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Question 1109132: The height of a ant mound is directly proportional to the square root of the number of ants. The height is 35 cm when there are 2000 termites. What is the height of this mound in cm when there are 10000 termites.
Found 2 solutions by josgarithmetic, Theo:
Answer by josgarithmetic(39620)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! the height is directly proportional to the square root of the number of ants.
the height is 35 cm when there are 2,000 ants.
what is the height when there are 10,000 ants.
you can use the direct variation formula on this.
the direct variation formula is y = k * x.
let y = the height of the ant mound.
let x = the number of ants.
k is the constant of proportionality.
since the height is directly proportion to the square root of the number of ants, then this formula becomes:
y = k * sqrt(x)
when y = 35, x = 2,000
formula becomes 35 = k * sqrt(2,000)
solve for k to get k = 35 / sqrt(2,000) = .7826237921.
since k is the constant of proportionality, it stays the same, regardless of the values of x.
therefore, when there are 10,000 ants, the formula becomes:
y = .78262379211 * sqrt(10,000).
this results in y = 78.26237921.
when there are 2,000 ants, the height of the ant mound is 35 centimeters.
when there are 10,000 ants, the height of the ant mound is 78.26237921 centimeters.
a direct variation formula can also be shown in ratio form as:
a/b = c/d
in this form, a is the original height of the ant mound and b is the square root of the original number of ants.
c is the new height of the ant mound and d is the square root of the new number of ants.
this formula would become:
35 / sqrt(2,000) = x / sqrt(10,000), where x is the new height of the ant mound.
cross multiply to get 35 * sqrt(10,000) = sqrt(2,000) * x
divide both sides of this equation by sqrt(2,000) to get 35 * sqrt(10,000) / sqrt(2,000) = x
solve for x to get x = 78.26237921 centimeters.
that's the new height of the mound when there are 10,000 ants in it.
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