# Calculus

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 = \frac{c}{x^2} + \frac{k}{(d-x)^2}
where c and k are positive constants which depend on the quantity of smoke each stack is emitting. If k = 9 c, find the point on the line joining the stacks where the concentration of the deposit is a minimum.

I know this question has to do with optimizing but I am having a hard time taking the derivative of the equation

1. 👍 0
2. 👎 0
3. 👁 2,007
1. You probably expected this to come out as

S = c/x^2 + k/(d-x)^2
= c x^-2 + k(d-x)^-2
dS/dx = -2cx^-3 - 2k(d-x)^-3
= 0 for a min

c/x^3 = k/(d-x)^3
but k = 9c
c/x^3 = 9c/(d-x)^3
1/x^3 = 9/(d-x)^3
(d-x)^3 = 9x^3
d-x = (9^(1/3))x
d = x( 9^(1/3) - 1)

x = d/(9^(1/3) - 1)

1. 👍 0
2. 👎 4

## Similar Questions

1. ### Math

The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one four times as strong as the

2. ### Calculus

A light is suspended at a height h above the floor. The illumination at the point P is inversely proportional to the square of the distance from the point P to the light and directly proportional to the cosine of the angle θ. How

3. ### Math: Calculus

The max. weight M that can be supported by a beam is jointly proportional to its width w and the square of its height h, and inversely proportional to its length L. a) write an equation that expresses the proportionality. b)

4. ### Physics

I need help on this question: You are given information on the intensity level (which is a measure of the sound intensity) at a point and its distance from the sound source. To make use of this information, you will need to

1. ### physics-speed

i need help with this one thanks A particle starts from rest and is acted on by a net force that does work at a rate that is proportional to the time t. The speed of the particle is proportional to: a. sq. root t b. t c. t^2

2. ### Math-inverse proportion

The variables a and b are inversely proportional. When the sum of a and b is 24, their difference is 6. What is b when a equals 5?

3. ### PRE-CALC still stuck

Write the statement as a power function equation. Use k for constant of variation if one is not given. a. Charles's law states the volume V of an enclosed ideal gas at a constant pressure varies directly as the absolute

4. ### Inverse Variation

Help me please.. Explain also :( 1. E is inversely proportional to Z and Z = 4 when E = 6. 2. P varies inversely as Q and Q = 2/3 when P = 1/2. 3.R is inversely proportional to the square of I and I = 25 when R = 100. 4. F varies

1. ### algebra

Suppose p and q are inversely proportional. If p=28 when q=7, find the value of p when q= 49.

2. ### Physics

I have a hard time solving this problem. The weight of an object at the surface of a planet is proportional to the planet's mass and inversely proportional to the square of the radius of the planet. Jupiter's radius is 11 times

3. ### physics

if a stone at the end of a string is whirled in a circle, the inward pull on the stone A) is known as the centrifugal force B) is inversely proportional to the speed of the object C) is inversely proportional to the square of the

4. ### calculus

The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one three times as strong as