In 2004 the population in Morganton, Georgia, was 43,000. The population in Morganton doubled by 2010. If the growth rate remains the same, what is the expected population in Morganton in 2020?

Answer:

Therefore the measure of∠ A is 60.07.

Step-by-step explanation:

Given:

In Right Angle Triangle ABC

∠ B = 90°

BC = 13   ....Side opposite to angle A

AC = 15  .... Hypotenuse

To Find:

m∠A = ?

Solution:

In Right Angle Triangle ABC ,Sine Identity,

[tex]\sin A = \dfrac{\textrm{side opposite to angle A}}{Hypotenuse}\\[/tex]

Substituting the values we get

[tex]\sin A = \dfrac{BC}{AC}=\dfrac{13}{15}=0.8666\\\\A=\sin^{-1}(0.8666)=60.065\\\\m\angle A=60.07\°[/tex]

Therefore the measure of∠ A is 60.07

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:

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:

a) The two populations were i) the pregnant mothers who took the drug ii) the pregnant mothers who did not take the drugs

b) The data must have been obtained in a survey because experiment was not done.

c) 63 out of 3980 developed abnormalities in I case.

Hence out of 1000 abnormalities estimated = [tex]\frac{63}{3980} *1000\\=15.829\\[/tex]

i.e. approximately 16

d) Mothers who did not take drug

(information incomplete)

e) Medical hypothesis testing requires accurate results and hence sample sizes should be very large.

The mentioned study compares two populations: women exposed to DES during their mother's pregnancy and women who weren't. The data seems to be from a survey, is calculated with available information to be around 15.8 per 1000 women for the first population and half that for the second. Medical studies use large samples for higher statistical reliability.

a. The two populations in this study are women whose mothers took the drug DES during pregnancy and women whose mothers did not take the drug DES during pregnancy.

b. The data is most likely obtained through a survey, since medical research often relies on observations, health histories, and existing data rather than conducting an experiment. This would also protect subjects' safety and uphold ethical considerations.

c. To provide a descriptive statistic, we would use the rate of occurrence in the sample to estimate the rate in the overall population. In the sample of 3980 women, 63 developed tissue abnormalities. This is a rate of (63/3980) * 1000 ≈ 15.8 per 1000 women.

d. This question does not provide specific data for the second population, but based on the statement that women from the first group are twice as likely to develop abnormalities, we can estimate that the occurrence of abnormalities in the second population would be half as frequent. This would be approximately 7.9 out of 1000 women.

e. Large sample sizes are often used in medical studies to ensure the results are statistically significant and more reliable. This helps to avoid anomalies and provides a more accurate representation of the population.

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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:

[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 ^.^

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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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:

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:

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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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:

[tex]m\angle CED= 64\°[/tex]  

[tex]m\angle ACD=124\°[/tex]  

Step-by-step explanation:

In the figure given:

∠ABC = 93°

∠BAC = 31°

∠CDE = 60°

To find ∠CED and ∠ACD.

Solution:

In triangle ABC, we are given two vertex angles. We can find the third angle as angle sum of triangle = 180°.

∠ABC = 93° , ∠BAC = 31°

∠BCA=  [tex]180\°-(93\°+31\°)[/tex]

∠BCA = 56°

[tex]m\angle BCA+m\angle ACD=180\°[/tex]    [Supplementary angles forming a linear pair]

[tex]m\angle ACD=180\°-56\°[/tex]

[tex]m\angle ACD=124\°[/tex]   (Answer)

In triangle CDE:

[tex]m\angle CDE+m\angle CED = m\angle ACD[/tex]   [Exterior angle theorem :Exterior angle of a triangle is equal to sum of opposite interior angles ]

[tex]60\°+m\angle CED = 124\°[/tex]

[tex]m\angle CED= 124\°-60\°[/tex]

[tex]m\angle CED= 64\°[/tex]      (Answer)

Answer:

m\angle CED= 64\°  

m\angle ACD=124\°  

Step-by-step explanation:

get an A!

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:

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.

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

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: 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

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: D

Step-by-step explanation:

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.

Question was Incomplete;Complete question is given below;

Trey drove to the mountains last weekend. There was heavy traffic on the way there, and the trip took 12 hours. When Trey drove home, there was no traffic and the trip only took 8 hours. If his average rate was 20miles per hour faster on the trip home, how far away does Trey live from the mountains?

Do not do any rounding.

Answer:

Trey lives 480 miles from the mountain.

Step-by-step explanation:

Given:

Time taken to drove the mountain =12 hours

Time taken to return back from mountain = 8 hours.

Let the speed at which he drove to mountain be denoted by 's'.

Speed on the trip to home = [tex]s+20[/tex]

We need to find the distance Trey live from the mountains.

Solution:

Let the distance be denoted by 'd'.

Now we know that;

Distance is equal to speed times Time.

framing in equation form we get;

distance from home to mountain [tex]d=12s[/tex]

Also distance from mountain to home [tex]d = (s+20)8=8s+160[/tex]

Now distance is same for both the trips;

so we can say that;

[tex]12s=8s+160[/tex]

Combining the like terms we get;

[tex]12s-8s=160\\\\4s=160[/tex]

Dividing both side by 4 we get;

[tex]\frac{4s}{4}=\frac{160}{4}\\\\s=40\ mph[/tex]

Speed while trip to mountain = 40 mph

Speed while trip to home = [tex]s+20=420+20=60\ mph[/tex]

So Distance [tex]d=12s=12\times40 = 480\ miles[/tex]

Hence Trey lives 480 miles from the mountain.

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

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:

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:

the gcf is 21r^2t^3

Step-by-step explanation:

21r^2t^3(3+2rt^2)

The greatest common factor of the expression 63r²t³+42r³t⁵ is 21r²t³, determined by finding the highest common power of each factor.

To find the greatest common factor (GCF) of the expression 63r²t³+42r³t⁵, we need to identify the highest powers of each factor that divide both terms.

Thus, the GCF of the expression 63r²t³+42r³t⁵ is 21r²t³.

The similarity statement relating the three triangles in the diagram is 'YHB ~ YDH ~ HDB'. This is because they are all similar triangles, sharing the same shape but differing in sizes due to scale.

In this case, the answer would be 'YHB ~ YDH ~ HDB'. We are looking for a similarity statement, which states that all three triangles are similar. Similar triangles are triangles that have the same shape, but can be different sizes, i.e. they are scaled versions of each other. Since HD is a perpendicular drawn from right angle H in triangle YHB, this creates two triangles (YDH, HDB) that are respectively similar to the original triangle YHB as each of the two triangles include one of the acute angles of triangle YHB and their own right angles. Therefore, YHB ~ YDH ~ HDB.

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Answer:

In this case the Central Tendency measure that would be appropriate to report is Mode.

Step-by-step explanation:

Central Tendency measures are listed as follows:

i.) Mean

ii) Median

iii) Mode.

In the case that the data collected of a population is qualitative and not quantitative then the best Central Tendency measure to qualify the data is Mode of the data.

In the given example the data collected is of the students' racial classification which is not quantitative and purely qualitative. Therefore in case it is proper to take the Central Tendency measure to be reported as the Mode.

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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