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Ne4ueva [31]
3 years ago
13

In 1950, scientists estimated a certain animal population in a particular geographical area to be 6,400. In 2000, the population

had risen to 7,200. If the animal population experiences the same percent increase over the next 50 years, what will the approximate population be?
A) 8,000
B) 8,100
C) 8.400
D) 8.600
Mathematics
1 answer:
nekit [7.7K]3 years ago
5 0

Answer:8600

Step-by-step explanation:

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\frac{8}{\left(-x+2\right)^2} & x = 1, x = 3

Step-by-step explanation:

I'm sure you are familiar with the product rule,

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u=3x+2,\:v=\left(2-x\right)^{-1},\\=>\frac{d}{dx}\left(3x+2\right)\left(2-x\right)^{-1}+\frac{d}{dx}\left(\left(2-x\right)^{-1}\right)\left(3x+2\right)

Now remember the sum rule:

\frac{d}{dx}\left(3x+2\right) = \frac{d}{dx}\left(3x\right)+\frac{d}{dx}\left(2\right),\\\frac{d}{dx}\left(3x\right) = 3,\\\frac{d}{dx}\left(2\right)  = 0\\\frac{d}{dx}\left(3x+2\right) = 3

For this second bit we apply the chain rule:

\frac{d}{dx}\left(\left(2-x\right)^{-1}\right) = -\frac{1}{\left(2-x\right)^2}\frac{d}{dx}\left(2-x\right),\\\frac{d}{dx}\left(2-x\right) = -1,\\\\=> -\frac{1}{\left(2-x\right)^2}\left(-1\right)\\=> \frac{1}{\left(2-x\right)^2}

If we substitute these values back into the expression...\frac{d}{dx}\left(3x+2\right)\left(2-x\right)^{-1}+\frac{d}{dx}\left(\left(2-x\right)^{-1}\right)\left(3x+2\right)

...we get the following:

3\left(2-x\right)^{-1}+\frac{1}{\left(2-x\right)^2}\left(3x+2\right)

The rest is just pure simplification:

3\left(2-x\right)^{-1}+\frac{1}{\left(2-x\right)^2}\left(3x+2\right)\\= \frac{3}{-x+2}+\frac{3x+2}{\left(-x+2\right)^2}\\= \frac{3\left(-x+2\right)}{\left(-x+2\right)^2}+\frac{3x+2}{\left(-x+2\right)^2}\\\\= \frac{3\left(-x+2\right)+3x+2}{\left(-x+2\right)^2}\\\\= \frac{8}{\left(-x+2\right)^2}

Now let's equate this to equal 8 for the second bit and solve for x:

\frac{8}{\left(-x+2\right)^2}=8,\\\frac{8}{\left(-x+2\right)^2}\left(-x+2\right)^2=8\left(-x+2\right)^2,\\8=8\left(-x+2\right)^2,\\\left(-x+2\right)^2=1,\\x = 1, x = 3

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Step-by-step explanation:

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