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3. Assume that a bear rummaging through an abandoned campsite will eat food at a rate proportional to the cube root of the amount of food left at the campsite. Throughout this problem, use $k$ as the constant of proportionality. (a) Assume that at $t = 0$ the bear has just arrived at the campsite, which initially contains $125 = (5)^3$ pounds of food, and has eaten no food. Write and solve an initial value problem for the amount of food $F(t)$ the bear has eaten at any point in time. This solution will involve the parameter $k$. (b) Assume that after 80 minutes, the bear has completely cleaned out the campsite (so there is no food left). Use this to determine the value of $k$ and what fraction of the initial food is left at the campsite after 20 minutes of the bear eating.

          3. Assume that a bear rummaging through an abandoned campsite will eat food at a rate proportional to the cube root of the amount of food left at the campsite. Throughout this problem, use $k$ as the constant of proportionality.
(a) Assume that at $t = 0$ the bear has just arrived at the campsite, which initially contains $125 = (5)^3$ pounds of food, and has eaten no food. Write and solve an initial value problem for the amount of food $F(t)$ the bear has eaten at any point in time. This solution will involve the parameter $k$.
(b) Assume that after 80 minutes, the bear has completely cleaned out the campsite (so there is no food left). Use this to determine the value of $k$ and what fraction of the initial food is left at the campsite after 20 minutes of the bear eating.

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$3. Assume that a bear rummaging through an abandoned campsite will eat food at a rate proportional to the cube root of the amount of food left at the campsite. Throughout this problem, use k as the constant of proportionality. (a) Assume that at t = 0 the bear has just arrived at the campsite, which initially contains 125 = (5)^3 pounds of food, and has eaten no food. Write and solve an initial value problem for the amount of food F(t) the bear has eaten at any point in time. This solution will involve the parameter k. (b) Assume that after 80 minutes, the bear has completely cleaned out the campsite (so there is no food left). Use this to determine the value of k and what fraction of the initial food is left at the campsite after 20 minutes of the bear eating.$

Added by Rosa C.

Calculus: Early Transcendentals

James Stewart 8th Edition

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Solved by Expert Eduard Sanchez

07/17/2023

Step 1

The rate of food eaten is proportional to the cube root of the amount of food left, so we have: $\frac{dA}{dt} = -k \sqrt[3]{A}$ where $k$ is the constant of proportionality. Show more…

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3. Assume that a bear rummaging through an abandoned campsite will eat food at a rate proportional to the cube root of the amount of food left at the campsite. Throughout this problem, use $k$ as the constant of proportionality. (a) Assume that at $t = 0$ the bear has just arrived at the campsite, which initially contains $125 = (5)^3$ pounds of food, and has eaten no food. Write and solve an initial value problem for the amount of food $F(t)$ the bear has eaten at any point in time. This solution will involve the parameter $k$. (b) Assume that after 80 minutes, the bear has completely cleaned out the campsite (so there is no food left). Use this to determine the value of $k$ and what fraction of the initial food is left at the campsite after 20 minutes of the bear eating.

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Transcript

00:01 Hi there, so for this problem, we are told that p represents the number of grizzly birds in a population at times t years, when t is greater or equal than zero.

00:14 Now the population b is increasing at a rate derivative proportional to 600 minus b.

00:19 So that will be that the rate of change of the population with respect to time is a proportional, so there will be a constant of proportionality, then this time 600 minus b.

00:29 And then this, we're given also the initial population that is 400.

00:35 We need to find the population in terms of the time and the constant of proportionality gain.

00:40 Now what we need to do in this case is to separate the variables, okay? so let me just write this in the following way.

00:52 That will be 600 times the constant of proportionality minus the population times the constant of proportionality.

00:58 We can pass this term to the other side in here, so we will have this, then this is equal to 600 times the constant of proportionality.

01:10 Now we need to write a integrating factor that is related to this second term in here, so that will be the exponential of constant of proportionality times the time.

01:23 And we need to multiply everything by this.

01:25 So we can now write this, that this is the derivative with respect to time of the population times the exponential of the constant of proportionality times the time, then this is equal to 600 times k times the exponential of minus the constant of proportionality times the time.

01:48 Once we have this, we need to integrate both sides of this.

01:51 Well, we first need to separate the variables...

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