A building contractor needs to compute the volume of air contained in a building with a rectangular floor, as shown in the figure below...

eddy2017

eddy2017

Elite Member
Joined
Oct 27, 2017
Messages
2,525
A building contractor needs to compute the volume of air contained in a building with a rectangular floor, as shown in the figure below. The building is 20 feet long, 4 yards wide, 9 12 feet high on two sides 9 and a half feet high on two sides and 15 feet high from the floor to the peak. What is the volume of air in the building in cubic feet?
1644765184016.png
the great FIND = Volume of air in the building in cubic ft.
measurements of the building thaat have been given
l =20 ft
w=4 yards ( I am converting to a single unit so 1f=3yds = 4* 3= 12 ft)
height on the sides=9 1/2 ft
height from the floor to the top of the ceiling =15 ft

The formula for volume applies here
[math]V = l * w * h[/math]
I have two heights so that is what got me stuck, any hints?
thanks,
 
what to do when you have to find the volume (of a structure) and you are giving two heights.
My guess: find two volumes: the volume for the floor to the top of the sides of the structure
then, the volume from the floor to the top of the ceiling then add them together.
Is this assumption correct?
 
What does 9 12 feet mean? Are two walls supposed to have a different height from two other walls?

Yes, your assumption is correct. You will add the volumes of a rectangular prism and a triangular prism.
 
V(rectangular prism)=l × w × h
=20 × 12 × 9.5
=2280ft^3

Vtriangular prism= Bh ,
where B represents the area of the base.
The base of this triangular prism is made up of two triangles.
The area of a triangle is equal to 1/2 the base times the height.
B(triangle)=1/2bh
So, I can substitute B on the formula for a triangular prism for 1/2*b*h
So

V(tp)=(1/2bh)h

But now I see that I have the same variable for two different things because I have two different heights..
I will distinguish them by naming them h1 and h2.
V(tp)=(1/2bh1)h2
h1 = the height of our triangle, the height of our base.
h2=is the height of our original formula. This h represents the height of our prism.

Let's go thru our triangular prism and see if I can identify these variables.
V(tp)=(1/2bh)h
Let's find the base of our triangle
b=12ft
h1= this is the height of our triangle.
To find it, I am subtracting the height from floor to peak from the height from both sides of our rectangle prism.
So,15ft - 9.5ft=5.5 ft
So h1 = 5.5 ft

Now to h2 which is the height of our prism.

To find the height of our prism I want to find the distance between our two bases.
The distance between the two bases of our triangular prism is 20 ft.
So,h2= 20ft,
Now I have values for each one of the variables in my formula.

Let's plug'em in.
V(tp)=(1/2bh1)h2
Vtp=(1/2(12ft*5.5ft))20ft
Simplifying right hand side,
Vtp=(1/2(66ft))20ft
=33ft*20ft
=660f^3
Now adding the two volumes together,
Vrp + Vtp
2280ft^3 +660ft^3
=2940 ft^3$$
The volume of air in cubic feet in the building is 2940ft^3.

Does this make sense to you?.
thanks,
 
Last edited:
Eddy

It may be right. But I am having trouble following you. And you never answered the questions in my previous post. What are the exact words of the problem? Is 9 12 supposed to be [imath]9\frac{1}{2}?[/imath] How can the walls have different heights?
 
It is 9 and 1/2

How can the walls have different heights?.
Well, according to the shape given the height of the walls is the same on all sides ( talking about the rectangular shape) are 9 ft and a half.

The front wall has a height of 15 ft but from floor to peak of ceiling

You said adding the volume of the triangular prism which is the top part and the rectangular prism which is the bottom part was good. So, that is what I tried to do applying the formulas for each one.
 
Last edited:
Eddy,

You do see that the "composite" volume is made by adding:
  1. a rectangular prism (RP) and

  2. a triangular right prism (TP) with a triangular base
Can you draw those two solids - separated ? Make sure you think about how you are going to put those two solids together to get back the composite figure.

  1. TP is made from two triangles (base and roof) and 3 rectangles (2 of those make the slanted roof)

  2. RP is made from 6 rectangles ( 1 floor and 4 sidewalls)

What is the height of the rectangular prism?
 
okay, But, please, can you review what I posted #4 and tell me what is good and what is bad?. Or maybe nothing is right. I need to have a feedback on my work @4. I studied working with both formulas and and I typed it all that up on my cell and I would like, if possible, you to give me a feedback on that.
Jeff wrote that it might be right, but that i needed to answer a couple of questions he had asked me. I would like you to check up on that work and tell me what is wrong about it.
 
you said adding was cor
Subhotosh Khan said:
Eddy,

You do see that the "composite" volume is made by adding:
  1. a rectangular prism (RP) and

  2. a triangular right prism (TP) with a triangular base
Can you draw those two solids - separated ? Make sure you think about how you are going to put those two solids together to get back the composite figure.

  1. TP is made from two triangles (base and roof) and 3 rectangles (2 of those make the slanted roof)

  2. RP is made from 6 rectangles ( 1 floor and 4 sidewalls)

What is the height of the rectangular prism?
Click to expand...
you said, like Jeff, adding the two solids volume was correct.
my question is before I go on, are the formulas employed wrong, or the process is wrong?
 
eddy2017 said:
Let's plug'em in.
V(tp)=(1/2bh1)h2
Vtp=(1/2(12ft*5.5ft))20ft
Simplifying right hand side,
Vtp=(1/2(66ft))20ft
=33ft*20ft----Should be 33ft^2
=660f^3
Now adding the two volumes together,
Vrp + Vtp
2280ft^3 +660ft^3
=2940 ft^3$$
The volume of air in cubic feet in the building is 2940ft^3.
Click to expand...
See the comment above, but I think your calculation is correct.
 
Your numbers are correct. Are you visualizing the problem correctly? My problem is your question in #2:
eddy2017 said:
what to do when you have to find the volume (of a structure) and you are giving two heights.
Click to expand...
That is why I wanted you to take the "composite" solid apart and then put it back together. As far as I can see, you have done that correctly.
 
Subhotosh Khan said:
Your numbers are correct. Are you visualizing the problem correctly? My problem is your question in #2:

That is why I wanted you to take the "composite" solid apart and then put it back together. As far as I can see, you have done that correctly.
Click to expand...
Thanks for confirming it, Doc. I was a good problem to do. the triangular prism gave a little bit of a hard time but it was good after I watched a couple of tutorials
 
Thanks to BBB. The first one to confirm it. I was watching another video and it showed an easier way. I may show it here.
 
A Math teacher also told me this. I am bringing it up for your approval.
he drew two solid figures out of my shape and told me this:

''In your case the base of the prism is the "house shaped" pentagon which can be seen in my drawing on the left side
You may see it as a combination of two trapezoids. Whatever way you look at it, its should be not much effort to calculate its area. My thought is that the easiest way would be to combine the two trapezoids in your mind to form a rectangle whose area obviously is :


12/2ft⋅(15ft+9.5ft)
And so the volume of the whole object is
V=12/2ft⋅(15ft+9.5ft)⋅20ft

which operation amounts to the same result I got. What do you think?
he practically did this,

V(total)= w/2⋅ (height1 + height2)* width
 
You must log in or register to reply here.
Top