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• Nov 19th 2012, 05:59 PM
ophischauss
a staircase has stairs with a width of 11 inches and a height of 8 inches per step. a storage closet is to be built under the staircase between the 5th and 13th steps.

1. draw a model of the back wall of the closet showing the stairs.
2. if the ceiling of the closet is the bottom of the stairs, what is the area of back wall of the closet?
3. if the length of the step is 38 inches, then what is the volume of the closet?
4. instead, if a board is placed over the bottom of the stairs so that the top of the closet is flat but slanted, draw a new model of the back wall of the closet.

5. using geometry formulas, find the area of the back wall of the closet.
6. find an equation for the position of the board. use calculus to find the area of the back wall of the closet.
7. if the length of the step is 38 inches, the what is the volume of the closet with slanted ceiling?
• Nov 19th 2012, 06:04 PM
MarkFL
Please post what you have so far or what your thoughts are on what needs to be done, so we can assist you where you are stuck.
• Nov 19th 2012, 06:11 PM
ophischauss
Ok so I believe that this is a riemann sums problem. So far I figured that there were 9 steps and since each one is 8 inches tall, I multiplied 8 by the 9 steps. This gave me a total of 72 in. for how deep the closet is.
So I thought that maybe the formula for the closet was 72(x). I thought that maybe I should use riemann sums with the end points 5 and 13, but I am not even sure if this is the right approach.
• Nov 19th 2012, 06:29 PM
MarkFL
If you are to include the 5th and 13th steps, then you are correct about there being 9 steps over the closet, since (13 - 5) + 1 = 9. The way the problem is worded, I was unsrue if the ends are included. Let's assume they are.

To find the area A of the back wall in square inches, we could use the Riemann sum type of approach as you suggested:

$\displaystyle A=11\sum_{k=1}^9(32+8k)=88\sum_{k=1}^9(4+k)$

We could also deconstruct the back wall into a trapezoid and 9 congruent right triangles. Both give the same result.

Now, once you have the area of the back wall, multiply this area by the depth of the closet which is 38 inches, to get the volume in cubic inches.

Once you have correctly done this, we will move on to the next part.
• Nov 19th 2012, 06:50 PM
ophischauss
Where is the 32 and 8 coming from
• Nov 19th 2012, 07:00 PM
MarkFL
The measures are in inches, and 32 is the elevation of the 4th step and we want to add 8 for each step 1 through 9 of the included steps.
• Nov 19th 2012, 07:18 PM
ophischauss
So would the area be 6023.11
• Nov 19th 2012, 07:25 PM
MarkFL
No, you should get an integral answer somewhat larger than that. I suggest computing the area using both methods I suggested, and the second method is simpler to use and can be used as a method to check your result.
• Nov 19th 2012, 07:30 PM
ophischauss
I'm not sure what you mean by your second method, but I resolved it and got a value of 7128, is that correct?
• Nov 19th 2012, 07:45 PM
ophischauss
or do I evaluate it like this:
88*(4k+k^2/2)from 1 to 9
this would give an answer of 6336
• Nov 19th 2012, 07:50 PM
MarkFL
The second method was deconstructing the back wall into a trapezoid and 9 congruent right triangles to get:

$\displaystyle A=\frac{99}{2}(32+104)+\frac{9}{2}\cdot8\cdot11=71 28$

We may even use the area of the trapezoid in the next part of the problem where we are instructed to use a geometric method to compute the area of the back wall when a board is placed over the bottom of the stairs.

Okay, so you have correctly found the area of the back wall, so what is the volume in cubic inches of the closet?
• Nov 19th 2012, 07:54 PM
MarkFL
Quote:

Originally Posted by ophischauss
or do I evaluate it like this:
88*(4k+k^2/2)from 1 to 9
this would give an answer of 6336

No, you would want to use:

$\displaystyle A=88\sum_{k=1}^9(4+k)=88\left(4(9)+\frac{9(10)}{2} \right)=88(36+45)=792\cdot9=7128$
• Nov 19th 2012, 07:54 PM
ophischauss