A solid S is made up of a cylindrical part and conical part. The height of the Soild is 4.5m. The common radius of the cylindrical part and the conical part is 0.9m. The height of the conical part is 1.4m.

a. Calculate the volume,correct to 1 decimal place,of soild S.

b. Calculate the total surface area of soild S.

c. A square base pillar offside 1.6m has the same volume as soild S. Determine the height of the pillar, correct to 1 decimal place.

r is cylinder radius and the radius of the base of the cone

H is the cylinder height
h is the cone height

The volume of a cylinder is the height times the base area
Vcyl = pi r^2 H
The volume of a anything with straight sides and apointy top is (1/3)base area*height
so
Vcone = (1/3) pi r^2 h

add them
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the surface area of the sides of a cylinder is 2 pi r H
the surface area of the bottom of the cylinder is pi r^2
the surface area of a cone is (1/2)perimeter * slant height
slant height = sqrt (r^2+h^2)
perimeter = 2 pi r
so total area = 2 pi rH + pi r^2 + (1/2)(2 pi r)(sqrt(r^2+h^2)

1.6^2 * height = that volume you got in part a above

a. To calculate the volume of the solid S, we need to find the volume of the cylindrical part and the volume of the conical part, and then add them together.

First, let's calculate the volume of the cylindrical part:
Volume of a cylinder = π * r^2 * h
where π is a constant approximately equal to 3.14159, r is the radius of the cylindrical part, and h is the height of the cylindrical part.

In this case, the radius (r) is 0.9m and the height (h) of the cylindrical part is given as half of the total height of the solid, which is 2.25m (since the total height is 4.5m). Therefore, the volume of the cylindrical part is:
Volume of cylindrical part = π * (0.9^2) * 2.25

Next, let's calculate the volume of the conical part:
Volume of a cone = (1/3) * π * r^2 * h
where r is the radius of the conical part and h is the height of the conical part.

In this case, the radius (r) is the same as the radius of the cylindrical part, which is 0.9m, and the height (h) of the conical part is given as 1.4m. Therefore, the volume of the conical part is:
Volume of conical part = (1/3) * π * (0.9^2) * 1.4

Finally, we can find the total volume of solid S by adding the volumes of the cylindrical and conical parts:
Total volume of solid S = Volume of cylindrical part + Volume of conical part

b. To calculate the total surface area of solid S, we need to find the curved surface area of the cylindrical part, the curved surface area of the conical part, and the base area of the conical part, and then add them together.

First, let's calculate the curved surface area of the cylindrical part:
Curved surface area of a cylinder = 2 * π * r * h
where r is the radius of the cylindrical part and h is the height of the cylindrical part.

In this case, the radius (r) is 0.9m and the height (h) of the cylindrical part is 2.25m, as mentioned earlier. Therefore, the curved surface area of the cylindrical part is:
Curved surface area of cylindrical part = 2 * π * 0.9 * 2.25

Next, let's calculate the curved surface area of the conical part:
Curved surface area of a cone = π * r * l
where r is the radius of the conical part and l is the slant height of the conical part.

In this case, the radius (r) is 0.9m and the slant height (l) can be found using the Pythagorean theorem:
l^2 = r^2 + h^2
l^2 = 0.9^2 + 1.4^2
l^2 = 0.81 + 1.96
l^2 = 2.77
l = √2.77

Therefore, the curved surface area of the conical part is:
Curved surface area of conical part = π * 0.9 * √2.77

Finally, let's calculate the base area of the conical part (which is the same as the curved surface area of the circular base):
Base area of conical part = π * r^2
Base area of conical part = π * 0.9^2

Now, we can find the total surface area of solid S by adding the curved surface areas of the cylindrical and conical parts, as well as the base area of the conical part:
Total surface area of solid S = Curved surface area of cylindrical part + Curved surface area of conical part + Base area of conical part

c. To find the height of the square base pillar with the same volume as solid S, we need to first calculate the volume of solid S using the formula from part (a). Then, we can use the volume formula for a square base pillar to find its height.

The volume formula for a square base pillar is:
Volume of square base pillar = side length^2 * height

In this case, the side length of the square base pillar is given as 1.6m, and we want the volume of the pillar to be the same as the volume of solid S. Therefore, we can set up the following equation:
Volume of square base pillar = Volume of solid S

By substituting the values and solving for the height of the pillar, we can determine its height.