I\'ve tried many times, just cannot seem to get it right.... For each of the fol

Question

I've tried many times, just cannot seem to get it right....

For each of the following forms determine whether the following limit type is indeterminate, always has a fixed finite value, or never has a fixed finite value. In the first case answer IND, in the second case enter the numerical value, and in the third case answer DNE.

those are the answers i inputted and they're not correct..

also... A smokestack deposits soot on the ground with a concentration inversely proportional to the square of the distance from the stack. With two smokestacks

d miles apart, the concentration of the combined deposits on the line joining them, at a distance x from one stack, is given by

where c and k are positive constants which depend on the quantity of smoke each stack is emitting. If k=2c, find the point on the line joining the stacks where the concentration of the deposit is a minimum.

I've tried many times, just cannot seem to get it right.... For each of the following forms determine whether the following limit type is indeterminate, always has a fixed finite value, or never has a fixed finite value. In the first case answer IND, in the second case enter the numerical value, and in the third case answer DNE. those are the answers i inputted and they're not correct.. also... A smokestack deposits soot on the ground with a concentration inversely proportional to the square of the distance from the stack. With two smokestacks d miles apart, the concentration of the combined deposits on the line joining them, at a distance x from one stack, is given by S = c/x2 + k/(d - x)2 where c and k are positive constants which depend on the quantity of smoke each stack is emitting. If k=2c, find the point on the line joining the stacks where the concentration of the deposit is a minimum.

Explanation / Answer

1- correct

2- correct

3- correct

4-correct

5-correct

6-correct

7-IND

8-correct

9-correct

10-0

11-correct

12-correct

13-correct

14-correct

15-correct

16-correct

17-correct

18-correct

19-correct

20-correct

S = c/x^2 + k/(d - x)^2

Plugging in k = 2c, we get:
S = c/x^2 + 2c/(d - x)^2

Now, differentiate S with respect to x:
dS/dx = -2c/x^3 + 4c/(d - x)^3

Set dS/dx = 0 to find where the minimum occurs:
-2c/x^3 + 4c/(d - x)^3 = 0
-(d - x)^3 + 2x^3 = 0
x = (d/8)(1 - 2^(1/3) + 2^(2/3))

So the point at which the concentration is minimum is a distance of (d/8)(1 - 2^(1/3) + 2^(2/3)) away from one stack.

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