Then, length , width , and height . Step 2: Use the given volume to find The volume of the solid is given as . Using the formula : Divide by 6: Take the cube root of both sides: Step 3: Calculate the actual dimensions Now that we have :
Length
Width
Height Step 4: Calculate the Total Surface Area (TSA) Using the formula : The total surface area of the solid is .

Concept: The whole surface area (total surface area) of a closed right circular cylinder includes the area of its two circular bases and the area of its curved surface. Formula for Total Surface Area (TSA) of a cylinder: , where is the radius of the base and is the height. Step 1: Relate diameter, radius, and height Let be the diameter of the base and be the radius. We know that diameter . The problem states that the diameter of the base is equal to its height, . So, . Since , we have . This implies that the radius . Step 2: Substitute into the TSA formula The TSA formula is . Substitute : Step 3: Simplify the expression Simplify the fractions: To add these terms, find a common denominator (which is 2): So, This can also be written as .

Concept: The volume of a right circular cone is given by the formula , where is the radius of the base of the cone and is the height of the cone. Step 1: Identify the given dimensions of the cone Let the radius of the base of this specific cone be and its height be .
Radius of the base, .
Height of the cone, , is "equal to the radius of the base." This means . Step 2: Substitute these dimensions into the volume formula The volume formula for a cone is . Substitute and : Step 3: Simplify the expression First, calculate : Now substitute this back into the volume equation: Multiply the terms: Now, multiply by : The volume of the cone is .

Concept: This problem involves the formulas for the volume and surface area of a sphere.
Volume of a sphere ( ) = , where is the radius.
Surface Area of a sphere ( ) = , where is the radius. Step 1: Set up the ratio of volumes Let the two spheres have radii and , volumes and , and surface areas and . We are given that the ratio of their volumes is 1:8. Substitute the volume formula: The term cancels out from the numerator and denominator: This can be written as: Step 2: Find the ratio of their radii To find the ratio of the radii , take the cube root of both sides: Since and (because ), So, the ratio of their radii is 1:2. Step 3: Find the ratio of their surface areas The surface area of a sphere is . We need to find the ratio : The term cancels out: This can be written as: Step 4: Substitute the ratio of radii From Step 2, we found . Substitute this into the surface area ratio: So, the ratio of their surface areas is 1:4.

Concept: This problem involves relating the dimensions and perimeters of a square and an equilateral triangle, and then calculating the area of the equilateral triangle. Formulas needed:
For a square with side : Diagonal , Perimeter .
For an equilateral triangle with side : Perimeter , Area . Step 1: Find the side of the square Let the side of the square be . The diagonal of the square is given as cm. We know . So, . Dividing by , we get cm. Step 2: Calculate the perimeter of the square Perimeter of the square cm. Step 3: Find the side of the equilateral triangle Let the side of the equilateral triangle be . The problem states that the square and the equilateral triangle have equal perimeters. So, Perimeter of equilateral triangle cm. We also know . Therefore, . cm. Step 4: Calculate the area of the equilateral triangle The area of an equilateral triangle with side is . Substitute cm: The area of the equilateral triangle is .