{Use of Tech} Growth rate of spotted owlets The rate of growth...

Question

{Use of Tech} Growth rate of spotted owlets The rate of growth (in g/week) of the body mass of Indian spotted owlets is modeled by the function r(t) = 10,147.9e⁻²·²ᵗ/(37.98e⁻²·² + 1), where t is the age (in weeks) of the owlets. What value of t > 0 maximizes r? What is the physical meaning of the maximum value?

Accepted Answer

A certain bacteria's population growth rate in a controlled lab environment is given by GOT equals 820e rates to -1.5% divided by 18 E to the negative 1.5% plus 1 all squared. Where T is the time and hours since the start of the observation. Determine the value of T greater than 0 that maximizes the growth rate of the bacteria. We have four possible answers, being 1.04, 1.93, 2.04, and 5.48 all in hours. Now to solve this, we're going to find the critical point. Of our function To find an equivocal point, we will set our first derivative equals a zero. So we have GFT. equals 820. E raises the -1.5 T. Divided by 18 E, rates the negative 1.5 T, plus 1 all squared. Now we can use the quotient rule to solve this derivative. We have G in our numerator, H in our denominator. And this derivative is given by the quotient rule. G T H of T minus G of T H T. All divided by H of T squared. Let's go ahead and find our derivatives now. You know, G is our numerator. And we can take this derivative by using a small chain rule. G will be 820 multiplied by -1.5, E rates -1.5 T. H is our denominator. And so this is another chain rule. H prime will be equivalent. 22. Multiplied by 18 E to -1.5 T plus 1. And by the chain rule, we multiply by the inside derivative. This would be 18 multiplied by 1.5 E to the negative 1.5 T. Now, let's put this into the quotient rule. We will get G T -1230 E to the -1.5 T. Multiplied by H. 18 E's -1.5. T + 1 All squared Minus, I'll actually move this over a little. This is all minus. G of T 820, E to the negative 1.5 T. Multiplied By 2, multiplied by 18 E to the negative 1.5 T plus 1. Multiplied by 18, 1.5 E to -1.5 T. This is all divided. By 18 E. 1.5 T plus 1. Raise to the 4th. Now, if we were to simplify this entire numerator, You can use calculator for this, but this simplifies down to negative 1230. E to the 3 T minus 324. Multiplied by E To the 1.5T. All divided By E to be 1.5 T plus 18, raised to the fourth. Now, we will set this equals 0. Insult Now we have a denominator which can be multiplied. So we end up getting 0, equals just the numerator, -1230. E 3 T minus 324 multiplied by E to the 1.5 T. Now, each of the 1.5 T can never equal 0, so you can actually divide out. This term on both sides. We end up getting 0. Equals -1230. E3T minus 324. And because we can divide 12:30. We get 0 equals E 3T minus 324. Finally, we can solve this for T by adding 324. To get E to the 3 T equals 324. And if we saw for tea, We end up getting 3 T equals natural log, 324, or T equals the natural log of 324 divided by 3. We can approximate this in the calculator. To get 1.93. This is an ours So we see the answer to our problem, 1.93 hours. Is answer B. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

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