Find the volume of the spherical sector if its total surface area is 132 cm², height is 7 cm and radius of the bounding cone is 4 cm.

To find the volume of the spherical sector, we need to follow these steps:

Step 1: Calculate the slant height of the spherical sector. The slant height can be found using the Pythagorean theorem. The height of the spherical sector is given as 7 cm, and the radius of the bounding cone is given as 4 cm. We can use these values to calculate the slant height.

Using the Pythagorean theorem, we have: slant height² = height² + radius² slant height² = 7² + 4² slant height² = 49 + 16 slant height² = 65

Taking the square root of both sides, we get: slant height = √65

Step 2: Calculate the radius of the spherical sector. The radius of the spherical sector is the same as the radius of the bounding cone, which is given as 4 cm.

Step 3: Calculate the volume of the spherical sector. The volume of a spherical sector can be calculated using the formula: Volume = (2/3) * π * radius³ * (1 - cos(θ))

In this case, the total surface area of the spherical sector is given as 132 cm². We can use this information to find the angle (θ) of the sector.

The total surface area of a spherical sector is given by the formula: Total surface area = 2πrh + πr²(1 - cos(θ))

Substituting the given values, we have: 132 = 2π * 4 * 7 + π * 4²(1 - cos(θ)) 132 = 56π + 16π(1 - cos(θ)) 132 = 56π + 16π - 16πcos(θ) 132 = 72π - 16πcos(θ)

Simplifying the equation, we get: 16πcos(θ) = 72π - 132 16πcos(θ) = 72π - 132 cos(θ) = (72π - 132) / (16π) cos(θ) = 9 - 33/8 cos(θ) = 3/8

Using the inverse cosine function, we can find the value of θ: θ = cos⁻¹(3/8)

Now that we have the value of θ, we can substitute it into the volume formula to find the volume of the spherical sector: Volume = (2/3) * π * radius³ * (1 - cos(θ)) Volume = (2/3) * π * 4³ * (1 - cos(cos⁻¹(3/8)))

Simplifying further, we get: Volume = (2/3) * π * 64 * (1 - 3/8) Volume = (2/3) * π * 64 * (5/8) Volume = (2/3) * π * 40 Volume = 80π

Therefore, the volume of the spherical sector is 80π cubic units.