Unconstrained population growth is controlled by the differential equation P' = KP.(a) Solve by separation of variables. Your answer will have two constants (K and the integration constant C).(b) Consider a culture of bacteria undergoing unconstrained growth. The initial population is 3M and after 5 hours the population is 7M. Find the exact function describing the population growth.(c) Under the assumptions of the previous part, what will the population be after 10 hours? Give an exact expression using your function from the previous part, then use a computer or calculator to get a number.(d) The population after 10 hours should be exactly 49/3 . Check this, and if it isn’t true correct your work. Can you see why this is true without using any differential equations?

The Department of Energy was also eager to sign on, in part because it wanted to learn about how radiation causes genetic mutations.

An modification in the nucleic acid sequence of an organism's, virus's, or extrachromosomal DNA is referred to as a mutation. DNA or RNA can be found in the viral genome.

Any alteration to a cell's DNA sequence. Mistakes in cell division can result in mutations, as can exposure to environmental DNA-damaging substances.

Ionizing radiation may harm genetic material (mutations) in gonads or germ cells, which can result in disorders with a genetic component (hereditary defects). Malformations, metabolic issues, immune system deficits, etc. might occur from them.

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Answer:

The slope of the line that is perpendicular to y = 4x + 6 is the negative reciprocal of the slope of y = 4x + 6, which is -1/4.

We can use the point-slope form of a line, y - y1 = m(x - x1), to find the equation of the line that passes through the point (8, -4) and has a slope of -1/4, where m = -1/4 and (x1, y1) = (8, -4).

Plugging in the values, we get:

y - (-4) = -1/4 (x - 8)

y + 4 = -1/4 x + 2

Multiplying both sides by 4, we get:

4y + 16 = -x + 8

Adding x to both sides, we get:

4y + 16 + x = 8

y = -x/4 - 2

So the equation of the line that is perpendicular to y = 4x + 6 and passes through the point (8, -4) is y = -x/4 - 2.

Step-by-step explanation:

this means a triangle with one angle 30°, a second angle 60°, and the third angle 90°.

the sides of such a 30-60-90 triangle are always in the ratio of 1 : √3 : 2

that means the Hypotenuse is twice the length of the shorter leg. and the longer leg is square root of 3 times longer than the shorter leg.

30-60-90 Theorem: If a triangle has sides in the ratio x: x 3: 2 x and angles of 30, 60, and, then it is a triangle. The longer leg, the hypotenuse, and the shorter leg are always the same. The Pythagorean Theorem can be used in their place if you ever forget these theorems.

Triangle with sides of 30-60-90

Sides of a triangle with angles of 30–60–90:

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The equation for direct variation showing the relationship between x and y  is   y = 8x

What is a direct variation?

A sort of proportionality known as "direct variation" occurs when one quantity directly changes in response to a change in another quantity. This suggests that if one quantity increases, the other quantity will also increase proportionately. Similar to the last example, if one quantity declines, the other amount also declines. The link between direct variation and the graph will be linear, resulting in a straight line.

Given when x = 8/7, y = 1/7

The general equation for direct variation is

y = mx

m = x/y = ( 8/7 ) / ( 1/7) = 8

Therefore the equation for direct variation for the given question is

 y = 8x

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The possible additional amount he will spend more than 15.

An inequality is a relationship that denotes a non-equal comparison between two numbers or mathematical expressions.

According to  the question,

Let Amy, will spend $c additional amount.

Amy will spend more than $32 on gifts.

He spent $17.

We solved this question by using inequality.

The inequality equation is:

  c + 1 7 > 32

    c > 32 -17  ( by changing the sign)

     C > 15

So, the possible additional amount he will spend more than 15.

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The conclusion is that it is not necessary to learn how to drive in order to enjoy the smell of pizza.

In order to draw a conclusion from two given premises, you must identify the relationship between them. In this case, the two premises are unrelated. The first premise states that you don't know how to drive and the second states that you don't like the smell of pizza. Since there is no relationship between these two premises, there is no conclusion that can be drawn from them. However, you can draw a conclusion based on the absence of a relationship. In this case, the conclusion is that it is not necessary to learn how to drive in order to enjoy the smell of pizza. This is because the two premises have no bearing on each other and, therefore, learning how to drive does not necessarily lead to enjoying the smell of pizza.

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Josh washed a total of 5 plates if he want to do the dishes after work.

To find the total number of plates washed by Josh:

The scenario describes a person named Josh who had to wash dishes after work.

The method used to find the total number of plates washed by Josh was simple addition. The number of small plates (1) and the number of large plates (4) were added together to find the total number of plates (1 + 4 = 5). This type of problem is usually taught in grade 1 or grade 2, as part of early mathematics education that focuses on basic addition and counting skills.

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A quadratic function can have two irrational roots.

but A quadratic equation can have no real number solutions.

What is quadratic function?

A quadratic function is a polynomial function with one or more variables in which the highest exponent of the variable is two. Since the highest degree term in a quadratic function is of the second degree, therefore it is also called the polynomial of degree 2. A quadratic function has a minimum of one term which is of the second degree. It is an algebraic function.

The parent quadratic function is of the form f(x) = x^2 and it connects the points whose coordinates are of the form (number, number^2). Transformations can be applied on this function on which it typically looks of the form f(x) = a (x - h)^2 + k and further it can be converted into the form f(x) = ax^2 + bx + c.

Now,

According to the question,

A quadratic function can have two irrational roots. we can understand this by taking a example.

example:

x^2-x-2=0

This equation have two irrational roots  1+√3 and 1 - √3.

A quadratic function can have no real number solutions.

Hence,

             A quadratic function can have two irrational roots.

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The value of the composition function f ( g ( 4 ) ) = 31

Evaluation of a function at the value of another function is known as Composition of function. A function composition is a process in which two functions, f and g, form a new function, h, in such a way that h(x) = g(f(x)). This signifies that function g is being applied to the function x. So, in essence, a function is applied to the output of another function.

Given data ,

Let the first function be represented as f ( x )

Now , the value of f ( x ) = 3x + 1

Let the second function be represented as g ( x )

Now , the value of g ( x ) = x² - 6

And , the composition function f ( g ( x ) ) is calculated as

when x = 4

Substitute the value of x = 4 in the equation , we get

g ( x ) = x² - 6

g ( 4 ) = 4² - 6

g ( 4 ) = 10

f ( g ( 4 ) ) = f ( 10 )

Substitute the value of x = 10 in the equation , we get

f ( x ) = 3x + 1

f ( 10 ) = 3 ( 10 ) + 1

f ( 10 ) = 31

Hence , the equation is f ( g ( 4 ) ) = 31

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Statements i. and ii. are true while statement iii. is incorrect because, in an experiment, the control group is usually selected and assigned by the researchers, not self-selected by the participants.

i. True. In an experiment, the researchers have control over how participants are placed into different groups, usually through random assignment.

ii. True. In an observational study, participants are usually not manipulated by the researchers and are free to choose which group they belong to.

iii. False. A control group in an experiment is usually not self-selected. The control group is typically determined by the researchers through random assignment. The purpose of the control group is to provide a comparison to the experimental group in order to determine if the treatment has a significant effect.

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Equations and inequalities are both mathematical constructions made by joining two expressions.

A connection in mathematics that compares two numbers variables  or other mathematical expressions in an unequal way is known as an inequality. It is most frequently used to compare the sizes of two numbers on the number line. Less than or equal to, larger than, less than, and more than or equal to are the four fundamental inequalities.

Here,

Given :

-1 to 2 in the graph ,

To find the solution for the inequality ,

We get option -1 as the inequality

=> -1 ,0 ,1,2

=> -1 is the solution for this

These symbols for inequality are not equal (≠), less than (<), greater than (>), less than or equal (≤), and more than or equal (≥). Comparing numbers and identifying the range or ranges of values that satisfy the requirements of a given variable are both done using inequalities.

Both mathematical phrases, equations and inequalities, are created by connecting two expressions. The equal sign (=) indicates that two expressions in an equation are believed to be equivalent. The symbols >, <, ≤ or ≥ indicate that the two expressions in an inequality are not always equal.

Therefore, the correct answer is option b.

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The total number of boys is 19, out of them 1 spring, 7 summers, 10 autumn leaves 1 to pick winter.

The ratio can be defined as the number that can be used to represent one quantity as a percentage of another. Only when the two numbers in a ratio have the same unit can they be compared. Ratios are used to compare two objects.

Given that there are 17 total summer lovers and 7 boys/summer, we know 10 girls like summer.

girls' summer then, so 10/2 is 5 girls' choice.

Since there are 17 total summer lovers and 7 boys/summer, we know 10 girls like summer and Two girls picked summer over spring,

so, 10/2 is 5 girls picking spring.

The last clue says that if you pick a spring lover, there is an 18.75% chance of it being a boy, so x/(x+5)=0.1875 x=1.

38% of interviewed students preferred summer, so: 17/x = 0.38 Total students= 45

We know that for every 8 guys, there are 11 girls.

So if we fiddle around with scaling that ratio to find the combination that adds up to 45, we get a scale of 2.35.

2.35*8=19 and 2.35*11=26

So the total number of girls is 26, meaning the number of girls/in autumn is 26-5-4-10=7

Three more boys chose autumn than the girls, so 10 boys/autumn.

Therefore, The total number of boys is 19, out of them 1 spring, 7 summers, 10 autumn leaves 1 to pick winter.

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Alan has to sell 540 boxes to earn less than $2050

A linear model is an equation that describes a relationship between two quantities that show a constant rate of change.

Given is graph that represent the relation between the total sales and number of boxes sold,

The linear model for same is given by =

f(b) = 3.75b + 25

We need to find, how many boxes would Alan have to sell to earn less than $2050,

Here, we have, f(b) = $2050

Therefore,

$2050 = 3.75b + 25

3.75b = 2025

b = 540

Hence, Alan has to sell 540 boxes to earn less than $2050,

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X1+X2 have distribution Bi(n1+n2, p) then the probability is P(X1+X2 = 1 | X2 = 0) =  np(1-p)ⁿ¹⁻¹ ,P(X1+X2 = 1| X2 = 1) = (1-p)ⁿ¹and P(X1 + X2 = 1) = (1-p)ⁿ¹ * np(1-p)ⁿ²⁻¹+ (1-p)ⁿ²*np(1-p)ⁿ¹-¹

Since both variables are independent but they have the same probability parameter, you can interpret that like if the experiment that models each try in both variables is the same. When you sum both random variables toguether, what you obtain as a result is the total amount of success in n1+n2 tries of the same experiment, thus X1+X2 have distribution Bi(n1+n2, p).

Note that, if X2 = k, then X1+X2 = 1 is equivalent to X1 = 1-k. Since X1 and X2 are independent, then P(X1+X2 = 1| X2 = K) = P(X1=1-k|X2=k) = P(X1 = 1-k).

If k = 0, then this probability is equal to P(X1 = 1) = np(1-p)ⁿ¹⁻¹

If k = 1, then it is equal to P(X1 = 0) = (1-p)ⁿ¹

Thus,

P(X1+X2 = 1) = P(X1+X2 = 1| X2 = 1) * P(X2=1) + P(X1+X2 = 1| X2 = 0) * P(X2 = 0) = (1-p)ⁿ¹ * np(1-p)ⁿ²⁻¹+ (1-p)ⁿ²*np(1-p)ⁿ¹-¹

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Answer:

If angle B is 60 and angle C is 70, We can make an equation by:

180 - (60+70)

The answer for this equation will be 50.

According to the upper equation, angle A is 50, angle B is 60, and angle C is 70.

The linear equation for this scenario could be: y = 5x + 77, where y represents the total amount of money Finn has after cooking x number of meals and 77 is the initial amount owed to his mom.

A linear equation is an algebraic equation with simply a constant and a first-order (linear) component of the form y=mx+b, where m is the slope and b is the y-intercept. The above is sometimes referred to as a "linear equation with two variables," where y and x are the variables.

Ax+By=C is the typical form for linear equations in two variables. 2x+3y=5, for example, is a simple linear equation. It is rather simple to get both intercepts when an equation is stated in this way (x and y).

As per the data given,

The total amount Finn has = $ 77.

Amount spent on meal = $5 per meal.

Total number of meal = x

So, the amount spent on taking x meal will be = 5*x = 5x

Hence, the required linear equation will be: y = 5x + 77

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Logarithm ln3 is approximately 1.682.

Logarithm, the exponent or power to which a base must be raised in order to produce a specific number.

The linear approximation of the natural logarithm function near x=e is given by y ≈ ln(e) + (y'(e))(x-e), where y'(e) is the derivative of the logarithm function at x=e.

Since ln'(x) = 1/x, we have y'(e) = 1/e.

Substituting x=3 and e=2.71828... into the approximation, we get:

ln(3) ≈ ln(e) + (1/e)(3-e) = ln(e) + (3-e)/e

ln(e) = 1, so the result is:

ln(3) ≈ 1 + (3-e)/e ≈ 1 + 0.682 ≈ 1.682.

Hence, ln3 = 1.682.

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She need to contribute to the account every month, to the nearest dollar is  $26,195.00.

An annual interest rate of 3% compounded monthly. If Gianna would like to end up with $26, 000 after 3 years,

A = $26,195.00

I = A - P = $195.00

A is equal to P(1 + rt).

Calculation:

To start, multiplying R percent by 100 gives us 3%/100, or 0.03 each year.

For ease of calculation, let's convert time to years: 3 months / 12 months / year = 0.25 years.

How to resolve our equation

A = 26000(1 + (0.03 × 0.25)) = 26195\sA = $26,195.00

The total amount accrued, principal plus interest, from simple interest on a principal of $26,000.00 at a rate of 3% per year for 0.25 years (3 months) is $26,195.00.

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Answer:

18.84 inches

Step-by-step explanation:

circumference of a circle = diameter x pi

diameter = radius x 2

==> circumference = radius x 2 x pi = 3 in x 2 x 3.14 = 18.84 in

In this study, the group of students who take the exam in a regularly lit room can be considered the control group.

The control group provides a baseline or comparison for the experimental group (the group of students who take the exam in a dimly lit room). The control group helps to establish a cause-and-effect relationship between the independent variable (room lighting) and the dependent variable (test performance), as any observed differences in test performance between the two groups can be attributed to the difference in lighting conditions.

By having a control group, the researcher can ensure that any observed effects are due to the independent variable and not due to extraneous variables.

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The third table shows a proportional relationship. Its graph is shown below.

A table of a proportional relationship is a set of pairs of values of the two variables involved in the relationship, where one variable is proportional to the other.

If the relationship between the variables on a table is proportional, the values in the table will form a straight line when plotted on a graph.

The constant of proportionality (k) for the third table = y/x = 2. Therefore it represents a proportional relationship.

The graph of the table is shown below.

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(i)  Area of the floor = 120m²

(ii) Cost of carpeting the floor = Rs 7200

Given:

Amount of the air of a rectangular room (Cuboidal shaped) =Volume of rectangular room= 600m³

Height of the rectangular room = 5m

(i)   According to question we have to find area of the rectangular floor.

  We know Area of Rectangle = length × breadth

 here, length and breadth is not known to us let's find.

We Know, Volume of Cuboid = length × breadth × height

                        600m³             = length × breadth × 5m

                        600 ÷ 5            = length × breadth

                           120                = length × breadth

                length × breadth     = 120

Area of the rectangular floor = length × breadth

                                                = 120m²

(ii) Now we have to calculate the cost of carpeting rectangular room floor at Rs 60 per sq m.

Area of the rectangular floor = 120m²

Given:

Cost of carpeting the floor at Rs 60

Cost of carpeting the floor= cost × Area of floor

                                              = 60 × 120

Cost of carpeting the floor = Rs 7200

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By taking a quotient we will see that he has saved $450.

On the first month Lance earnesd $546, and on the second month he earned $354.

Adding these we get:

$546 + $354 = $900

So he earned a total of 900 dollars between April and June. And we know that he saves half of that, then the amount saved is equal to that divided by 2:

$90/2 = $450

He has saved 450 dollars.

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The required simplified solution of the given radical expression is  [tex]\sqrt{15} \times x^{1/6} / \sqrt[3]{10} \times y[/tex].

The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.

Here,
Given a radical expression,
[tex]= \sqrt{15xy} / \sqrt[3]{10xy^3}[/tex]

Now, [tex]\sqrt{15xy} = [15xy]^{1/2}[/tex] and [tex]\sqrt[3]{10xy^3} = [10xy^3]^{1/3}[/tex]
[tex]= [15xy]^{1/2}/ [10xy^3]^{1/3}\\[/tex]

After simplifying the above expression,
= [tex]\sqrt{15} \times x^{1/6} / \sqrt[3]{10} \times y[/tex]

Thus, the required simplified solution of the given radical expression is  [tex]\sqrt{15} \times x^{1/6} / \sqrt[3]{10} \times y[/tex].

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The function y = 3x + 7 is a linear function and the second function that is represented in table is an exponential function.

Here the function y = 3x + 7 is a linear function  and the second function is an exponential function because powers of 2.

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According to the information, on the 11th day, half of the substance should remain.

To calculate what day half of the substance will remain, we must carry out the following mathematical procedure.

First of all we must find the range of decrease from one value to another.

4, 4.5, 5.5, 4 and 3

We average these values and it gives us 4.4.

We then subtract 78 - 50 to identify the difference between these two mass values. and then we divide the result in the decreasing range (4.4).

According to the above, it would take another 6.3 days to reach 50%, so more or less on day 11 half of the substance would remain.

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The stopping distance of regular tires on glare ice is estimated by the function D(F) = 2F+115, where D(F) is the stopping distance in feet and F is the air temperature in degrees Fahrenheit.

(a) D(0°) = 115 feet, D(-20°) = 95 feet, D(32°) = 147 feet

(b) The domain should be restricted to [-57.5°,32°] because the function D(F) gives a negative value when F is less than -57.5°, which is not physically possible. The function is also not defined for temperatures greater than 32° because the stopping distance is not a linear function of the temperature at higher temperatures.

The stopping distance of regular tires on glare ice is estimated by the function D(F) = 2F+115, where D(F) is the stopping distance in feet and F is the air temperature in degrees Fahrenheit. For specific temperatures, D(0°) = 115 feet, D(-20°) = 95 feet, and D(32°) = 147 feet. The domain should be restricted to [-57.5°,32°] because the function is not physically possible for temperatures less than -57.5° and is not a linear function of temperature at higher temperatures.

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The set of side measurement that will form a triangle is 8, 9 and 15.

A triangle is a polygon with 3 sides. The sum of angles in a triangle is 180 degrees.

Therefore, the triangle inequality theorem can be used to know the length of sides that can form a triangle.

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

If the length are a, b and c. Therefore,

a + b > c

a + c > b

b + c > a

Therefore, applying this principle the measurement that will form a triangle is  8, 9 and 15.

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Answer:

We can use a system of equations to solve this problem. Let x be the number of 1-point shots and y be the number of 2-point shots.

The first equation represents the total number of shots taken:

x + y = 55 (1)

The second equation represents the total number of points scored:

x + 2y = 91 (2)

To solve for x and y, we can use the first equation to substitute for one of the variables in the second equation. Using (1) to substitute for x in (2):

y = 55 - x

Then substitute this value of y in equation (1)

x + (55 - x) = 55

Solving for x, we get:

x = 20

So the team made 20 1-point shots and 35 2-point shots.

Answer:

There are:

19 1-point shots

36 2-point shots

Step-by-step explanation:

Solve for the system of equations. The first equation will give you the value based on the total amount of points, while the other equation will be based on the amount of shots in total:

It is given that the team scores 91 points in all, and that they are all 1- and 2- point shots (no 3 pointers). Set the equation. Let x = 1-point shots, and y = 2-point shots:

x + y = 55

1x + 2y = 91

First, isolate the variable, y, in the first equation. Subtract x from both sides of the equation:

y + x = 55

y + x (-x) = 55 (-x)

y = 55 - x

Next, plug in 55 - x for y in the second equation:

x + 2y = 91

x + 2(55 - x) = 91

Then, simplify. First, distribute 2 to all terms within the parenthesis:

2(55 - x)

= (2 * 55) - (2 * x)

= 110 - 2x

x + 110 - 2x = 91

First, combine like terms. Like terms are terms that share the same amount of the same variable:

(x - 2x) + 110 = 91

-x + 110 = 91

Then, isolate the variable, x. First, subtract 110 from both sides of the equation:

-x + 110 (-110) = 91 (-110)

-x = 91 - 110

-x = -19

Isolate the variable, x, completely, by dividing -1 from both sides of the equation:

(-x)/-1 = (-19)/-1

x = -19/(-1)

x = 19

~

Plug in 19 for x in one of the given equations in the system of equations:

x + y = 55

19 + y = 55

Isolate the variable, y, by subtracting 19 from both sides of the equation:

y + 19 = 55

y + 19 (-19) = 55 (-19)

y = 55 - 19

y = 36

~

There are 19 1-point shots & 36 2-point shots

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Answer: 98

Step-by-step explanation:

63 + 35

= 63 + 30 + 5

= 93 + 5

= 98

Answer:

63+35=98

Step-by-step explanation: