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lozanna [386]
3 years ago
14

The number of salmon swimming upstream to spawn is approximated by the following function:

Mathematics
1 answer:
umka2103 [35]3 years ago
5 0

Answer:

The water temperature that produces the maximum number of salmon swimming upstream is approximately 12.305 degrees Celsius.

Step-by-step explanation:

Let S(x) = -x^{3}+2\cdot x^{2}+405\cdot x +4965, for 6 \leq x \leq 20. x represents the temperature of the water, measured in degrees Celsius, and S is the number of salmon swimming upstream to spawn, dimensionless.

We compute the first and second derivatives of the function:

S'(x) = -3\cdot x^{2}+4\cdot x +405 (Eq. 1)

S''(x) = -6\cdot x +4 (Eq. 2)

Then we equalize (Eq. 1) to zero and solve for x:

-3\cdot x^{2}+4\cdot x +405 = 0

And all roots are found by Quadratic Formula:

x_{1} \approx 12.305\,^{\circ}C, x_{2}\approx -10.971\,^{\circ}C

Only the first root is inside the given interval of the function. Hence, the correct answer is:

x \approx 12.305\,^{\circ}C

Now we evaluate the second derivative at given result. That is:

S''(12.305) = -6\cdot (12.305)+4

S''(12.305) = -69.83

According to the Second Derivative Test, a negative value means that critical value leads to a maximum. In consequence, the water temperature that produces the maximum number of salmon swimming upstream is approximately 12.305 degrees Celsius.

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The expression is given as:

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Expand the expression

2\sqrt{30} \times (\sqrt5+ \sqrt6+ \sqrt{10} + \sqrt{15}) = 2\sqrt{30} \times \sqrt5+  2\sqrt{30} \times \sqrt6+  2\sqrt{30} \times \sqrt{10} +  2\sqrt{30} \times \sqrt{15}

Factor out 2

2\sqrt{30} \times (\sqrt5+ \sqrt6+ \sqrt{10} + \sqrt{15}) = 2(\sqrt{30} \times \sqrt5+  \sqrt{30} \times \sqrt6+  \sqrt{30} \times \sqrt{10} +  \sqrt{30} \times \sqrt{15})

Combine the radicals

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Expand the expression

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Evaluate the roots

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Expand

2\sqrt{30} \times (\sqrt5+ \sqrt6+ \sqrt{10} + \sqrt{15}) =10\sqrt{6} +  6\sqrt{20}+  20\sqrt{3} +  30\sqrt{2}

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Read more about simplified expressions at:

brainly.com/question/8008182

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Hello.

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