Mathematics, 20.10.2019 21:30 adrianayepez8

What is all of the surface area and volume of this castle? find the surface area and volume of all the figures below, then out of all of the surface areas and volumes together. explain your work and round your final answers to the nearest tenth.

What is all of the surface area and volume of this castle? find the surface area and volume of all

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Answers

Answer from: QueenMiah16

Step-by-step explanation:

There are a few formulas that are useful for this:

lateral area of a pyramid or cone: LA = 1/2·Ph, where P is the perimeter and h is the slant heightlateral area of a cylinder: LA = π·dh, where d is the diameter and h is the heightarea of a rectangle: A = lw, where l is the length and w is the widthvolume of a cone or pyramid: V = 1/3·Bh, where B is the area of the base and h is the heightvolume of a cylinder or prism: V = Bh, where B is the area of the base and h is the height

You will notice that for lateral area purposes, a pyramid or cone is equivalent to a prism or cylinder of height equal to half the slant height. And for volume purposes, the volume of a pyramid or cone is equal to the volume of a prism or cylinder with the same base area and 1/3 the height.

Since the measurements are given in cm, we will use cm for linear dimensions, cm^2 for area, and cm^3 for volume.

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The heights of the cones at the top of the towers can be found from the Pythagorean theorem.

(slant height)^2 = (height)^2 + (radius)^2

height = √((slant height)^2 - (radius)^2) = √(10^2 -5^2) = √75 = 5√3

The heights of the pyramids can be found the same way.

height = √(13^2 -2^2) = √165

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Area

The total area of the castle will be ...

total castle area = castle lateral area + castle base area

These pieces of the total area are made up of sums of their own:

castle lateral area = cone lateral area + pyramid lateral area + cylinder lateral area + cutout prism lateral area

and ...

castle base area = cylinder base area + cutout prism base area

So, the pieces of area we need to find are ...

cone lateral area (2 identical cones)pyramid lateral area (2 identical pyramids)cylinder lateral area (3 cylinders, of which 2 are the same)cutout prism lateral areacylinder base area (3 cylinders of which 2 are the same)cutout prism base area

Here we go ...

Based on the above discussion, we can add 1/2 the slant height of the cone to the height of the cylinder and figure the lateral area of both at once:

area of one cone and cylinder = π·10·(18 +10/2) = 230π

area of cylinder with no cone = top area + lateral area = π·1^2 +π·2·16 = 33π

area of one pyramid = 4·4·(13/2) = 52

The cutout prism outside face area is equivalent to the product of its base perimeter and its height, less the area of the rectangular cutouts at the top of the front and back, plus the area of the inside faces (both vertical and horizontal).

outside face area = 2((23+4)·11 -3·(23-8)) = 2(297 -45) = 504

inside face area = (3 +(23-8) +3)·4 = 84

So the lateral area of the castle is ...

castle lateral area = 2(230π + 52) +33π + 504 + 84 = 493π +692

≈ 2240.805 . . . . cm^2

The castle base area is the area of the 23×4 rectangle plus the areas of the three cylinder bases:

cylinder base area = 2(π·5^2) + π·1^2 = 51π

prism base area = 23·4 = 92

castle base area = 51π + 92 ≈ 252.221 . . . . cm^2

Total castle area = (2240.805 +252.221) cm^2 ≈ 2493.0 cm^2

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Volume