Question 208577: Hi, I was hoping to get some help with this problem......
The gravitational attraction between two masses varies inversely as the square of the distance between them. The force of attraction is 4 lbs when the masses are 3ft apart, What is the attraction when the masses are 6ft apart?
Thank you so much!!!
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! variation equations can be found here:
http://www.purplemath.com/modules/variatn.htm
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stick with the basic definitions of directly proportional or inversely proportional and this website should help. i'll explain the difference below.
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if y is directly proportional to x, then y = k*x where k is a constant of proportionality.
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if y is inversely proportional to x, then y = k/x where k is a constant of proportionality.
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let's see how these work in practice.
if you want to know how it applies to your problem and skip the tutorial, then just jump to the bottom, otherwise continue.
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y is directly proportional to x with a constant factor of 1
y = x
x = 1,2,3 means y = 1,2,3
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y is directly proportional to x with a constant factor of 10
y = 10x
x = 1,2,3 means y = 10,20,30
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y is directly proportional to x with a constant factor of .5
y = .5x
x = 1,2,3 means y = .5,1,1.5
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y is directly proportional to x with a constant factor of -10
y = -10x
x = 1,2,3 means y = -10,-20,-30
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in all of the above, an increase in the value of x resulted in an increase in the value of y if you disregard the direction of the increase (just look at absolute values).
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now let's look at the same examples using inverse proportion.
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y is inversely proportional to x with a constant factor of 1
y = 1/x
x = 1,2,3 means y = (1),(1/2),(1/3)
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y is inversely proportional to x with a constant factor of 10
y = 10/x
x = 1,2,3 means y = (10/1),(/10/2),(10/3)
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y is inversely proportional to x with a constant factor of .5
y = .5/x
x = 1,2,3 means y = (.5/1),(.5/2),(.5/3)
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y is inversely proportional to x with a constant factor of -10
y = -10/x
x = 1,2,3 means y = (-10/1),(-10/2),(-10/3)
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in all of the above, an increase in the value of x resulted in a decrease in the value of y if you disregard the direction of the increase (just look at absolute values).
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with direct proportion, the absolute value of y increases as the absolute value of x increases, and decreases as the absolute value of x decreases.
if abs(x) goes up, abs(y) goes up, if abs(x) goes down, abs(y) goes down.
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with inverse proportion, the absolute value of y decreases as the absolute value of x increases, and increases as the absolute value of x decreases.
if abs(x) goes up, abs(y) goes down, if abs(x) goes down, abs(y) goes up.
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NOW TO YOUR PROBLEM
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The gravitational attraction between two masses varies inversely as the square of the distance between them.
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The force of attraction is 4 lbs when the masses are 3ft apart.
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What is the attraction when the masses are 6ft apart?
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let d = distance between them.
let g = the gravitational attraction between them.
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since this is an inverse relationship, the general form of our equation should be:
g = k/d^2
where g is the gravitational attraction between them and d is the distance between them and k is the constant of proportionality.
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the problem states that the force of attraction is 4 pounds when the masses are 3 feet apart.
putting this in our equation, we get:
4 = k/3^2
to solve for k, we multiply both sides of this equation by 3^2 to get:
4 * 3^2 = k
this means that k = 4 * 9 = 36
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we now know what k is and we can solve further.
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the question is what is the attraction when the masses are 6 feet apart.
k = 36
d = 6
formula is:
g = 36/d^2
this becomes:
g = 36/36
this becomes:
g = 1
this says that the gravitational attraction between them is 1 pound.
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at 3 feet it's 4 pounds and at 6 feet it's 1 pound. as they get further apart the gravitational attraction gets weaker.
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let's go back to the original equation to see if we got it right.
what is the gravitational pull at 3 feet?
k = 36
d = 3
g = 36/9 = 4
this means the value of k is confirmed as good.
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a word about units.
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g is in pounds and d is in feet.
k = g*d^2 means that k is in square feet pounds.
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our equation of 4 = 36 / 9 is really:
4 pounds = 36 square feet pounds / 9 square feet
when you divide 36 square feet pounds by 9 square feet, the square feet part of the equation cancels out and you are left with 4 pounds.
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you don't normally track the types of units as you go along, but sometime it helps to understand what is going on.
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