1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
anzhelika [568]
3 years ago
12

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 o

f all of the surface areas and volumes together. Explain your work and round your final answers to the nearest tenth. ​

Mathematics
1 answer:
motikmotik3 years ago
4 0

Answer:

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 height
  • lateral area of a cylinder: LA = π·dh, where d is the diameter and h is the height
  • area of a rectangle: A = lw, where l is the length and w is the width
  • volume of a cone or pyramid: V = 1/3·Bh, where B is the area of the base and h is the height
  • volume 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.

___

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

___

<u>Area</u>

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 area
  • cylinder 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

___

<u>Volume</u>

The total castle volume will be ...

  total castle volume = castle cylinder volume + castle cone volume + castle pyramid volume + cutout prism volume

As we discussed above, we can combine the cone and cylinder volumes by using 1/3 the height of the cone.

  volume of one castle cylinder and cone = π(5^2)(18 + (5√3)/3)

  = 450π +125π/√3 ≈ 1640.442 . . . . cm^3

 volume of flat-top cylinder = π·1^2·16 = 16π ≈ 50.265 . . . . cm^3

The volume of one pyramid is ...

  (1/2)4^2·√165 = 8√165 ≈ 102.762 . . . . cm^3

The volume of the entire (non-cut-out) castle prism is the product of its base area and height:

  non-cutout prism volume = (23·4)·11 = 1012 . . . . cm^3

The volume of the cutout is similarly the product of its dimensions:

  cutout volume = (23 -8)·4·3 = 180 . . . . cm^3

so, the volume of the cutout prism is ...

  cutout prism volume = non-cutout prism volume - cutout volume

  = 1012 -180 = 832 . . . .  cm^3

Then the total castle volume is ...

  total castle volume = 2·(volume of one cylinder and cone) + (volume of flat-top cylinder) +2·(volume of one pyramid) +(cutout prism volume)

  = 2(1640.442) + 50.265 +2(102.762) +832 ≈ 4368.7 . . . . cm^3

You might be interested in
9.3+b=−42 , you would subtract 9.3 to isolate the variable.
Rama09 [41]

Answer:

yeh you are correct so it will be b is equal to minus 42 minus 9.3 that will be equal to -51.3 so that is the value of b

5 0
3 years ago
Please help me this is for a grade
IgorC [24]

Answer:

b

Step-by-step explanation:

4 0
3 years ago
I NEED THE CORRECT ANSWER TO THIS ASAP FOR A TEST!!!!! What is the solution to the inequality?
garik1379 [7]
The correct answer is c , m<6
6 0
3 years ago
Read 2 more answers
Helpppppppnmeeeeeeee plsss I will give you brainliest
Anon25 [30]

5: 6/5

6: -3

7: 2/5

8: 4/2 or 2

6 0
2 years ago
Read 2 more answers
If you sleep an average of 7.5 hours each night, how many hours do you sleep in a year?
satela [25.4K]

Answer:

\boxed{\sf 2737.5 \ hours}

Step-by-step explanation:

The average is 7.5 hours of sleep each night.

There are 365 nights in 1 year.

\sf Multiply \ the \ value \ by \ 365.

7.5 \times 365

2737.5

5 0
3 years ago
Read 2 more answers
Other questions:
  • 1. A group of eight girl's guides raised $590 for their group by selling cookies. Three of the
    15·1 answer
  • These are math problems can you pls help me
    11·1 answer
  • The first answer is GH. But if the lenght of EF is a units, then the lenghts of GH is
    9·1 answer
  • Complete the sentence to make true statements 5 is 100 times
    11·1 answer
  • Between 1 and 49 what number has the most factors
    5·1 answer
  • Please help asap 25 pts + brainliest to right/best answer
    14·1 answer
  • The graph shows a linear relationship between x and y. Based on the graph, what is the value of x when the value of y is 10? Wil
    9·1 answer
  • The average of 3 numbers is 15 two of the numbers are 7 and 10 what is the third number?
    15·1 answer
  • What is the perimeter, in feet, a rectangle whose length is twice its width and whose width is 8 feet?
    10·1 answer
  • Pls help me if I cabbnnnnsbdhdhdjdjxjkxx
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!