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Thread: Riemanns Sum Problem

  1. #1
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    Riemanns Sum Problem

    The Question:
    Riemanns Sum Problem-question.png
    What I have so far:
    Riemanns Sum Problem-line-1.png
    Riemanns Sum Problem-line-2.png
    Riemanns Sum Problem-line-3.png
    I have also proven that the vertical cross-sections result in the same formulae for Volume:
    Riemanns Sum Problem-line-4.png
    The Questions:
    State one assumption that must be made and its associated effect in relation to finding a formula for the lightweight ‘pop-up’ tent.
    If the safety capacity for the benign use of a camping gas lamp inside a confined space is (20.8 m^3), calculate whether it would be safe to use the lamp within the lightweight ‘pop-up’ tent.
    We Know the height of the tent is 1.89m, and that the maximum volume for the torch is 20.8m^3, however in my volume equation I have an unknown variable, how do I solve?
    Any suggestions would be great thanks,
    MathsKid007
    Thanks from mburgess
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  2. #2
    MHF Contributor

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    Re: Riemanns Sum Problem

    You know that the height is 1.89 meters? I did not see that anywhere in the problem. Model the tent as a hemisphere with center at (0, 0, 0) and radius 1.89. The cross section at y= 0 is $\displaystyle x^2+ z^2= 1.89^2$. At a given height, z, x goes from $\displaystyle -sqrt{1.89^2- x^2}$ to $\displaystyle sqrt{1.89^2- z^2}$. So a cross section is a hexagon with $\displaystyle s= sqrt{1.89^2- z^2}$. As your picture shows, that can be considered six triangles each have base s and height $\displaystyle \frac{s\sqrt{3}}{2}$ so area $\displaystyle \frac{s^2\sqrt{3}}{4}$. The area of the hexagon is 6 times that, $\displaystyle \frac{3s^2\sqrt{3}}{2}= \frac{3(1.89^2- z^2)}{2}$.

    Taking the "thickness" of each horizontal "slab" to be $\displaystyle \Delta z$ the Riemann sum is $\displaystyle \frac{3}{2}\sum (1.89^2- z^2)\Delta z$. In the limit as we take more and more thinner and thinner that becomes the integral $\displaystyle \frac{3}{2}\int (1.89^2- z^2)dz$.
    Thanks from Vinod
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  3. #3
    Senior Member Vinod's Avatar
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    Re: Riemanns Sum Problem

    Quote Originally Posted by MathsKid007 View Post
    The Question:
    Click image for larger version. 

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ID:	38841
    What I have so far:
    Click image for larger version. 

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ID:	38842
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    I have also proven that the vertical cross-sections result in the same formulae for Volume:
    Click image for larger version. 

Name:	line 4.PNG 
Views:	10 
Size:	25.2 KB 
ID:	38845
    The Questions:
    State one assumption that must be made and its associated effect in relation to finding a formula for the lightweight ‘pop-up’ tent.
    If the safety capacity for the benign use of a camping gas lamp inside a confined space is (20.8 m^3), calculate whether it would be safe to use the lamp within the lightweight ‘pop-up’ tent.
    We Know the height of the tent is 1.89m, and that the maximum volume for the torch is 20.8m^3, however in my volume equation I have an unknown variable, how do I solve?
    Any suggestions would be great thanks,
    MathsKid007
    Hello,
    What is the answer provided to you? From where did you take height of the tent and maximum volume of torch?. It is not given any where in the problem?
    Last edited by Vinod; Jul 17th 2018 at 09:05 AM.
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