1. Riemanns Sum Problem

The Question:

What I have so far:

I have also proven that the vertical cross-sections result in the same formulae for Volume:

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

2. 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 $\displaystyle \frac{3}{2}\int (1.89^2- z^2)dz$.

3. Re: Riemanns Sum Problem

Originally Posted by MathsKid007
The Question:

What I have so far:

I have also proven that the vertical cross-sections result in the same formulae for Volume:

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?