Please help due in 5 minutes

Answer:

a

Step-by-step explanation:

If the growth rate of bacteria at any time is proportional to the number of bacteria present at t then the population after 20 weeks will be 403.42\(x_{0}\) in which \(x_{0}\) is the initial population.

Given that the growth rate of bacteria at any time t is proportional to the number present at t and triples in 1 week.

We are required to find the number of bacteria present after 10 weeks.

let the number of bacteria present at t is x.

So,

dx/dt∝x

dx/dt=kx

1/x dx=k dt

Now integrate both sides.

\(\int\limits {1/x} \, dx\)=\(\int\limits{k} \, dt\)

log x=kt+log c----------1

Put t=0

log \(x_{0}\)=0 +log c    (\(x_{0}\) shows the population in beginning)

Cancelling log from both sides.

c=\(x_{0}\)

So put c=\(x_{0}\) in 1

log x=kt+log \(x_{0}\)

log x=log \(e^{kt}\)+log \(x_{0}\)

log x=log \(e^{kt}x_{0}\)

x=\(e^{kt}x_{0}\)

We have been given that the population triples in a week so we have to put the value of x=2\(x_{0}\) and t=1 to get the value of k.

2\(x_{0}\)=\(e^{k} x_{0}\)

2=\(e^{k}\)

log 2=k

We have to now put the value of t=20 and k=log 2 ,to get the population after 20 weeks.

x=\(e^{20log 2}\)\(x_{0}\)

x=\(e^{0.30*2}\)\(x_{0}\)

x=\(e^{6}x_{0}\)

x=403.42\(x_{0}\)

If the growth rate of bacteria at any time is proportional to the number of bacteria present at t then the population after 20 weeks will be 403.42\(x_{0}\) in which \(x_{0}\) is the initial population.

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The given question is incomplete as the question incudes the following:

Calculate the population after 20 weeks.

Answer:

im guessing 15

Step-by-step explanation:

To calculate the income Alice will receive in her first year of retirement, we need to consider her pension fund, the state pension, and the annuity rates.

Given: Alice's salary: £55,000

Her contributions to the pension scheme: 8% of salary

Employer contributions and tax relief: 4% of salary

Retirement age: 66

State pension: £9,000 per year

First, let's calculate Alice's pension fund. She has been in the scheme from age 35 to 66, which is a total of 31 years. During this period, she has made contributions of 8% of her salary, topped up by 4% from the employer and tax relief. So, her annual contributions are (8% + 4%) of £55,000 = £5,500.

Her total pension fund would be £5,500 x 31 = £170,500.

Next, let's consider the annuity rates. Annuity rates determine the income that can be purchased with the pension fund. The rates vary based on factors such as age, gender, and prevailing market conditions.

Using the annuity rates, Alice can calculate the income she will receive from her pension fund. Let's assume the annuity rate is 5% per year. So, the income from her pension fund would be £170,500 x 5% = £8,525.

Finally, we add the state pension to calculate Alice's total income in her first year of retirement:

Total income = Pension fund income + State pension

Total income = £8,525 + £9,000 = £17,525.

Therefore, Alice will receive £17,525 as her income in the first year of retirement.

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Answer:

Step-by-step explanation:

4 + 6z - 15 = 6z - 11

2(3z -4) = 6z - 8

Here's the tricky part. You are only taking 8 away from 6z

The result is going to be larger than when you take away 11 from 6x

Try it

suppose z = 7

6*7 - 8 = 42 - 8 = 34

6*7 - 11 = 42 - 11 = 31

34>31

(a) The normal mode shapes are defined by the eigenvectors of the matrix and are the shapes the system will take when it vibrates with only one natural frequency active. The two normal mode shapes are as follows: Normal mode 1:x1(t) = 0.909x1 + 0.416x2Normal mode 2:x2(t) = -0.416x1 + 0.909x2.

(b) The principal coordinates, u1 and u2, are defined as follows: ui = vT qi Where v is the eigenvector matrix and q is the generalized coordinate vector. Therefore,u1 = 0.909q1 - 0.416q2u2 = 0.416q1 + 0.909q2.

(c) To solve for the equation of motion with respect to the principal coordinates, we will first need to derive the mass matrix. As a result of the transformation from the generalized coordinates to the principal coordinates, this is the diagonal matrix M:diag(m1, m2) = diag(76, 28)From the equation, we get,DvTq" + Kvtq = 0
vTMvTq" + vTKvTq = 0
Md" + Kvq = 0
Therefore,d1u1" + k1u1 = 0
d2u2" + k2u2 = 0Where k1 and k2 are the diagonal entries of the stiffness matrix K, which can be obtained from the dynamic matrix (D = K - ω2M). k1 = 204,800 N/m, k2 = 28,000 N/m.

(d) We must first convert the initial conditions from the generalized coordinates to the principal coordinates:

q1 = 0.909x1 + 0.416x2 = 0.909(0.5) + 0.416(40) = 16.82q2 = -0.416x1 + 0.909x2 = -0.416(0.5) + 0.909(80) = 72.14.

(e) The free vibration can be determined using the following equations:u1(t) = A1 cos(w1t + φ1)u2(t) = A2 cos(w2t + φ2)To solve for A1, A2, φ1, and φ2, we must use the initial conditions in the principal coordinates.

The initial conditions for u1 are:u1(0) = A1 cos(0 + φ1) = 16.82u1'(0) = -100A1 sin(0 + φ1) = 0.

Therefore, φ1 = 0 and A1 = 16.82The initial conditions for u2 are:u2(0) = A2 cos(0 + φ2) = 72.14u2'(0) = -20A2 sin(0 + φ2) = 0.

Therefore, φ2 = 0 and A2 = 72.14Hence, the free vibration with respect to the principal coordinates is as follows:u1(t) = 16.82 cos(100t)u2(t) = 72.14 cos(20t).

Therefore, the normal mode shapes for each natural frequency were explained, the principal coordinates in terms of the generalized coordinates were determined, the equation of motion was determined with respect to the principal coordinates, the initial conditions with respect to the principal coordinates were determined, and the free vibration of the system with respect to the principal coordinates in response to the initial conditions listed in (d) were also determined.

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Answer:

y=2x+5

Step-by-step explanation:

The height of the tunnel, h, is approximately 2.65 meters, correct to 2 decimal places.

We're given that the cross-section of the tunnel is part of a circle with a radius of 4 meters and the width at road level is 6 meters. To find the height, h, we can use the Pythagorean theorem in relation to the radius and half of the width.

Since the tunnel width is 6 meters, half of it would be 3 meters. This forms a right triangle with the radius as the hypotenuse and half of the width and the height as the other two sides. Applying the Pythagorean theorem:

4^2 = 3^2 + h^2
16 = 9 + h^2
h^2 = 7

Now, taking the square root of both sides:

h = √7 ≈ 2.65

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a. The graph of the budget line would have hamburgers on the x-axis and bags of figs on the y-axis. b. the budget line equation represents Rachel's affordability based on her income and the prices of hamburgers and figs. Changes in prices, income, or additional resources can affect the slope, position, or rotation of the budget line on a graph.

a) Rachel's budget line equation can be written as follows:

Budget = (Price of Hamburger * Quantity of Hamburgers) + (Price of Figs * Quantity of Figs)

Since the price per hamburger is $3 and the price per bag of figs is $2, the equation becomes:

Budget = 3x + 2y

Where x represents the quantity of hamburgers and y represents the quantity of bags of figs. The graph of the budget line would have hamburgers on the x-axis and bags of figs on the y-axis.

b) If the price of figs increases to $3, the new budget line equation becomes:

Budget = 3x + 3y

The graph of the new budget line would show a steeper slope compared to the original budget line. This indicates that the relative price of figs has increased, making them relatively more expensive compared to hamburgers.

c) In this scenario, Rachel has an income of $50, the price per hamburger is $3, the price per bag of figs is $3, and she receives an additional $10 in cash from a friend. The new budget line equation can be written as:

Budget = (3x + 3y) + 10

The graph of the new budget line would shift upward parallel to the original budget line. The additional cash from Rachel's friend increases her purchasing power, allowing her to afford more hamburgers and/or bags of figs.

d) Now, Rachel's friend gives her a gift basket containing 3 free bags of figs. In this case, the budget line equation remains the same as in part c:

Budget = (3x + 3y) + 10

However, since Rachel receives 3 free bags of figs, she can allocate more of her budget towards purchasing hamburgers. This would cause the budget line to rotate outward from the y-intercept, resulting in a flatter slope. The new budget line would reflect Rachel's ability to purchase more hamburgers with the same income and price of figs.

In summary, the budget line equation represents Rachel's affordability based on her income and the prices of hamburgers and figs. Changes in prices, income, or additional resources can affect the slope, position, or rotation of the budget line on a graph.

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Amrit had £859.98 in her account after 3 years as a result.

Any loans and borrowings come with interest. the percentage of a loan balance that lenders use to determine interest rates. Consumers can accrue interest through lending money (via a bond or deposit certificate, for example), or by making a deposit into a bank account that pays interest.

We can apply the following formula to determine Amrit's three-year compound interest:

\(A = P(1 + r/n)^(n*t)\)

Where:

A is the sum of money at the end of the given time.

P stands for the principal sum (initial investment)

A annual interest rate is expressed as r, where r is a decimal, and n is the number many times that interest is compounded annually.

T is the age in years.

In Amrit's case, we have:

P = £750

r = 0.05 for the first year, and 0.04 for each subsequent year

Considering that interest is compounded annually, n = 1.

t = 3 years

Applying the technique, we can determine Amrit's account balance after three years as follows:

A = 750(1 + 0.05/1)¹¹ × (1 + 0.04/1)²¹

A = 750(1.05)¹ × (1.04)²

A = 750(1.05) × (1.0816)

A = £859.98 (rounded to 2 decimal places)

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Therefore , the solution of the given problem of probability comes out to be the result of adding these five times is: 5 x (1/32) = 5/32.

Probability theory is a branch of mathematics that deals with estimating the likelihood of an event occurring or of a claim being true. A probability is just a value between 0 and 1 and 1, with 0 approximately denoting the possibility of an event occurring and 1 denoting certainty. The likelihood or chance that an event will occur is expression numerically as a probability.

Here,

There are five options since you need one additional set and four even numbers:

5 C 4 ==5

OEEEE, EOEEE, EOEOE, EEEOE, and EEEEO

Three-sixths of the time, you'll obtain an odd number.

Three-sixths of the time, you'll obtain an even number.

As a result, the likelihood of OEEEE is (3/6)5=1/32.

This also represents the likelihood of each additional possibility.

The result of adding these five times is: 5 x (1/32) = 5/32.

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Answer:

When you reflect a point across the x-axis, the x-coordinate remains the same, but the y-coordinate is transformed into its opposite so just move it over to the opposite side

Step-by-step explanation:

A frustum is one of two pieces of a double cone divided at the vertex. A double cone is a three-dimensional shape that is created by connecting two cones with their vertices touching.

When the double cone is cut through the vertex, it creates two pieces known as frustums. A frustum has a circular base and a smaller circular top, which are parallel to each other. The height of the frustum is the distance between the two circular bases.

The volume of a frustum can be calculated using the formula V = (1/3)h(a^2 + ab + b^2), where h is the height, a is the radius of the larger base, and b is the radius of the smaller top. Frustums are commonly found in architecture and engineering, such as in the design of buildings and bridges.

A "napped cone" is one of two pieces of a double cone divided at the vertex. When a double cone is bisected through its vertex, it results in two identical, mirror-image napped cones. These geometric shapes have various applications in mathematics, engineering, and design due to their unique properties.

Napped cones share some characteristics with regular cones, such as having a circular base, but their pointed vertex is replaced by a flat plane where the double cone was divided. This creates a shape that is both symmetrical and easy to manipulate for various purposes.

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Answer:what

Step-by-step explanation:what does this mean

The solution of the systems of equations is (1, -1,667).

A set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought

Given that, a system of equations, we need to solve them graphically,

The equation are =

4x-3y = 9....(i)

x-3y = 6....(ii)

To solve by graphically, we will first plot the graph,

Finding the coordinates for same,

Equation (i)

4x-3y = 9

Put x = 0,

y = -3

a) (0, -3)

Put x = 3,

y = 1

b) (3, 1)

Equation (ii)

x-3y = 6

Put x = 0,

y = -2

c) (0, -2)

Put x = 3,

y = -1

d) (3, -1)

Therefore, we get the coordinates of the equations, now when we plot the graph, both the lines will intersect at a point and that point will be the solution of the systems of equations,

The lines are intersecting at (1, -1,667).

Hence, the solution of the systems of equations is (1, -1,667).

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The area of inscribed regular hexagon is 6\(\sqrt{3\\\) unit square using the properties of equilateral triangles which combined, form the hexagon.

In a circle, a regular hexagon can be inscribed with all the six vertices of hexagon touching the circumference of the circle. The center of the circle is the center of the hexagon itself.

Now, if we join the center with all the vertices of the hexagon, we see six equilateral triangles are formed. The other two sides of each triangle apart from the ones joining the vertices of hexagon, are radii of the circle and all the sides are equal in length. The attached image can be referred for better understanding.

To find the area of the hexagon, we need to find the area of single equilateral triangle and multiply it by 6, as six triangles make a regular hexagon.

Radius of the circle (\(r\)) = 2 units

Area of a single equilateral triangle = \(\sqrt{3} /4\) \(r^{2}\)

                                                           =( \(\sqrt{3} /4\) ) × \(2^{2}\)

                                                           = \(\sqrt{3}\)

Area of hexagon = Area of 6 equilateral triangles = 6 × \(\sqrt{3}\)

Area of hexagon = \(6\sqrt{3}\) \(units^{2}\)

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1. C

2. D

3. A

4. A

5. C

6. A

7. Do yourself

Answer:

option 6b):) is correct

Given system is: dx1/dt = 2x1 + 2x2 + tdx2/dt = x1 + 3x2 - 2tNow we will use matrix notation, let X = [x1 x2] and A = [2 2; 1 3]. Then the given system can be written in the form of X' = AX + B, where B = [t - 2t] = [t, -2t].Now let D = |A - λI|, where λ is an eigenvalue of A and I is the identity matrix of order 2.

Then D = |(2 - λ) 2; 1 (3 - λ)|= (2 - λ)(3 - λ) - 2= λ² - 5λ + 4= (λ - 1)(λ - 4)Therefore, the eigenvalues of A are λ1 = 1 and λ2 = 4.Now let V1 and V2 be the eigenvectors of A corresponding to eigenvalues λ1 and λ2, respectively. Then AV1 = λ1V1 and AV2 = λ2V2. Therefore, V1 = [1 -1] and V2 = [2 1].Now let P = [V1 V2] = [1 2; -1 1]. Then the inverse of P is P⁻¹ = [1/3 2/3; -1/3 1/3]. Now we can find the matrix S(t) = e^(At) = P*diag(e^(λ1t), e^(λ2t))*P⁻¹, where diag is the diagonal matrix. Therefore,S(t) = [1 2; -1 1] * diag(e^(t), e^(4t)) * [1/3 2/3; -1/3 1/3])= [e^(t)/3 + 2e^(4t)/3, 2e^(t)/3 + e^(4t)/3; -e^(t)/3 + e^(4t)/3, -e^(t)/3 + e^(4t)/3].Now let Y = [y1 y2] = X - S(t).

Then the given system can be written in the form of Y' = AY, where A = [0 2; 1 1] and Y(0) = [x1(0) - (1/3)x2(0) - (e^t - e^4t)/3, x2(0) - (2/3)x1(0) - (2e^t - e^4t)/3].Now let λ1 and λ2 be the eigenvalues of A. Then D = |A - λI| = (λ - 1)(λ - 2). Therefore, the eigenvalues of A are λ1 = 1 and λ2 = 2.Now let V1 and V2 be the eigenvectors of A corresponding to eigenvalues λ1 and λ2, respectively.  Therefore, V1 = [1 -1] and V2 = [2 1].Now let P = [V1 V2] = [1 2; -1 1]. Then the inverse of P is P⁻¹ = [1/3 2/3; -1/3 1/3]. Now we can find the matrix Y(t) = e^(At) * Y(0) = P*diag(e^(λ1t), e^(λ2t))*P⁻¹ * Y(0), where diag is the diagonal matrix. Therefore,Y(t) = [1 2; -1 1] * diag(e^(t), e^(2t)) * [1/3 2/3; -1/3 1/3]) * [x1(0) - (1/3)x2(0) - (e^t - e^4t)/3, x2(0) - (2/3)x1(0) - (2e^t - e^4t)/3]= [(e^t + 2e^(2t))/3*x1(0) + (2e^t - e^(2t))/3*x2(0) + (e^t - e^4t)/3, -(e^t - 2e^(2t))/3*x1(0) + (e^t + e^(2t))/3*x2(0) + (2e^t - e^4t)/3].Therefore, the general solution of the system is X(t) = S(t) + Y(t), where S(t) = [e^(t)/3 + 2e^(4t)/3, 2e^(t)/3 + e^(4t)/3; -e^(t)/3 + e^(4t)/3, -e^(t)/3 + e^(4t)/3] and Y(t) = [(e^t + 2e^(2t))/3*x1(0) + (2e^t - e^(2t))/3*x2(0) + (e^t - e^4t)/3, -(e^t - 2e^(2t))/3*x1(0) + (e^t + e^(2t))/3*x2(0) + (2e^t - e^4t)/3].

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An extraneous solution is the root of a transformed equation that isn't the root of the original equation.

Your information is incomplete. Therefore, an overview will be given. It should be noted that an extraneous solution simply means a solution that is gotten from the solving of a problem but isn't a valid solution to the problem.

In order to find whether a solution is extraneous, you'll bed to plug it back into the equation and see if it works.

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To find the slope from a table, you need to find the difference in the y-values (the dependent variable) between two points and divide it by the difference in the x-values (the independent variable) between those same two points.

This is known as the slope formula:

Here is an example of finding the slope from a table:

Suppose we have the following table:

x y

3 7

4 9

To find the slope between these two points, we can use the slope formula as follows:

slope = (9 - 7)/(4 - 3) = 2/1 = 2

So the slope between these two points is 2.

It's important to note that the slope is a measure of the slope of the line that passes through the two points. If you want to find the slope of the line that is the best fit for a larger set of data, you will need to use a different method, such as linear regression.

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Answer:

  2)  c)  (x-3)² + (y+2)² = 25

  5)  x^2 +y^2 -8x -16y +54 = 0

  6)  x^2 +y^2 -10x -12y +36 = 0

Step-by-step explanation:

2) The standard form equation for a circle is ...

  (x -h)^2 +(y -k)^2 = r^2

You are given the center: (h, k) = (3, -2) and a point on the circle. So, the equation will be ...

  (x -3)^2 +(y +2)^2 = r^2

Since we know a point on the circle we know that ...

  (7 -3)^2 +(1 +2)^2 = r^2 = 16 +9 = 25

So, the circle's equation is ...

  (x -3)^2 +(y +2)^2 = 25 . . . . . matches choice C

__

5) As in the previous problem, the standard form equation is ...

  (x -4)^2 +(y -8)^2 = (-1-4)^2 +(7-8)^2 = 25+1 = 26

To put this in general form, we need to subtract 26 and eliminate parentheses.

  x^2 -8x +16 +y^2 -16y +64 -26 = 0

  x^2 +y^2 -8x -16y +54 = 0

__

6) A circle tangent to the y-axis will have a radius equal to the x-value of the center point.

  (x -5)^2 +(y -6)^2 = 5^2

  x^2 -10x +25 +y^2 -12y +36 = 25

  x^2 +y^2 -10x -12y +36 = 0

Answer: 9500000

Step-by-step explanation: 95 × 100000 = 9500000

*Multiply the value by 100000*

Answer: Jack made $898.25 in commission

Step-by-step explanation:

Jack made $898.25 in commission. ($2,000 x 15%) + ($3,975 x 5%) = $898.25

\(5975-2000=3975\) this is the amount jack earns 5% commission on

Answer:

10980 is the correct answer

We have the following:

with a proportionality rule we can solve the question, since if 40% are 60 pieces, how many will be 100% :

The answer is 150 pieces of bubble in total

The standard deviation is best described as a measure of the spread of a distribution around its mean or expected value.

What is standard deviation?

Standard deviation (SD) is a numerical measurement of how spread out or dispersed a set of data is. It can be expressed as the square root of variance (s²), which is the average deviation of a data point from the mean.

What is the formula for standard deviation?

The formula for standard deviation can be written as:

s = √(Σ (Xi - µ)² / (n - 1))

where:s = standard deviation

           Σ = sum

           Xi = each data point

           µ = the mean of the data points

           n = number of data points

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Answer:

To solve the given system of equations using the substitution method, we need to solve one equation for one variable and then substitute that expression into the other equation for that same variable. Let's solve the second equation for y.

-x + y = 4

y = x + 4

Now we can substitute this expression for y into the first equation and solve for x.

-2x + 2(x + 4) = 7

-2x + 2x + 8 = 7

8 = 7

The equation 8 = 7 is not true, which means the system of equations has no solution. We can see this visually by graphing the two lines. They are parallel and will never intersect, which means there is no point that satisfies both equations.

Therefore, the solution to the system of equations is "No solution."

Note: Please be sure to double-check your work to avoid mistakes.

Step-by-step explanation:

Hope this helps you!! Have a wonderful day/night!!

The homogeneous and particular solutions to obtain the general solution to the differential equation.

The solution to the differential equation d²y/dt² + y = cos(t) with initial conditions y(0) = 0 and y'(0) = 0 can be found using the method of Variation of Parameters. The solution is:

y(t) = (1/2)*cos(t) - (1/2)tsin(t) + (1/2)*∫[0,t] sin(τ)*cos(τ)dτ

where the integral represents the convolution of the functions cos(t) and sin(t) with the Green's function of the differential equation, which is sin(t). The first two terms are the homogeneous solution of the differential equation, while the third term is the particular solution obtained using the method of Variation of Parameters.

To obtain this solution, we first find the homogeneous solution by assuming y(t) = Acos(t) + Bsin(t) and substituting it into the differential equation. We then solve for the constants A and B using the initial conditions. Next, we use the method of Variation of Parameters to find the particular solution, which involves finding two functions u(t) and v(t) and using them to construct the particular solution. Finally, we add the homogeneous and particular solutions to obtain the general solution to the differential equation.

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Answer:The result obtained by differentiating the result of an integral.

The derivative of an integral is given by the fundamental theorem of calculus. More specifically, if f(x) is a continuous function on the interval [a, b], then the derivative of the integral of f(x) from a to x is given by f(x).

In other words, if F(x) is an antiderivative of f(x), then the derivative of the integral of f(x) from a to x is F'(x) = f(x).

Symbolically, we can write:

d/dx ∫[a,x] f(t) dt = f(x)

where the integral sign ∫ represents the integral operation and d/dx represents the derivative operation.

This result is very useful in calculus, as it allows us to easily compute derivatives of functions that are defined as integrals.