Volumes of Revolution Word Problem

[xcgirl]

Homework Statement

Assume that the Earth is a sphere with circumference of 24,900 miles.
a. Find the volume of the Earth north of latitude 45 degrees. (hint: integrate with respect to y)
b. Find the volume of the Earth between the equator and latitude 45

Homework Equations

circle: x^2 + y^2 = r^2

The Attempt at a Solution

so far, I have just been working on A. I took a cross section of the sphere from latitude 45 and up and drew it on a graph. I realized that if i revolved it around the y-axis that it would form the shape I need, a dome.

I found the radius using the circumference and set up my integral.
i have: pi * int(124500pi - y^2 dy after simplifying.

I think I'm all set to integrate and find the answer, but I can't figure out what to use for the upper and lower bounds. I thought i might try and use sin(45) or cos(45) or tan(45), in some way, but I can't really wrap my head around the problem from this point forward.

 

[tiny-tim]

… avoid using huge numbers … !

i have: pi * int(124500pi - y^2 dy after simplifying.

I think I'm all set to integrate and find the answer, but I can't figure out what to use for the upper and lower bounds. I thought i might try and use sin(45) or cos(45) or tan(45), in some way, but I can't really wrap my head around the problem from this point forward.

Hi xcgirl!

Hint: with big numbers like this, just put the radius = r throughout the calculation, and then put the number for r in at the end - you're much less likely to make a mistake (like forgetting to square something!) - and you won't have five-digit limits for the integral sign!

Yes, your approach seems fine.

Integrate over y, from y = r/√2 to r.

(If in doubt as to whether it's sin or tan, draw a diagram!)

 

[xcgirl]

when i do this, i am getting very large numbers, larger even then the actual volume of the earth. I'm not sure what's going on.

 

[Tedjn]

The integral you might be using, which seems to be

$$V = \pi\int_{\frac{r}{\sqrt{2}}}^r (124500\pi - y^2)dy$$

doesn't look right. You may have simplified wrong; from where did you get $124500\pi$?

 

[tiny-tim]

… let's see …

when i do this, i am getting very large numbers, larger even then the actual volume of the earth. I'm not sure what's going on.

Hi xcgirl!

Show us the integral you used, before putting any numbers in (ie just using r), so we can see what is going wrong.

 

[thewave]

http://www4a.wolframalpha.com/Calculate/MSP/MSP103319a0269e42g61e9e00000ga3250hih542d42?MSPStoreType=image/gif&s=35&w=114&h=48
This is the integral I used and I get 1.6376*10^12 miles cubed which doesn't seem right.

 

[Mark44]

If the Earth were a cube 8000 miles on each side, its volume would be (8,000)³ mi³ = 512 x 10⁹ mi³ = 5.12 x 10¹¹ mi³. Being roughly spherical, the Earth would fit inside such a box, so its volume would be less than this. That makes the value too big by maybe two orders of magnitude, since you're calculating the volume above 45 degrees N.