Estimating Seating Capacity

Date: 05/10/2000 at 20:54:24
From: Rosie
Subject: area of a convention center

The floor of a convention center meeting room is being prepared for a 
gathering. You must estimate about how many chairs will be needed. The 
convention center floor measures 80 feet across at the front, and 120 
feet across at the rear. The back row will be 150 feet from the front. 
Allow for three 4-foot-wide aisles running from front to rear.

a. What is the total floor area?

b. What is the area of the aisles?

c. Subtract the area of the aisles from the total floor area to find 
the area available for seating. Allow a 2' x 3' space for each seat. 
Approximately how many chairs can be placed in the seating area?

d. What are some reasons why this calculation may be slightly 
incorrect? Do you think that the estimated seating capacity is a 
little high or a little low?

I don't know where to begin with this problem, so please HELP!

Date: 05/11/2000 at 03:09:16
From: Doctor Jesse
Subject: Re: area of a convention center

Stay cool! This problem is only about counting chairs. You can do 
this, but you have to take it one step at a time. Let's look at what 
you've got, one sentence at a time:

>The floor of a convention center meeting room is being prepared for 
>a gathering. You must estimate about how many chairs will be needed.

The key word here is "estimate."

>The convention center floor measures 80 feet across at the front, 
>and 120 feet across at the rear.

Okay... It is time to start drawing a picture of this place, but we 
need more information.

>The back row will be 150 feet from the front.

Now we have enough information to draw a picture of the hall. It looks 
something like:

         120ft
     ------------   |
     \          /   |
      \        /  150ft
       \      /     |
        ------      |
         80ft

My picture is horrible, so you should draw a better one. Imagine the 
shape of the room in your mind. Imagine you are standing in it. Where 
are the aisles?

>Allow for three 4-foot-wide aisles running from front to rear.

That won't fit in my diagram, but put the three aisles in yours.

>a. What is the total floor area?

The floor is the shape of a trapezoid. For now, forget the aisles, 
forget the chairs, and just think about the floor. Look in your book, 
and figure out the area of the trapezoid you have drawn in your 
diagram. I'll give you a hint:

     (top+bottom)/2 x height

That should give you an answer in square feet. That is how much carpet 
it would take to cover the whole floor of this convention hall.

>b. What is the area of the aisles?

You can figure that. They are just three rectangles, 150ft x 4ft., 
assuming they run straight along the room. Even if they don't, we're 
only estimating. The answer you get here (in square feet) is the 
amount of carpet it would take to carpet the aisles.

>c. Subtract the area of the aisles from the total floor area to 
>find the area available for seating.

Take the answer from a and subtract the answer from b. This is how 
much carpet it would take to cover everything BUT the aisles. We don't 
want chairs in the aisles (the Fire Marshal would close us down) so 
this is the square footage we are going to use.

>Allow a 2'x 3' space for each seat. Approximately how many chairs 
>can be placed in the seating area?

Each chair will take up 2 ft. x 3 ft., or 6 square feet. To figure out 
how many chairs will fit in the seating area, divide the number you 
just calculated at the start of part c by this number (6). This is the 
number of chairs that will fit in the hall.

>d. What are some reasons why this calculation may be slightly 
>incorrect? 

Going back to the diagram, we can see that the chairs can't really fit 
all the space. Each one is a 2x3 rectangle, and the space is a 
trapezoid. If you try to draw in a bunch of chairs as little 
rectangles, you can see that there will be little bits of wasted space 
on the sides. And for that reason...

>Do you think that the estimated seating capacity is a little high 
>or a little low?

...our estimate is a little high. Because of the irregular shape of 
the hall, there is going to be some wasted space, and to get an 
accurate count, we would need to account for that. But we're not 
trying to get an accurate count - we're only estimating.

I hope this helps you out, Rosie. Remember: the keys to word problems 
are:

   1. Stay cool.
   2. Take things one step at a time.
   3. Visualize the problem by drawing it and picturing yourself 
      there.

Keep up the good work!

- Doctor Jesse, The Math Forum
  http://mathforum.org/dr.math/