Surface Areas and Volumes

The surface area of a cuboid is calculated using the formula 2(lb + bh + hl), where l is the length, b is the breadth, and h is the height of the cuboid. For example, a cuboid with dimensions 5cm, 4cm, and 3cm has a surface area of 2(5*4 + 4*3 + 3*5) = 94 cm².

The volume of a cylinder is found using the formula πr²h, where r is the radius of the base and h is the height. For instance, a cylinder with a radius of 7 cm and height of 10 cm has a volume of 22/7 * 7² * 10 = 1540 cm³.

Total surface area (TSA) includes all the surfaces of the object, while curved surface area (CSA) only includes the curved part. For a cylinder, TSA is 2πr(h + r) including the top and bottom circles, whereas CSA is 2πrh, excluding them.

To calculate the surface area of combined solids, break them down into individual solids, find their surface areas, and then add or subtract overlapping areas. For example, a cone on a hemisphere has a TSA of CSA of cone + CSA of hemisphere.

The volume of a sphere is given by 4/3πr³, where r is the radius. A sphere with a 7 cm radius has a volume of 4/3 * 22/7 * 7³ = 1437.33 cm³ approximately.

The slant height (l) of a cone can be found using the Pythagorean theorem: l = √(r² + h²), where r is the radius and h is the height. For a cone with r=3 cm and h=4 cm, l = √(3² + 4²) = 5 cm.

The curved surface area of a cone is πrl, where r is the radius and l is the slant height. For a cone with r=7 cm and l=25 cm, CSA = 22/7 * 7 * 25 = 550 cm².

The volume of combined solids is the sum of the volumes of individual solids. For example, a toy made of a cone and hemisphere has a volume of 1/3πr²h (cone) + 2/3πr³ (hemisphere).

The total surface area of a solid hemisphere is 3πr², including the base. The curved surface area is 2πr². For a hemisphere with r=10.5 cm, TSA = 3 * 22/7 * 10.5² = 1039.5 cm².

Rearrange the volume formula V=πr²h to h=V/(πr²). For a cylinder with V=1540 cm³ and r=7 cm, h = 1540 / (22/7 * 7²) = 10 cm.

The volume of a cone is 1/3πr²h, where r is the radius and h is the height. A cone with r=3 cm and h=4 cm has a volume of 1/3 * 22/7 * 3² * 4 = 37.71 cm³ approximately.

Rearrange the volume formula V=4/3πr³ to r=³√(3V/4π). For a sphere with V=38808 cm³, r = ³√(3*38808 / (4*22/7)) = 21 cm.

A cylinder has two parallel circular bases and a uniform cross-section, while a cone has one circular base and tapers to a point. The volume of a cylinder is πr²h, and a cone is 1/3 of that.

First, find the surface area to be painted, then multiply by the cost per unit area. For example, painting a cuboid of TSA 94 cm² at ₹0.50/cm² costs 94 * 0.50 = ₹47.

The TSA of a cone is πr(l + r), where r is the radius and l is the slant height. For a cone with r=7 cm and l=25 cm, TSA = 22/7 * 7 * (25 + 7) = 704 cm².

Capacity is the volume of the container. For a cylindrical tank with r=1.4 m and h=3 m, capacity is πr²h = 22/7 * 1.4² * 3 = 18.48 m³ or 18480 liters (since 1 m³=1000 liters).

The volume of a hollow cylinder is π(R² - r²)h, where R is the outer radius, r is the inner radius, and h is the height. For R=14 cm, r=7 cm, h=20 cm, volume = 22/7 * (14² - 7²) * 20 = 9240 cm³.

The TSA of the toy is CSA of cone + CSA of hemisphere. For a cone with l=5 cm and r=3.5 cm, and hemisphere with r=3.5 cm, TSA = πrl + 2πr² = 22/7 * 3.5 * 5 + 2 * 22/7 * 3.5² = 55 + 77 = 132 cm².

The volume of a frustum is 1/3πh(R² + Rr + r²), where R and r are the radii of the two bases, and h is the height. For R=8 cm, r=3 cm, h=12 cm, volume = 1/3 * 22/7 * 12 * (8² + 8*3 + 3²) = 1232 cm³.

Multiply the volume by the density. For an iron pole with volume=111532.8 cm³ and density=8g/cm³, weight = 111532.8 * 8 = 892262.4 g or 892.2624 kg.

The lateral surface area (LSA) of a cuboid is 2h(l + b), excluding the top and bottom. For a cuboid with l=10 cm, b=5 cm, h=8 cm, LSA = 2*8*(10 + 5) = 240 cm².

The diameter of the hemisphere equals the side of the cube. For a cube of side 7 cm, the largest hemisphere has a diameter of 7 cm, hence radius 3.5 cm.

The TSA of a capsule is CSA of cylinder + 2 * CSA of hemispheres = 2πrh + 2 * 2πr² = 2πr(h + 2r). For r=1.4 cm and h=5 cm, TSA = 2 * 22/7 * 1.4 * (5 + 2.8) = 68.64 cm².

The volume of water displaced equals the volume of the submerged part of the solid. For a sphere of radius 3 cm fully immersed, displaced water volume is 4/3πr³ = 113.14 cm³.

The TSA of a hollow cylinder is 2π(R + r)(R - r + h), where R is outer radius, r is inner radius, and h is height. For R=14 cm, r=7 cm, h=20 cm, TSA = 2 * 22/7 * 21 * 27 = 3564 cm².