A chicken broth container is in the shape of a rectangular prism, with a length of 9.5 centimeters, a width of 6 centimeters, and a height of 16.5 centimeters. The container is 90% full. How many liters of chicken broth are in the container? ( ) 3 1 L 1000 cm = Round your answer to the nearest hundredth.

Mr. Sanchez expects to earn $54,806.25 this year.

To find the salary Mr. Sanchez expects to earn this year, we need to calculate 11% of his salary from last year and add it to his previous salary.

The 11% increase can be found by multiplying Mr. Sanchez's salary from last year by 0.11: $49,375 * 0.11 = $5,431.25

Adding this increase to his previous salary gives us the salary Mr. Sanchez expects to earn this year: $49,375 + $5,431.25 = $54,806.25

Answer:

2/5

Step-by-step explanation

Answer: The answer is 8/5, or 1 3/5

Step-by-step explanation: The first step is to represent the unknown number by an alphabet, for example let's say the number is x. That means one quarter of x becomes

X × ¼

Or X/1 × 1/4

Which equals x/4

One quarter of a number increased by 2/5 can now be written as

x/4 + 2/5

One quarter of a number increased by 2/5 gives 4/5 can now be expressed as

x/4 + 2/5 = 4/5

The first step is to subtract 2/5 from both sides of the equation

x/4 = 4/5 - 2/5

Note that 5 is the common denominator, hence

x/4 = (4-2)/5

x/4 = 2/5

When you cross multiply, 4 moves to the right hand side and 5 moves to the left hand side. You now have

5x = 4×2

5x = 8

Divide both sides of the equation by 5

x = 8/5 (1 3/5)

Answer:

Answer in explanation

Step-by-step explanation:

The ratio of their ownership is 4:3 I.e J to T

Now we know that Terrence has 36 model cars. To find the number of model cars Jim own, we need to find the unit ownership. This is the same as 36/3 which is 12 cars.

This means Jim has 12 * 4 = 48 model cars.

Now we are looking at them selling 10 of their cars each. This would bring the number of model cars owned to be 38 and 26 respectively.

The ratio here would now be 38:26 which is same as 19:13. Of course this is different from 4:3, hence we can conclude that the ratio will indeed change

Final answer:

Jim owns 48 model cars.

If both Jim and Terrence sell ten cars each, Jim will have 38 cars, Terrence will have 26, and the ratio of Jim's cars to Terrence's cars will change to 19:13.

Explanation:

The ratio of the number of model cars that Jim owns to the number of cars Terrence owns is 4:3. If Terrence owns 36 model cars, we can set up a proportion to find out how many model cars Jim owns.

Because Terrence's part of the ratio corresponds to 36 cars, we have:

Jim's cars / Terrence's cars = 4/3
Jim's cars / 36 = 4/3

Cross-multiplying to solve for Jim's cars:

(Jim's cars) * 3 = 4 * 36
Jim's cars = (4 * 36) / 3
Jim's cars = 144 / 3
Jim's cars = 48

Therefore, Jim owns 48 model cars.

If Jim and Terrence each sell ten of their model cars, Jim will have 38 model cars and Terrence will have 26. The new ratio will be:

38 / 26, which simplifies to 19 / 13, different from the original ratio of 4/3.

So, yes, the ratio will change if they both sell ten of their cars.

Simple interest is $ 9

Solution:

Given that,

P = $ 400

[tex]R = 9 \% = \frac{9}{100} = 0.09[/tex]

T = 0.25 years

The formula for simple interest is:

[tex]I = P \times R \times T[/tex]

Where,

I is the simple interest earned

R is the rate of interest in decimal

T is the number of years

Substituting the values we get,

[tex]I = 400 \times 0.09 \times 0.25\\\\I = 36 \times 0.25\\\\I = 9[/tex]

Thus simple interest is $ 9

Answer:

Each knight will guard 288 minutes in one day.

Step-by-step explanation:

Given:

A castle has to be guarded 24 hours a day. Five knights are ordered to split each day's guard duty equally.

Now, to find the minutes will each night spend on guard duty in one day.

As, in 1 hour there are 60 minutes.

Thus, in 24 hour there are 60 × 24 = 1440 minutes.

Total minutes for guarding = 1440 minutes.

So, there are knights ordered for guarding are = 5.

And each day's guard duty equally.

Now, to get the minutes will each night spend on guard duty in one day we divide the total minutes for guarding by number of knights that is 5:

[tex]1440\div5[/tex]

[tex]=288\ minutes.[/tex]

Therefore, each knight will guard 288 minutes in one day.

Answer:

Step-by-step explanation:

The formula for determining the distance between two points on a straight line is expressed as

Distance = √(x2 - x1)² + (y2 - y1)²

Where

x2 represents final value of x on the horizontal axis

x1 represents initial value of x on the horizontal axis.

y2 represents final value of y on the vertical axis.

y1 represents initial value of y on the vertical axis.

From the points given,

x2 = 0

x1 = 9

y2 = - 33

y1 = 7

Therefore,

Distance = √(0 - 9)² + (- 33 - 7)²

Distance = √(- 9² + (- 40)² = √(81 + 1600) = √1681

Distance = 41

The formula determining the midpoint of a line is expressed as

[(x1 + x2)/2 , (y1 + y2)/2]

[(9 + 0) , (7 - 33)]

= (9, - 26]

The distance between the points (9, 7) and (0, -33) is 41 units, while the midpoint of the line segment joining them is (4.5, -13).

The subject of the question is Mathematics, specifically involving concepts in geometry. Given two points in 2-dimensional space - (9, 7) and (0, -33), we are asked to find the distance between these points and the midpoint of the line segment. The formula for the distance between two points (x1,y1) and (x2,y2) is  [tex]\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/tex] applying the values,

[tex]\sqrt{ (0 - 9)^2 + (-33 - 7)^2 } = \sqrt{81 + 1600} = \sqrt{1681} = 41 \text{ units}[/tex] The midpoint of the line segment between two points (x1, y1) and (x2, y2) is ((x1+x2)/2, (y1+y2)/2). So that would be ((9 + 0)/2, (7 - 33)/2) = (4.5, -13).

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Answer:

Median = 5

Lower Quartile = 2.5

Upper Quartile = 8.5

Step-by-step explanation:

- First of all, you need to order the numbers from lowest to greatest: 2 2 3 4 4 6 6 8 9 9

- Then, you will find at the number that sits exactly at the middle of this set of numbers. Because this is a set of numbers that has 10 numbers, you will have to look at the two middle points (4 and 6) and divide them by 2 (essentialy finding the average).

4+6=10, 10/2 = 5 = Mean

- To find the upper and lower quartiles, you basically have to find the medians of the set of numbers that are below and above the central median

- So, for the lower quartile, the set of numebrs is: 2 2 3 4. The median sits between 2 and 3, so we have to find the average of those: 2+3=5, 5/2=2.5

- For the upper quartile, the set of numbers is 6 8 9 9. The median is the average of 8+9. So 8+9=17, 17/2 = 8.5

Lower Quartile (Q1) = 3 Median (Q2) = 5 Upper Quartile (Q3) = 8

First, arrange the data in ascending order:

2, 2, 3, 4, 4, 6, 6, 8, 9, 9

The median is the middle value of the sorted data set. Since there are 10 values (an even number), the median will be the average of the 5th and 6th values.

The 5th value is 4

The 6th value is 6

Median (Q2) = (4 + 6) / 2

= 5

The lower quartile is the median of the lower half of the data set (excluding the overall median). For this data set, the lower half is:

2, 2, 3, 4, 4

Since there are 5 values in this half, the lower quartile is the 3rd value.

Lower Quartile (Q1) = 3

The upper quartile is the median of the upper half of the data set (excluding the overall median). For this data set, the upper half is:

6, 6, 8, 9, 9

Since there are 5 values in this half, the upper quartile is the 3rd value.

Upper Quartile (Q3) = 8

Answer: 321 adult tickets and 227 children tickets were sold.

Step-by-step explanation:

Let x represent the number of adult tickets that were sold.

Let y represent the number of children tickets that were sold.

The total number of tickets that the theatre sold is 548. This means that

x + y = 548

Adult tickets sell for $6.50 each, and children's tickets sell for $3.50 each. The total ticket sales was $2881. This means that

6.5x + 3.5y = 2881 - - - - - - - - - - -1

Substituting x = 548 - y into equation 1, it becomes

6.5(548 - y) + 3.5y = 2881

3562 - 6.5y + 3.5y = 2881

- 6.5y + 3.5y = 2881 - 3562

- 3y = - 681

y = - 681/ -3

y = 227

x = 548 - y = 548 - 227

x = 321

Answer:

313 ft

Step-by-step explanation:

If 12in = 12 ft

313in = x

= 313 ft

To find the actual length of the room, use the scale to set up a proportion and solve for the actual length.

To find the actual length of the room, we can use the scale provided. According to the scale, 12 inches on the drawing corresponds to 12 feet in the actual room. Since the length of the room in the drawing is 313 inches, we can set up a proportion to find the actual length:
12 inches on the drawing / 313 inches on the drawing = 12 feet in the actual room / actual length of the room
Cross-multiplying, we get:
12 inches x actual length of the room = 313 inches x 12 feet
Dividing both sides by 12 inches, we find:
Actual length of the room = (313 inches x 12 feet) / 12 inches
Calculating this, we get:
Actual length of the room = 313 feet

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Answer:

Through the expression, "let freedom ring", Martin Luther King Jr. was emphasizing on the need for community effort throughout the nation in order to counteract segregation. Let freedom ring is a metaphor for the action of spreading equality

Step-by-step explanation:

Answer:this isn’t in the right spot it’s in math. But when mlk says this he wants equality for all people through the country and this is the metaphor he uses. He wants justice and freedom to be granted to all.

Step-by-step explanation:

Sasha runs at a constant speed of 3.8 meters per second for 1/2 hour.Then she walks at a constant rate of 1.5 meters per second for 1/2 hour , 9,540 meters Sasha run and walk in 60 minutes

Given :

              V1 = 3.8 m/s

               t1 = 1/2 = 30 x 60 = 1,800 seconds.

Here,

V1 = Running speed of Shasha.

t1 = Time

→ She walks at distance V2 = 1.5 m/s for t1 = 1,800 seconds.

  Distance of Shasha :

  S1 = V1 x t1

  S1 = 3.8 x 1,800

  S1= 6,840 meters

→She walks at distance V2 = 1.5 m/s for t2 = 1,800 seconds.

 Distance of Shasha:

 S2 = V2 x t2

 S2= 1.5x 1,800

 S2= 2700 meters

Then, The total distance she runs & walks

S = S 1 + S 2

S= 6,840 + 2,700

S= 9540 meters

Therefore,  9,540 meters Sasha run and walk in 60 minutes

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Sasha runs 6840 meters and walks 2700 meters for a total distance of 9540 meters in 60 minutes.

To calculate the total distance Sasha ran and walked, you need to use the formula for distance which is speed x time. Sasha first runs at a speed of 3.8 m/s for 1/2 hour (or 1800 seconds), so her running distance is 3.8 x 1800 = 6840 meters. She then walks at a speed of 1.5 m/s for 1/2 hour (or 1800 seconds), so her walking distance is 1.5 x 1800 = 2700 meters. Combining both makes a total distance of 6840 + 2700 = 9540 meters.

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Answer:

3 Vases

4 Mason jars

Step-by-step explanation:

The vase costs $12 and the mason jar costs $8. She has $72 to spend. We know that she must at least buy 7 containers. Let vase be x₁ and mason jar x₂. We have two equations:

[tex]x_1+x_2=7[/tex]

[tex]72=12x_1+8x_2[/tex]

WE can solve the value by substitution:

[tex]x_1=7-x_2[/tex]

[tex]72=12(7-x_2)+8x_2[/tex]

[tex]x_2=3[/tex]

Therefore:

[tex]x_1=7-3=4[/tex]

Answer: The width of the rectangle is 9x^5y^5

Step-by-step explanation:

Area = 45x^8y^9 sq yard

Length = 5x^3y^4 yards

Area of rectangle = length * width

Width = Area/length

= 45x^8y^9/5x^3y^4

= 45/5 * x^8/x^3 * y^9/y^4

= 9x^5y^5yards

Width = 9x^5y^5yards

Answer:

[tex]FJ=18\ units[/tex]

Step-by-step explanation:

see the attached figure to better understand the problem

we know that

If two triangles are similar, then the ratio of its corresponding sides is proportional and its corresponding angles are congruent

In this problem

△FEG is similar with △FHJ -----> by AA Similarity Theorem

so

[tex]\frac{FE}{FH}=\frac{FG}{FJ}[/tex]

we have

[tex]FE=HF+EH=12+8=20\ units\\FH=HF=12\ units\\FG=30\ units[/tex]

substitute the given values

[tex]\frac{20}{12}=\frac{30}{FJ}\\\\FJ=12(30)/20\\FJ=18\ units[/tex]

Answer: Fj = 18

Step-by-step explanation:

Hello! So i just took the test and I got this wrong but the answer is 18 down below is a screenshot so you know 18 is the correct answer :) Hope this helps ^.^

The value of x is 3.8

Solution:

Given equation is:

[tex]x + 7.4 = 11.2[/tex]

We have to solve the equation for "x"

Move the terms so that you end up with only terms involving x on one side of the sign and all the numbers on the other

Therefore, we get

x + 7.4 = 11.2

When we move 7.4 from left side to right side of equation it becomes -7.4

x = 11.2 - 7.4

Subtract 7.4 from 11.2

x = 3.8

Thus value of x is 3.8

Answer:

Correct answer is D

Step-by-step explanation:

The concept of population census is applied in solving the question. Population as we know is the total number of inhabitants in a place or the combination of people dwelling in a place.

Sample is a unit or a part of population census and not the entirety of the population.

In this case, Our population of interest is the whole inhabitant in the country, as indicated that census bureau mails survey form to 250,000 households asking about some question. And In one month, responses were obtained from 240,000 of the households contacted.

As it is, our population of interest is not the household that can be contacted by telephone because it is presumed that the households with phones may be lesser than the total population of the sample been considered. Irrespective of those that were or were not contacted by telephones, our population of interest is ALL OF US HOUSEHOLDS. As the essence of a survey is to have an idea of an estimate of the population parameter.

Hence the correct option is D

Answer: the number of multiple choice questions in the test is 12.

the number of short response questions in the test is 8.

Step-by-step explanation:

Let x represent the number of multiple choice questions in the test.

Let y represent the number of short response questions in the test.

The total number of questions in the test is 20. It means that

x + y = 20

The multiple choice questions are worth 3 points each and the short response question are worth 8 points each. The total number of points is 100. It means that

3x + 8y = 100 - - - - - - - - - - 1

Substituting x = 20 - y into equation 1, it becomes

3(20 - y) + 8y = 100

60 - 3y + 8y = 100

- 3y + 8y = 100 - 60

5y = 40

y = 40/5 = 8

x = 20 - y = 20 - 8

x = 12

Option d: Angle A is 30 degrees, BC is 5 units

Option e: AC is 8 units, BC is 6 units

Explanation:

The triangle ABC has a right angle at C.

The length of the hypotenuse is 10 units.

The image of the triangle with this measurement is attached below:

Option a: Angle A is 20 degrees, BC is 2 units

[tex]\begin{aligned}\sin 20 &=\frac{2}{h y p} \\h y p &=\frac{2}{\sin 20} \\&=5.8476\end{aligned}[/tex]

Since, hypotenuse is 10 units, Option a is not correct answer.

Option b: AC is 7 units, BC is 3 units

[tex]\begin{aligned}A B &=\sqrt{7^{2}+3^{2}} \\&=\sqrt{49+9} \\&=\sqrt{58}\end{aligned}[/tex]

Since, hypotenuse is 10 units, Option b is not correct answer.

Option c: Angle B is 50 degrees, BC is 4 units

[tex]\begin{aligned}\cos 50 &=\frac{4}{h y p} \\h y p &=\frac{4}{\cos 50} \\&=6.222\end{aligned}[/tex]

Since, hypotenuse is 10 units, Option c is not correct answer.

Option d: Angle A is 30 degrees, BC is 5 units

[tex]\begin{aligned}\sin 30 &=\frac{5}{h y p} \\h y p &=\frac{5}{\sin 30} \\&=10\end{aligned}[/tex]

Since, hypotenuse is 10 units, Option d is the correct answer.

Option e: AC is 8 units, BC is 6 units

[tex]\begin{aligned}A B &=\sqrt{8^{2}+6^{2}} \\&=\sqrt{64+36} \\&=\sqrt{100} \\&=10\end{aligned}[/tex]

Since, hypotenuse is 10 units, Option e is the correct answer.

Thus, Option d and e are the correct answers.

In triangle ABC, Angle A is 30 degrees, BC is 5 units.

Trigonometric ratio is used to show the relationship between the sides and angles of a right angled triangle.

Triangle ABC has a right angle at C and a hypotenuse AB.

If angle A is 30°:

sin(30) = BC/10

BC = 5 units

In triangle ABC, Angle A is 30 degrees, BC is 5 units.

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Answer:

y = 0.5x - 5

Step-by-step explanation:

(0,-5) (6,-2)

m = (y2-y1)/(x2-x1)

= (-2-(-5))/(6-0)

= (-2+5)/6

= 3/6

= 1/2 or 0.5

Y-intercept is clearly -5, so c = -5

in y = mx + c

y = 0.5x - 5

Answer:

Step-by-step explanation:

The measure of the floor of the rectangular room that is 12 feet by 15 feet. The formula for determining the area of a rectangle is expressed as

Area = length × width

Area of the rectangular room would be

12 × 15 = 180 feet²

The tiles are square with side lengths 1 1/3 feet. Converting 1 1/3 feet to improper fraction, it becomes 4/3 feet

Area if each tile is

4/3 × 4/3 = 16/9 ft²

The number of tiles needed to cover the entire floor is

180/(16/9) = 180 × 9/16

= 101.25

102 tiles would be needed because the tiles must be whole numbers.

Answer:

  see below

Step-by-step explanation:

The formula for the amount resulting from P earning interest at rate r continuously compounded is ...

  A = Pe^(rt)

for P=2500 and r=0.12, this becomes ...

  A = 2500e^(0.12t)

Answer: We can be 95% confident that the true proportion of all college students gain weight in their freshman year.

Step-by-step explanation:

Given : A popular claim, nicknamed "freshman fifteen," states that many college students gain weight in their freshman year.

The 95% confidence interval as 55.9% < p < 78.4%.

Here : Population parameter = p , where p is the proportion of college students gain weight in their freshman year.

Interpretation of 95% confidence interval : We can be 95% confident that the true proportion of all college students gain weight in their freshman year.

Answer: D

Step-by-step explanation:

Answer: option C is the correct answer

Step-by-step explanation:

Triangle ABC is a right angle triangle.

From the given right angle triangle,

AC represents the hypotenuse of the right angle triangle.

With m∠C as the reference angle,

BC represents the adjacent side of the right angle triangle.

AB represents the opposite side of the right angle triangle.

To determine Tan m∠C, we would apply

the Tangent trigonometric ratio.

Tan θ = opposite side/adjacent side. Therefore,

Tan C = 24/10

Tan C = 12/5

Answer:

C

Step-by-step explanation:

tan(C) = opposite/adjacent

= 24/10

= 12/5

Answer:

36 men, 45 women

Step-by-step explanation:

36 men and 45 women attended the concert

Two or more expressions with an Equal sign is called as Equation.

Given,

There was a country concert held at the park.

For every 4 men there were 5 women that went to the concert

81 people attended the concert.

We need to find how many men and how many women each attended the concert.

4x+5x=81

9x=81

Divide both sides by 9

x=9

So 4(9)=36

5(9)=45

Hence, 36 men and 45 women attended the concert

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Answer:

The volume of the container is 729 cubic centimetres.

Step-by-step explanation:

Given:

Camren has a clear container in the shape of a cube. Each edge is 9 centimeters long he found the volume of the container in cubic centimeters by multiplying the edge length by itself 3 times.

Now, to find the volume of the container in cubic centimeters.

Edge of the cube = 9 centimeters.

So, to get the volume of container by putting formula as the container is in the shape of a cube:

[tex]Volume\ of\ cube=(edge)^3[/tex]

[tex]Volume=(9)^3[/tex]

[tex]Volume=9\times 9\times 9[/tex]

[tex]Volume=729\ cubic\ centimeters.[/tex]

Therefore, the volume of the container is 729 cubic centimetres.

Answer:

5° F

Step-by-step explanation:

According to Newton's law of cooling, the rate of change is proportional to the difference between the temperature and the ambient temperature.

dT/dt = k (T − T₀)

Solving this by separating the variables:

dT / (T − T₀) = k dt

ln (T − T₀) = kt + C

T − T₀ = Ce^(kt)

T = T₀ + Ce^(kt)

We're given that T₀ = 65.

T = 65 + Ce^(kt)

At t = 10, T = 35.

35 = 65 + Ce^(10k)

-30 = Ce^(10k)

At t = 20, T = 50.

50 = 65 + Ce^(20k)

-15 = Ce^(20k)

Squaring the first equation:

900 = C² e^(20k)

Dividing by the second equation:

-60 = C

Therefore:

T = 65 − 60e^(kt)

At t = 0:

T = 65 − 60e^(0)

T = 5

The initial temperature is 5° F.

Answer:

Interval data

Step-by-step explanation:

Brian's score is an interval data because it appears within the GMAT range of score, which is 200-800

Answer:Ashton used 390 centimeters of string for his project.

Step-by-step explanation:

The total length of the piece of string that Ashton has is 520 centimeters.

He cuts the string into 8 equal pieces. This means that the length of each string would be

520/8 = 65 centimeters.

If he 6 of the pieces for a project, it means that the number of centimeters of string that Ashton used for his project would be

65 × 6 = 390 centimeters

Answer:

a) [tex] t = 2 *ln(\frac{82}{5}) =5.595[/tex]

b) [tex] t = 2 *ln(-\frac{820}{p_0 -820}) [/tex]

c) [tex] p_0 = 820-\frac{820}{e^6}[/tex]

Step-by-step explanation:

For this case we have the following differential equation:

[tex] \frac{dp}{dt}=\frac{1}{2} (p-820)[/tex]

And if we rewrite the expression we got:

[tex] \frac{dp}{p-820}= \frac{1}{2} dt[/tex]

If we integrate both sides we have:

[tex]ln|P-820|= \frac{1}{2}t +c[/tex]

Using exponential on both sides we got:

[tex] P= 820 + P_o e^{1/2t}[/tex]

Part a

For this case we know that p(0) = 770 so we have this:

[tex] 770 = 820 + P_o e^0[/tex]

[tex] P_o = -50[/tex]

So then our model would be given by:

[tex] P(t) = -50e^{1/2t} +820[/tex]

And if we want to find at which time the population would be extinct we have:

[tex] 0=-50 e^{1/2 t} +820[/tex]

[tex] \frac{820}{50} = e^{1/2 t}[/tex]

Using natural log on both sides we got:

[tex] ln(\frac{82}{5}) = \frac{1}{2}t[/tex]

And solving for t we got:

[tex] t = 2 *ln(\frac{82}{5}) =5.595[/tex]

Part b

For this case we know that p(0) = p0 so we have this:

[tex] p_0 = 820 + P_o e^0[/tex]

[tex] P_o = p_0 -820[/tex]

So then our model would be given by:

[tex] P(t) = (p_o -820)e^{1/2t} +820[/tex]

And if we want to find at which time the population would be extinct we have:

[tex] 0=(p_o -820)e^{1/2 t} +820[/tex]

[tex] -\frac{820}{p_0 -820} = e^{1/2 t}[/tex]

Using natural log on both sides we got:

[tex] ln(-\frac{820}{p_0 -820}) = \frac{1}{2}t[/tex]

And solving for t we got:

[tex] t = 2 *ln(-\frac{820}{p_0 -820}) [/tex]

Part c

For this case we want to find the initial population if we know that the population become extinct in 1 year = 12 months. Using the equation founded on part b we got:

[tex] 12 = 2 *ln(\frac{820}{820-p_0}) [/tex]

[tex] 6 = ln (\frac{820}{820-p_0}) [/tex]

Using exponentials we got:

[tex] e^6 = \frac{820}{820-p_0}[/tex]

[tex] (820-p_0) e^6 = 820[/tex]

[tex] 820-p_0 = \frac{820}{e^6}[/tex]

[tex] p_0 = 820-\frac{820}{e^6}[/tex]

For the given case of increment of mice' population, we get following figures:

An equation containing derivatives of a variable with respect to some other variable quantity is called differential equations. The derivatives might be of any order, some terms might contain product of derivatives and the variable itself, or with derivatives themselves. They can also be for multiple variables.

For the considered case, the population of mice with respect to time passed in months is given by the differential equation:

[tex]\dfrac{dp}{dt} = 0.5p - 410[/tex]

Taking same variable terms on same side, and then integrating, we get:

[tex]\dfrac{dp}{0.5p - 410} = dt\\\\\int \dfrac{dp}{0.5p - 410} = \int dt\\\\\dfrac{\ln(|0.5p - 410|)}{0.5} = t + C_1\\\ln(|0.5p - 410|) = 0.5t + 0.5C_1 = 0.5t + C[/tex]

where C₁ is integration constant.

Since it is specified that at time t = 0, the population p = 770, therefore, putting these values in the equation obtained above, we get:

[tex]\ln(|0.5p - 410|) = 0.5t + 0.5C_1 = 0.5t + C\\\\\ln(|0.5 \times 770 - 410|) = 0.5 \times 0 + C\\\\\ln(|-25|) = C\\C = \ln(25) \approx 3.22[/tex]

Therefore, we get the relation between p and t as:

[tex]\ln(|0.5p - 410|) = 0.5t + 0.5C_1 = 0.5t + C\\\ln(|0.5p - 410|) \approx 0.5t + 3.22\\\\|0.5p - 410| \approx e^{0.5t + 3.22}\\\text{Squaring both the sides}\\\\(0.5p - 410)^2 \approx e^{t+6.44}\\(p-820)^2 \approx 4e^{t+6.44}\\\\p^2 -1640p + 672400 \approx 4e^{t+6.44}[/tex]

Calculating the needed figures for each sub-parts of the problem:

a): The time at which the population becomes extinct.

Let it be t at which p becomes 0, then, from the equation obtained, we get:

[tex]p^2 -1640p + 672400 \approx 4e^{t+6.44}\\\text{At p = 0}\\672400 \approx 4e^{t+6.44}\\\\t \approx \ln{(\dfrac{672400}{4}) - 6.44 = \ln(168100) - 6.44 \approx 5.59 \text{\: (In months)}[/tex]

Thus, after 5.59 months approx, the population of mice will extinct.

b) Find the time of extinction if p(0) = p0, where 0 < p0 < 820

From the equation [tex]\ln(|0.5p - 410|) = 0.5t + C[/tex]

putting [tex]p = p_0[/tex] when t = 0, we get the value of C as:

[tex]\ln(|0.5p_0 - 410|) = C[/tex]

Thus, the equation becomes

[tex]\ln(|0.5p - 410|) = 0.5t + \ln(|0.5p_0 - 410|)[/tex]

At time of extension t months, p becomes 0, thus,

[tex]\ln(|0.5p - 410|) = 0.5t + \ln(|0.5p_0 - 410|)\\\text{At p = 0, we get}\\\\\ln(410)=0.5t + \ln(|0.5p_0 - 410|)\\\\t = 2\ln(\dfrac{410}{0.5p_0 - 410}) = 2\ln(\dfrac{820}{|p_0-820|})\\\\\text{Since 0 } < p_0 < 820, \text{ thus, we get }\\\\t = 2\ln(\dfrac{820}{820-p_0})[/tex]

Thus, the extinction time (in months) of population of mice when its given that [tex]p(0) = p_0[/tex] is given by:

[tex]t = 2\ln(\dfrac{820}{820-p_0})[/tex]

c) Find the initial population [tex]p_0[/tex] if the population is to become extinct in 1 year.

Putting t = 12 (since t is measured in months, and that 1 year = 12 months) in the equation obtained in the second part, we get the value of initial population as:

[tex]t = 2\ln(\dfrac{820}{820-p_0})\\\\12 = 2\ln(\dfrac{820}{820-p_0})\\e^{6} = \dfrac{820}{820-p_0}\\1 - \dfrac{p_0}{820} = \dfrac{1}{e^6}\\p_0 \approx 820(1 - \dfrac{1}{e^6}}) \approx 818[/tex]

Thus, the initial population of mice for given conditions would be approx 818

Therefore, for the given case of increment of mice' population, we get following figures:

Learn more about differential equations here:

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