Question 1

A wildlife biologist is trying to determine the run size of salmon in a small stream in Western Alaska. The population of salmon in the stream is given by a differentiable function S, where S(t) is the population of salmon measured in thousands and t is measured in days after June 1st. The population of salmon increases with respect to t at a rate that is directly proportional to the date. At day 15, the population of salmon is 217,000 and is increasing at a rate of 30,000 per day. Which of the following is a differential equation that could model this scenario?

120

$\frac{dS}{dt}=30S$

$\frac{dS}{dt}=\frac{1}{2}S$

$\frac{dS}{dt}=2S$

$\frac{dS}{dt}=450S$

Accepted Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Question 2

The slope field for a specific differential equation is shown. Which of the following differential equations could the slope field be for?

Accepted Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Question 3

The slope field of a function is shown. Which of the following could be the solution to the differential equation satisfying y(0)=0?

Accepted Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Question 4

Find the general solution to the differential equation: $\frac{dy}{dx}=\frac{xy}{2}$ .

Accepted Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Question 5

Which of the following equations is the solution to the differential equation $\frac{dy}{dx}=\frac{xy}{2}$ for $y>2$ with the initial condition $y\left(8\right)=5$ ?

Accepted Answer

y=4\ +\ e^{\frac{1}{2}x^2-4x}

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Question 6

When Covid-19 was first detected in Alaska, 1000 people were already infected. The number of people infected (given by the function C) started increasing at a rate that satisfies the differential equation $y>2$ , where t is measured in days since the discovery of the first case and k is the constant of the contagion factor. WHich equation could model C(t)?

Accepted Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Question 7

After the snow melts, the population of mosquitoes grows at a rate proportional to the population. From June 1st to June 11th, the number of mosquitoes (in millions) grows from 10 to 40. If the population of mosquitoes continues to grow according to the same model, how many days after June 1st will the population of mosquitoes reach 50?

Accepted Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Question 8

Find the equation for $\frac{dy}{dx}=\frac{xy}{2}$ if $y>2$ and $y\left(8\right)=5$ .

Accepted Answer

\frac{dS}{dt}=30S

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Question 9

Find the equation for $\frac{dy}{dx}=\frac{xy}{2}$ if $y>2$ and $y\left(8\right)=5$

Accepted Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Question 10

Gargi, a bristlecone pine tree in the White Mountains of California, is the oldest known tree at age 5,065 years. If the half-life of carbon 14 is 5,750 years (100 units of carbon 14 will decay to 50 in 5,750 years), determine what percentage of carbon 14 you would expect to find in the innermost ring of Gargi.

Accepted Answer

54.304%

Answer

12.05%

Answer

64.35%

Answer

35.506%

Question 11

Solve: $\frac{dy}{dx}=\frac{xy}{2}$ given that $y>2$ Use the equation editor to type in your answer.

Accepted Answer

$\frac{dy}{dx}=\frac{xy}{2}$

Question 12

The growth of a given population is represented by $\frac{dy}{dx}=\frac{xy}{2}$ where $y>2$ represents the population in millions. When does the population grow fastest?

Accepted Answer

When $\frac{dy}{dx}=\frac{xy}{2}$ (population is 5 million)

Answer

When $\frac{dy}{dx}=\frac{xy}{2}$ (population is 5 million)

Answer

When $\frac{dy}{dx}=\frac{xy}{2}$ (population is 5 million)

Answer

When $\frac{dy}{dx}=\frac{xy}{2}$ (population is 5 million)

Question 13

The general solution to $\frac{dy}{dx}=\frac{xy}{2}$ is $y>2$ . Find a solution to the initial value problem $y\left(8\right)=5$ , $y\left(0\right)=4$ .

Accepted Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Question 14

$\frac{dS}{dt}=30S$ is a solution to $y'=y+\frac{y}{x}$ .

Accepted Answer

true

Answer

false

Question 15

A differential equation is separable if it can be written in the form:

Accepted Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Question 16

True or false. $\frac{dy}{dx}=\frac{xy}{2}$ is separable.

Accepted Answer

true

Answer

false

Question 17

The general solution to $\frac{dy}{dx}=\frac{xy}{2}$ (for x, y > 0) is:

Accepted Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Question 18

The solution to $\frac{dy}{dx}=\frac{xy}{2}$ with $y>2$ is:

Accepted Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Question 19

If a ball is thrown upward and reaches its maximum height when $\frac{dy}{dx}=\frac{xy}{2}$ seconds, give the function representing the ball's velocity. Use the equation editor to type in your answer.

Accepted Answer

y=4\ +\ e^{\frac{1}{2}x^2-4x}

Question 20

Assume that the world's population's interest in the new boy band, "Hunks o'Love," is growing at a rate proportional to the number of its fans. If the Hunks o'Love had 2,000 fans one year after they released their first album and 50,000 fans five years after their first album, how many fans did they have the moment the first album was released?

Accepted Answer

894.427

Question 21

The population of a species of jellyfish in a small harbor is directly proportional to $\frac{dy}{dx}=\frac{xy}{2}$ , where $y>2$ is the population (in thousands) and $y\left(8\right)=5$ is the time (in years). At $y\left(0\right)=4$ (1990) the population was 100,000, and in 1992, the population was 300,000. In what year did the population reach 400,000?

Accepted Answer

1994

Question 22

Water is leaking out of a large barrel at a rate proportional to the square root of the depth of the water at that time. If the water level starts at 36 inches and drops to 34 inches in 1 hour, how many hours will it take for all of the water to drain out of the barrel? Round to the nearest hundredth.

Accepted Answer

35.49

Question 23

The habitat of sea urchins in a Pacific tidal pool can support a maximum of $\frac{dy}{dx}=\frac{xy}{2}$ urchins. The habitat's population, $y>2$ , grows proportionally to the product of the current population and the difference between $y\left(8\right)=5$ and $y\left(0\right)=4$ . Which equation describes the relationship?

Accepted Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Question 24

If $\frac{dy}{dx}=\frac{xy}{2}$ and $y>2$ when $y\left(8\right)=5$ , then

Accepted Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Question 25

The solution curve of $\frac{dy}{dx}=\frac{xy}{2}$ that passes through the point (2, 4) is

Accepted Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

Answer

$\frac{dS}{dt}=30S$

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