Question 1

Solve the differential equation. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

A

Question 2

The masses  are located at the point  .Find the moments  and  and the center of mass of the system.  ; 
A) 
B) 
C) 
D) 
E)

Accepted Answer

A

Question 3

Calculate the center of mass of the lamina with density

Accepted Answer

(0,2)

Question 4

Find the coordinates of the centroid for the region bounded by the curves  , x = 0, and y = 72.

Accepted Answer

The answer of Find the coordinates of the centroid for...

Question 5

A tank contains 1050 L of brine with 20 kg of dissolved salt.Pure water enters the tank at a rate of 16 L/min.The solution is kept thoroughly mixed and drains from the tank at the same rate.How much salt is in the tank after 20 minutes?

Accepted Answer

The answer of A tank contains 1050 L of brine...

Question 6

A swimming pool is 10 ft wide and 36 ft long and its bottom is an inclined plane,the shallow end having a depth of 1 ft and the deep end,12 ft.If the pool is full of water,find the hydrostatic force on the shallow end.(Use the fact that water weighs 62.5 lb/  .)

Accepted Answer

The answer of A swimming pool is 10 ft wide...

Question 7

Find the exact coordinates of the centroid.

Accepted Answer

The answer of Find the exact coordinates of the centroid....

Question 8

Choose the differential equation corresponding to this direction field. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Choose the differential equation corresponding to this...

Question 9

Find the centroid of the region bounded by the graphs of the given equations. 
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find the centroid of the region bounded...

Question 10

Find the centroid of the region bounded by the given curves. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The region is bounded by the curves $y = x^2$, $x = 1$, and $y = 0$.

We can find the centroid $(\bar{x},\bar{y})$ using the following formulas:

$$\bar{x} = \frac{\int_{a}^{b} x f(x) dx}{\int_{a}^{b} f(x) dx}$$

$$\bar{y} = \frac{\int_{c}^{d} y g(y) dy}{\int_{c}^{d} g(y) dy}$$

where $f(x)$ is the distance of the curve from the $y$-axis, $g(y)$ is the distance of the curve from the $x$-axis, and $(a,b)$ and $(c,d)$ are the intervals of integration.

In this case, we have:

$$f(x) = x$$
$$g(y) = \sqrt{y}$$

and the limits of integration are:

$$a = 0, \quad b = 1, \quad c = 0, \quad d = 1$$

so we can calculate:

$$\bar{x} = \frac{\int_{0}^{1} x^3 dx}{\int_{0}^{1} x dx} = \frac{1}{4}$$

$$\bar{y} = \frac{\int_{0}^{1} y^{3/2} dy}{\int_{0}^{1} \sqrt{y} dy} = \frac{2}{5}$$

Therefore, the centroid is at $\left(\frac{1}{4}, \frac{2}{5}\right)$, which corresponds to choice (C).

Question 11

Solve the initial-value problem.

Accepted Answer

The answer of Solve the initial-value problem. ...

Question 12

One model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected people and the number of uninfected people.In an isolated town of 2000 inhabitants, 110 people have a disease at the beginning of the week and 1100 have it at the end of the week.How long does it take for 70% of the population to be infected?
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of One model for the spread of an...

Question 13

The Pacific halibut fishery has been modeled by the differential equation  where  is the biomass (the total mass of the members of the population)in kilograms at time t (measured in years),the carrying capacity is estimated to be  and  per year.If  ,find the biomass a year later.

Accepted Answer

The answer of The Pacific halibut fishery has been modeled...

Question 14

Find the centroid of the region bounded by the given curves. 
A) 
B) 
C) 
D) 
E) None of these

Accepted Answer

The answer of Find the centroid of the region bounded...

Question 15

Find the solution of the differential equation  that satisfies the initial condition  .

Accepted Answer

The answer of Find the solution of the differential equation...

Question 16

Solve the differential equation. 
A) 
B) 
C) 
D) 
E) None of these

Accepted Answer

The answer of Solve the differential equation. 
A) 
B) 
C)...

Question 17

A certain small country has $20 billion in paper currency in circulation,and each day $70 million comes into the country's banks.The government decides to introduce new currency by having the banks replace old bills with new ones whenever old currency comes into the banks.Let  denote the amount of new currency in circulation at time t with  .Formulate and solve a mathematical model in the form of an initial-value problem that represents the "flow" of the new currency into circulation (in billions per day).

Accepted Answer

The answer of A certain small country has $20 billion...

Question 18

Find the center of mass of a lamina in the shape of a quarter-circle with radius 9 with density

Accepted Answer

The answer of Find the center of mass of a...

Question 19

Find the centroid of the region shown,not by integration,but by locating the centroids of the rectangles and triangles and using additivity of moments.

Accepted Answer

The answer of Find the centroid of the region shown,not...

Question 20

Biologists stocked a lake with 700 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake)to be 10800 .The number of fish tripled in the first year.Assuming that the size of the fish population satisfies the logistic equation,find an expression for the size of the population after t years.

Accepted Answer

The answer of Biologists stocked a lake with 700 fish...

Quiz 7: Applications of Integration

Solve the differential equation.

The masses are located at the point .Find the moments and and the center of mass of the system. ;

Calculate the center of mass of the lamina with density

Find the coordinates of the centroid for the region bounded by the curves , x = 0, and y = 72.

A tank contains 1050 L of brine with 20 kg of dissolved salt.Pure water enters the tank at a rate of 16 L/min.The solution is kept thoroughly mixed and drains from the tank at the same rate.How much salt is in the tank after 20 minutes?

A swimming pool is 10 ft wide and 36 ft long and its bottom is an inclined plane,the shallow end having a depth of 1 ft and the deep end,12 ft.If the pool is full of water,find the hydrostatic force on the shallow end.(Use the fact that water weighs 62.5 lb/ .)

Find the exact coordinates of the centroid.

Choose the differential equation corresponding to this direction field.

Find the centroid of the region bounded by the graphs of the given equations.

Find the centroid of the region bounded by the given curves.

Solve the initial-value problem.

The Pacific halibut fishery has been modeled by the differential equation where is the biomass (the total mass of the members of the population)in kilograms at time t (measured in years),the carrying capacity is estimated to be and per year.If ,find the biomass a year later.

Find the centroid of the region bounded by the given curves.

Find the solution of the differential equation that satisfies the initial condition .

Solve the differential equation.

Find the center of mass of a lamina in the shape of a quarter-circle with radius 9 with density

Find the centroid of the region shown,not by integration,but by locating the centroids of the rectangles and triangles and using additivity of moments.