1. Consider the system described by the differential equation: d3y(t)/dt3 + 4d2y(t)/dt2 + 6dy(t)/dt + 8y(t) = 20u(t)
(i) Find the transfer function of the system.
(ii) Derive a state space representation for the system
2. A single input, single output system has the matrix equations:
x.(t)= [0 1 -3 -4] x(t) +[0 1]u(t) and y(t) = [3 0]x(t)
Determine the transfer function G(s) = Y(s)/U(s)

Answer:

355 ÷ 8 = 44 with remainder 3

Step-by-step explanation:

                              44

                         -------------------------

                    8   | 355

                        -  32

                       --------

                              35

                           -  32

                           ------

                                3

Answer:

1

Step-by-step explanation:

First, simplify the parentheses.

2 - 3 = -1

Then, put this simplified value into the original expression.

(2 - 3)^(-1)

= (-1)^(-1)

Finally, execute the exponent function.

(-1)^(-1) = 1/1 = 1

Answer:

[tex]y = \sqrt{34} \\\\x = \sqrt{106}[/tex]

Step-by-step explanation:

Answer:

Cans of soup = 10

Frozen dinners = 5

Step-by-step explanation:

To determine the number of cans of soup and frozen dinners Caleb purchased, set up and solve a system of equations based on the given information.

Let x be the number of cans of soup Caleb purchased.

Let y be the number of frozen dinners Caleb purchased.

Each can of soup has 200 mg of sodium and each frozen dinner has 500 mg of sodium. The total amount of sodium in the purchased cans and frozen dinners is 4500 mg. Therefore:

[tex]200x + 500y = 4500[/tex]

Caleb purchased a total of 15 cans of soup and frozen dinners. Therefore:

[tex]x + y = 15[/tex]

To solve the system of equations, rearrange the second equation to isolate x:

[tex]\begin{aligned}x + y &= 15\\x + y -y&= 15-y\\x&=15-y\end{aligned}[/tex]

Substitute this into the first equation to eliminate the term in x, and solve for y:

[tex]\begin{aligned}200(15-y) + 500y &= 4500\\3000-200y + 500y &= 4500\\3000+300y &= 4500\\3000+300y-3000 &= 4500-3000\\300y&=1500\\300y \div 300&=1500 \div 300\\y&=5\end{aligned}[/tex]

Therefore, Caleb purchased 5 frozen dinners.

Substitute the found value of y into the rearranged second equation and solve for x:

[tex]\begin{aligned}x&=15-y\\x&=15-5\\x&=10\end{aligned}[/tex]

Therefore, Caleb purchased 10 cans of soup.

Answer:

Caleb purchased 10 cans of soup and 5 frozen dinners.

Step-by-step explanation:

Caleb purchased 10 cans of soup and 5 frozen dinners.

Let no. of can of soup be x and no. of frozen dinner be y.

To solve this, we can set up the following system of equations:

[tex]\tt x + y = 15[/tex]......[i]

[tex]\tt 200x + 500y = 4500[/tex]......[2]

Let's find the value by elimination method:

Multiplying equation one by 200.

we get,

[tex]\tt 200 x+ 200 y = 15*200[/tex]

[tex]\tt 200x +200 y =3000[/tex]

Subtracting equation 2 with equation 1.

[tex]\tt (200x+500y)-(200x +200 y) =4500-3000[/tex]

[tex]\tt \tt 300y = 1500[/tex]

[tex]\tt y =\dfrac{1500}{300}[/tex]

[tex]\tt y = 5[/tex]

Substitute the value of y into the first equation to find x.**

[tex]\tt x + 5 = 15[/tex]

[tex]\tt x = 15 - 5[/tex]

[tex]\tt x = 10[/tex]

Therefore, Caleb purchased 10 cans of soup and 5 frozen dinners.

Required Answer -:

16cos³ x sin² x

2cos x - ( cos 3x + cos 5x )

= 2cos x - 2 cos 4x cos x

= 2cos x 2sin² 2x

= 4 cos x ( 2sin x cos x)²

= 16cos³ x sin² x

Thus,, the value of 2 cos x - cos 3x - cos 5x is = 16cos³ x sin² x

Answer:

g(3) is 24

Step-by-step explanation:

To find the value of g(3), we need to plug 3 into the function g(x) and evaluate the resulting expression. The function g(x) is defined as g(x) = 3(2)^x. When we plug 3 into this function, we get g(3) = 3(2)^3. To evaluate this expression, we need to calculate 2^3, which is 8. Therefore, g(3) = 3 * 8 = 24.

In summary, the value of g(3) is 24.

Answer:

g(3) = 24

Step-by-step explanation:

Given information,

→ g(x) = 3(2)^x

Then the value of g(3) will be,

→ g(x) = 3(2)^x

→ g(3) = 3(2)³

→ g(3) = 3(8)

→ [ g(3) = 24 ]

Hence, the value of g(3) is 24.

The slope of the line given in the graph will be [m] is  - 0.78.

The general equation of a straight line is -

y = mx + c

Where -

[m] is the slope of the line.

[c] is the y - intercept.

Given is the graph of a straight line.

The line passes through the coordinates as follows -

(- 1.8, 0)        [lying on x-axis]

(0, - 1.4)        [lying on y-axis]

The slope of the line can be calculated using the formula -

m = (y₂ - y₁)/(x₂ - x₁)

m = (- 1.4 - 0)/(0 + 1.8)

m = - 1.4/1.8

m = - 0.78

Slope of the line will be -0.78.

Therefore, the slope of the line given in the graph will be [m] = - 0.78.

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The quadratic is the left-most piece.  This exists on the interval (-infty, -2).

The linear piece is the center piece and exists on the interval (-2,6).

The square root piece is the right-most piece and exists on (6, infty).

These intervals are describing how each piece lines up with the x-axis, as illustrated in the picture.

The required description of the given piecewise function are as follows:

quadratic → (-∞, -2); linear → (-2, 6) and square root → (6, ∞).

A piecewise-defined function (also known as a piecewise function or a hybrid function) is a function defined by multiple sub-functions, each of which applies to a different interval of the main function's domain (a sub-domain).

The graph is given in the question, as shown

1. quadratic function:

This the sub-function define between the interval (-∞, -2).

2. linear function:

This the sub-function define between the interval (-2, 6).

3. square root function:

This the sub-function define between the interval (6, ∞).

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Answer:

$345 (PLEASE GIVE BRAINLYEST )

Step-by-step explanation:

1/3x=115

x=345

The function [tex]y = (\frac{1}{2})^x[/tex] is a decreasing function.

The function [tex]y = (\frac{1}{2})^x[/tex] is shallower from left to right.

The value of the function [tex]y = (\frac{1}{2})^x[/tex] is unity at the y-intercept.

A function can be defined as the outputs for a given set of inputs.

The inputs of a function are known as the independent variable and the outputs of a function are known as the dependent variable.

The given function is [tex]y = (\frac{1}{2})^x[/tex] which is an exponential function cause the exponent is the variable.

Now, At x = - 2 y = 4 ⇒ (- 2, 4),

x = - 1, y = 2 ⇒ (- 1, 2).

x = 0, y = 1 ⇒ (0, 1).

x = 1, y = 1/2 ⇒ (1, 1/2).

x = 2, y = 1/4 ⇒ (2, 1/4).

x = 3, y = 1/8 ⇒ (3, 1/8).

The function is a decreasing function as the value of x increases the value of the function is decreasing.

The function is shallower from left to right.

The value of the function is unity at y-intercept as anything raised to the zero power is 1.

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(a) The population of the state was 18.5 million in 2000. (b)The population of the state will reach 26.6 million in the year 2002

A population is a distinct collection of people, whether that group consists of a nation or a population that shares a specific trait.  A population is the group of people from which a statistical sample is taken in statistics. As a result, a population can be defined as any group of people who have something in common.

Since t = 0 ,

Then A = 18.5

(b) The population of the state will reach 26.6 million in the year 2002

2002, by using the formula is 18.5e^0.173*t

Given,

A= 26.6

Then,    

[tex]A = 18.5 x e^{(0.1708t)} 26.6[/tex]

  [tex]= 18.5 x e^{(0.1708t)} \frac{26.6}{18.5}[/tex]

  = [tex]e^{(0.1708t)} 1.437838[/tex]

   = [tex]e^{(0.1708t)} \ln (\frac{1.437838}{0.1708})[/tex]

t = 2.126116

t =2

Therefore, 2 is the round down to the nearest years. So 2002 is the approximate year the population reached 26.6 million.

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The inequality that shows how long the temperature in Austin will remain colder than that in San Antonio is 33.5 + 2.5t < 37.5 + 1.5t and the time is less than 4 hours

From the question, we have the following parameters that can be used in our computation:

Austin

Initial temperature = 33.5 degrees Fahrenheit

Rate of increment = 2.5 degrees Fahrenheit

San Antonio

Initial temperature = 37.5 degrees Fahrenheit

Rate of increment = 1.5 degrees Fahrenheit

The above parameters mean that

Current temperature = Initial temperature + Rate of increment x Number of hours

So, we have

Austin: A(t) = 33.5 + 2.5t

San Antonio: B(t) = 37.5 + 1.5t

When Austin is colder than San Antonio, we have

A(t) < B(t)

Substitute the known values in the above equation, so, we have the following representation

33.5 + 2.5t < 37.5 + 1.5t

Evaluate the like terms

t < 4

Divide both sides by 1

t < 4

Hence, the interval is less than 4 hours

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Answer:

I'll help

Step-by-step explanation:

the entrance fee to the skating rink=$5

each minute the skater uses the rink costs=$0.25

Step-by-step explanation:

your teacher could have explained this better by telling us that A(t) is the remaining amount of the original isotope.

(a)

we start with 200 g and let it do its thing for 25 years.

all we need to do is enter the number of years into the function formula (as this is a college question you are able to work with functions, right ?) and calculate :

A(25) = 200e^(-0.054 × 25) = 200e^-1.35 =

= 51.84805213... g ≈ 51.85 g

after 25 years, about 51.85 g of the sample will be left.

assuming you have a calculator (at least on your computer) that was really "difficult", don't you think ?

(b)

half of the original amount is 200/2 = 100 g.

so, we need to find the value for t so that A(t) = 100.

100 = 200e^(-0.054 × t)

100/200 = 1/2 = 0.5 = e^(-0.054 × t)

ln(0.5) = -0.054 × t

t = ln(0.5) / -0.054 = 12.8360589... years

I assume you need to round to the nearest hundredth here as well :

after about 12.84 years the sample will decay to half of the original amount.

The term that makes NO distinction between an explanatory variable X and a response variable Y is Correlation , the correct option is (a) .

In the question ,

four options are given ,

we have to select the option that makes no distinction between explanatory variable X and a response variable Y .

Response Variable is the (dependent variable) measures the outcome of the study.

Explanatory Variable is the (independent variable) helps to explain or influence changes in a response variable.

So , Correlation makes no distinction between explanatory and response variables.

So , it makes no difference in calculating the correlation , which variable you call x and which you call y  .

Therefore , the correct option is (a) .

The given question is incomplete , the complete question is

Which of the following makes NO distinction between an explanatory variable X and a response variable Y (i.e., you can interchange the roles of X and Y and get the same result) ?

(a) Correlation

(b) Regression

(c) Both correlation and regression make no distinction between X and Y.

(d) Both correlation and regression make distinction between X and Y.

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Answer:

1/2

Step-by-step explanation:

Slope is rise (height)/run(length)

Slope is 1 meter/2 meters

The required amount at the end of the 7 years is given as $32,340

Given that,
An amount of $250 is contributed at the end of each month, and the balance earns interest at an APR of 6.2%.

Investment is defined as when an individual spends money on something that returns him or her higher money than the money they or invest.

here,
According to the question,
The total amount accrued after 7 years is given as,

[tex]y\:=\:250\cdot12\cdot 7\left[1\:+\:\frac{0.062}{12}\right]^{12\cdot 7}\\y = $32,340[/tex]

Thus, The required amount at the end of the 7 years is given as $32,340.

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A) The lead executives statement has been justified below considering the fall and rise of the graph line.

B) Between the years 7 and 8.

A) Looking at the attached graph, we can trace that the printed ad's revenue peaked at $3 million and then began to fall steadily. The online ad revenue started  with nothing and thereafter gradually increased in value to millions of dollars. Now, since the printed ad revenue has been declining and online ad revenue is seen to be increasing, it means that the lead executive's assertion is accurate.

B) Both curves are seen to intersect between the seventh and eighth year, I would say around the middle of year 7, and both online ads and printed ads generated an approximate revenue of $2.2 million each

Between the years 7 and 8.

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The value of the expression upon evaluation by using the distributive property as required is; 28.

It follows from the task content that the distributive property is to be used to determine the value of the expression given.

Since the given expression is; 8 • 3 ½.

It follows that the expression can be written as follows;

8 ( 3 + ½ )

Hence, by the distributive property, it follows that we have;

( 8 × 3 ) + ( 8 × ½ )

= 24 + 4

= 28.

On this note, the value of the expression is; 28.

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By using various properties of triangles, we get the value of [tex]\angle[/tex]ABD is 47.5°.

What are the properties of a triangle?

A triangle's sides and angles serve as the foundation for all of its characteristics. Triangular closed polygons with three sides and three vertices are what the word "triangle" means. The three internal angles of a triangle add up to 180°.

Let us first name a few angles that we will be using throughout the solution.

Let, [tex]\angle[/tex]DEN=a

[tex]\angle[/tex]ENC=c

[tex]\angle[/tex]MEB=b

[tex]\angle[/tex]EMB=d

Now, since it is given that AD ≅ AE, let us assume that triangle ADE is an isosceles triangle, with AD=AE.

Hence we get

[tex]\angle[/tex]DAE+[tex]\angle[/tex]ADE+[tex]\angle[/tex]AED=180° (angle sum property)

35°+2[tex]\angle[/tex]ADE=180°

[tex]\angle[/tex]ADE=[tex]\angle[/tex]AED=72.5°

Next, we know that

[tex]\angle[/tex]AED+a+[tex]\angle[/tex]NEC=180° (linear pair property of angles on a straight line)

72.5°+a+30°=180°

a=77.5°

Since a=b (opposite angles)
we get, a=b=77.5°

Now in triangle ENC, if we take the sum of angles, we get

[tex]\angle[/tex]NEC+[tex]\angle[/tex]NCE+c=180° (angle sum property)

30°+25°+c=180°

c=125°

We also know that the sum of interior angles made by an intersecting line between 2 parallel lines (given shape is a trapezoid) is 180, hence we get

c+d=180°

d=55°

In triangle MBN, we have

d+b+[tex]\angle[/tex]MBD=180° (angle sum property)

55°+77.5°+[tex]\angle[/tex]MBD=180°

[tex]\angle[/tex]MBD=47.5°

Therefore, we get the value of [tex]\angle[/tex]ABD=47.5°

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Answer:

1100

Step-by-step explanation:

First, we start off by finding 10% which is $100. We divide that by 10 to get $10 which is 10%. Now we need to multiply 10% (which is $10) by 10 which is   $100. That gives you this answer

The total amount in the account on the first day of the 11th year is if Charlie deposits $1000 into his investment account. The account grows at a rate of 8% per year is $1880.

When the loan is given to you, then some amount is charged to you for the principal amount and that is called interest.

Given data:

Charlie deposits, P =  $1000 into his investment account.

The account grows at a rate, r =  8% per year.

The time, t = 11 year

Calculate the simple interest as shown below,

[tex]SI = (P\times r\times t) / 100[/tex]

SI = (1000 × 8 × 11) / 100

SI = $880

Calculate the total amount by the formula given below,

Amount = A

[tex]A = P + SI[/tex]

A = 1000  + 880

A = $1880

Therefore, the total amount in the account on the first day of the 11th year is if Charlie deposits $1000 into his investment account. The account grows at a rate of 8% per year is $1880.

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Using the fact that the angles are supplementary, we can write and solve a linear equation to find that x = 5°.

Remember that two angles are supplementary if the addition of their measures is equal to 180°.

Here we know that angles J and K are supplementary, then we know that:

J + K = 180°

Here we also know that:

J = 120°

K = 12x

So we can write a linear equation of the form:

120° + 12x = 180°

12x = 180° - 120°

12x = 60°

x = 60°/12 = 5°

x = 5°

We conclude that the value of x is 5°.

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Answer:

x ≤ 10

Step-by-step explanation:

3/2 x - 1 ≤ x +4

3/2 x - x ≤ 1 + 4

(3-2)/2 x ≤ 5

1/2 x ≤ 5

2 * 1/2 x ≤  2 * 5

x ≤ 10

Answer:

b) 2/9

c) 1/3

Step-by-step explanation:

The probability of a desired event happening is the number of ways the desired event can happen divided by the total number of possible outcomes.

a) Correct

b)

There are 9 possible outcomes.

2 outcomes are negative.

p(negative) = 2/9

c)

There are 3 outcomes that are more then 3.

p(more than 3) = 3/9 = 1/3

Answer:

49

hope this helps!!

I disagree with the student because y = x/9 represents a proportional relationship

Proportional relationships are relationships between two variables where their ratios are equivalent

A proportional relationship can we written as follows:-

y = mx

Where 'y' is dependent variable, 'x' is independent variable, and 'm' is the slope or we can say 'm' is constant of proportionality.

'm' can take any value whether negative or positive, decimal or whole number. 'm' is a real number.

The given problem is y = x/9 and student says we can only multiply constant, and not divide. We disagree with this comment because m= 1/9 is correct.

So y = 1/9 x makes sense.

Hence, y = x/9 represents a proportional relationship.

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the total area of the two triangles as shwon in the image is 70 in²

Area can be defined as the total portion occuppied by a plane fiqure.

To calculate the total area of the two triangles, we use the formula below.

Formula:

Where:

Substitute these values into equation 1

Similarly,

Where:

Substitute these values into equation 2

Therefore, the total area of the two triangles is 40+30 = 70 in².

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