The Fourier series representation of the output y(t) is;
(a) y(t) = 0.097 cos(3πt - 67)
(b) y(t) = 0.097 sin(3πt - 67) + 0.051 cos(6πt + π/4 -78)
Given;
A causal continuous-time LTI system whose input x(t) and output y(t) are related by the following differential equation;
[tex]\frac{d}{dt}y(t) + 4y(t) = x(t)[/tex]
By taking Laplace's Transform;
sY(s) + 4Y(s) = X(s)
Y(s)/X(s) = H(s) = 1/(s + 4)
(a) When input x(t) = cos3πt
∵ ω = 3π
H(jω) = 1/(jω + 4)
H(jω₁) = 1/[tex]\sqrt{w^2+4^2}[/tex] = 0.097
∠H(jω) = -tan⁻¹(ω/4) = -67° = Ф₁
y(t) = H(jω₁)*cos(3πt + Ф₁)
→ y(t) = 0.097 cos(3πt - 67)
(b) When input x(t) = sin3πt + cos(6πt + π/4)
∵ ω = 6π
H(jω) = 1/(jω + 4)
H(jω₂) = 1/[tex]\sqrt{w^2+4^2}[/tex] = 0.051
∠H(jω) = -tan⁻¹(ω/4) = -78° = Ф₂
y(t) = H(jω₁) * sin(3πt + Ф₁) + H(jω₂) * cos(6πt + π/4 + Ф₂)
→ y(t) = 0.097 sin(3πt - 67) + 0.051 cos(6πt + π/4 -78)
The Fourier series representation of the output y(t) is;
y(t) = 0.097 cos(3πt - 67)
y(t) = 0.097 sin(3πt - 67) + 0.051 cos(6πt + π/4 -78)
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Instructions solve n given the equation -3(2n-3)=25 -8n
Answer:
n=7
Step-by-step explanation:
-6n-9=25-8n
2n=34
n=17
Find the image of 4(3, 7) under a translation along the vector (-4, 2).
A'(7,5)
A'(1,-9)
A'(-1,9)
4'(-7,-5)
The image of A(3, 7) under a translation along the vector (-4, 2) is A'(-1,9).
Given:
A(3,7) under a translation along the vector (-4, 2).
A' = A + (-4,2)
= (3,7) + (-4,2)
= (3+(-4), 7+2)
= (3-4, 9)
= (-1, 9)
Therefore the image of A(3, 7) under a translation along the vector (-4, 2) is A'(-1,9).
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Find the length of the third side. If necessary, round to the nearest tenth.
Answer: 17.9
Step-by-step explanation:
Answer: 17.9
Step-by-step explanation:
a = 11c = 21b = ?[tex]a^{2} +b^{2} =c^{2} \\b^{2}=c^{2}-a^{2} =21^2-11^2=441-121=320\\b=\sqrt{320} =17.8885438[/tex]
Rounded to the nearest tenth would be 17.9.
When written in scientific notation, the number 1,240,000 will have a
Choose... as the exponent on the power of 10.
Using scientific notation, the number 1240000 can be expressed as 1.24×10⁶
Scientific NotationScientific notation is a form of presenting very large numbers or very small numbers in a simpler form. As we know, the whole numbers can be extended till infinity, but we cannot write such huge numbers on a piece of paper. Also, the numbers which are present at the millions place after the decimal needed to be represented in a simpler form. Thus, it is difficult to represent a few numbers in their expanded form. Hence, we use scientific notations.
Converting the number 1,240,000 to scientific notation, we will get
1,240,000 = 1.24×10⁶
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The difference between the digits of a two digit
number is 1. The number itself is 1 more than 5 times
the Sum of Its digit. If the units digit is greater than
the tens digit, find the number.
Answer:
56
Step-by-step explanation:
Number with two digits = 10y + x
x > y
x-y = 1
10y + x = 5(x+y) + 1
x = y + 1
10y + y + 1 = 5(y+1 +y) + 1
11y + 1 = 5(2y+1) + 1
11y + 1 = 10y + 5 + 1
11y - 10y = 5
y = 5
x = 5 + 1 = 6
the united nations stores data about energy production from animal waste and the use around the world. the following data set gives information for converting animal waste into electricity. country or area commodity - transaction year quantity unit united states animal waste - transformation in electricity 2014 171907 terajoules united states animal waste - transformation in electricity 2013 149233 terajoules united states animal waste - transformation in electricity 2012 104847 terajoules united states animal waste - transformation in electricity 2011 141347 terajoules united states animal waste - transformation in electricity 2003 185912 terajoules united states animal waste - transformation in electricity 2002 197120 terajoules united states animal waste - transformation in electricity 2001 104725 terajoules united states animal waste - transformation in electricity 2000 126745 terajoules united states animal waste - transformation in electricity 1999 133206 terajoules united states animal waste - transformation in electricity 1998 224425 terajoules united states animal waste - transformation in electricity 1997 129113 terajoules united states animal waste - transformation in electricity 1996 5274 terajoules united states animal waste - transformation in electricity 1993 21748 terajoules united states animal waste - transformation in electricity 1992 242924 terajoules united states animal waste - transformation in electricity 1991 210448 terajoules united states animal waste - transformation in electricity 1990 215437 terajoules what is the slope of the linear equation of the entire data set? round to nearest hundredth. provide your answer below:
The slope of the linear equation of the entire data set is -720.98
1. Bring data in to excel sheet
2.Select both x and y variable and insert scatter plot chart. Select any data point in the chart and add trend line. In the format trend line option choose "Linear" and also select "Display an equation on chart" & "Display R-squared value on chart"
To find: What is the slope of the linear equation of the entire data set? Round to nearest hundredth.
Regression equation : y = -720.98x + 15,90,277.32
Slope = -720.98 (Round to nearest hundredth.)
Intercept = 15,90,277.32
So, the slope of the linear equation of the entire data set is -720.98
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A radioactive compound with mass 100 grams decays at a rate of 5.8% per hour. Which equation represents how many grams of the compound will remain after 2 hours?
[tex]\qquad \textit{Amount for Exponential Decay} \\\\ A=P(1 - r)^t\qquad \begin{cases} A=\textit{current amount}\\ P=\textit{initial amount}\dotfill &100\\ r=rate\to 5.8\%\to \frac{5.8}{100}\dotfill &0.058\\ t=\textit{hours}\dotfill &2\\ \end{cases} \\\\\\ ~\hfill A=100(1 - 0.058)^{2} ~\hfill[/tex]
The equation represents the grams of the compound will remain after 2 hours will be;
⇒ C = 100 (1 - 0.058)²
What is an expression?
Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
A radioactive compound with mass 100 grams decays at a rate of 5.8% per hour.
And, Time = 4 hours
Now,
Since, We know that;
Amount for Exponential Decay is,
⇒ [tex]C = P (1 - r)^t[/tex]
Where, C is current amount, P is initial amount, r is rate and t is timeeee
Substitute all the values, we get;
The equation represents the grams of the compound will remain after 2 hours will be;
⇒ C = 100 (1 - 0.058)²
Thus, The equation represents the grams of the compound will remain after 2 hours will be;
⇒ C = 100 (1 - 0.058)²
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this was due last week pls help me
Answer:
X=-2
Step-by-step explanation:
5+0.1x=3(1-0.3x)
5+0.1x=3-0.9x
+0.9x +0.9x
5+x=3
-5. -5
x=-2
Hopes this helps
Answer:
the answer is B
Step-by-step explanation:
5+0.1x=3(1-0.3x)=}
5+0.1x=3-0.9x=}
0.1x+0.9x=3-5=}
x= - 2
John walks at a constant rate of 8 miles every 3 hours. What is the rate written as miles per hours
The rate written in miles per hour is 2.67 mph.
What is constant rate?
A rate of change is constant when, at any given point along the function, the ratio of the output to the input remains constant. The slope is another name for the rate of change that is constant. A linear function will change at a constant rate.
Let, John walks at a constant rate of 8 miles every 3 hours.
Therefore, the rate per hour is,
[tex]\frac{8}{3} = 2.6666\\ = 2.67 miles per hour.[/tex]
Therefore, the rate written in miles per hour is 2.67 miles per hour.
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Express this equation
Can someone help me with this?
Using multiplication or division, we can have either of the numbers from the operation.
Multiplication and Division of Numbersmultiplication is the repeated addition of numbers. But the rules for multiplication of integers are different from that of addition. It includes three possibilities. They are:
Multiplication between two positive numbers,Multiplication between two negative numbers; andMultiplication between a positive number and a negative number.The product of two integers with similar sign numbers will always be positive. This means the product of two positive numbers or two negative numbers will always be positive. While the product of a positive number and a negative number (integers with different signs) will always be negative.
In this case, the three numbers are interconnected via arithmetic operation.
135 ÷ 15 = 9
9 × 15 = 135
135 ÷ 9 = 15
So, which ever way we go with, these numbers are connected to each other.
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a rectangular container with a square base and open top must have volume of 16 \text{} cubic meters. if material for the base costs $8 per square meter, and material for the sides costs $2 \text{} per square meter, find the dimensions of the container so that the cost of material to make it will be a minimum. as your answer, please input the value of the minimal cost.
The minimal cost of the box is $96.
Given data;
The volume of the box (v) = x²y
Given that, volume = 16 m³
So, 16 = x²y
y = 16/x² → (I)
Material for the base costs $8 per square meter, and material for the sides costs $2 per square meter.
Total cost (C) = 8 (x²) + 2*(xy)(4)
= 8x² + 8xy
Using (I),
C(x) = 8x² + 8x(16/x²)
C (^) = 8x² + 128/x
Now find critical points from
∴ [tex]\frac{d}{dx} = [8x^2+\frac{128}{x} ]=0[/tex]
x³ = 8
x = 2
So, the minimal cost will be given by x= 2,
Now, y = 16/x²
= 16/4
= 4
Dimension of the box that minimizes the cost is;
height = 4m, width=2m and breadth = 12m
Now,
Minimum cost = 8 (2)² + 128/2 = 32 + 64
= 96
Hence, The minimal cost of the box is $96.
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or
The radius of a semicircle is 2 millimeters. What is the semicircle's area?
The area of the semi circle with radius 2mm is 2[tex]\pi[/tex][tex]mm^{2}[/tex]
What is a semi-circle?A semicircle is formed when a lining passing through the centre touches the two ends on the circle. Thus, by joining two semicircles we get a circular shape.
When a circle is cut into two halves or when the circumference of a circle is divided by 2, we get a semicircular shape.
Area of a semi-circle = [tex]\frac{\pi r^{2} }{2}[/tex]
but radius r = 2mm
substitute r into the area equation, we have
Area of semicircle = [tex]\frac{\pi 2^{2} }{2}[/tex]
Area of semicircle is = 2[tex]\pi[/tex][tex]mm^{2}[/tex]
In conclusion the area of the semicircle is 2[tex]\pi[/tex][tex]mm^{2}[/tex]
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A server at a restaurant is hoping for a 15% tip from a large party. The
party's bill at the end of the night was $312.00. Write a proportion and
describe how you would solve it to calculate the amount of the tip the
server hopes to receive. Be sure to state your final answer as an amount of
money, rounded to the nearest cent.
*Use the "/" on your keyboard as a fraction bar for ratios when writing the
proportion. For example: I can show that one-half and two-fourths are
proportional by typing "1/2 = 2/4" for the proportion.
$57 is the amount of the tip the server hopes to receive.
What is Percentage?percentage, a relative value indicating hundredth parts of any quantity.
Given,
A server at a restaurant is hoping for a 15% tip from a large party.
The party's bill at the end of the night was $312.00.
Now we need to find 15% of $312
15% is converted to decimal by dividing with 100
15/100=0.15
Now multiply 0.15 with 380
0.15×380
57
Hence $57 is the amount of the tip the server hopes to receive.
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Alex invests $300 in a company. He will receive 6% simple interest annually for the next 5 years.
After 5 years, Alex will have earned in interest.
Answer: After 5 Years Alex will receive 90 Dollar.
Step-by-step explanation 5 x 6% of 300 = 30% is 90 Dollar.
2.5 ( x + 8 ) = 3.5x - 4
Answer:
x = 20
Step-by-step explanation:
Expanding the bracket on the right gives us :
2.5x + 20 = 3.5x - 4
Subtract 2.5x from both sides :
20 = x - 4
Add 4 to both sides :
20 = x
x = 20
Hope this helped and have a good day
In ΔHIJ, h = 730 cm, i = 390 cm and ∠J=35°. Find the area of ΔHIJ, to the nearest square centimeter.
The area of the triangle ΔHIJ is 42805 square centimeters.
What is a triangle?A triangle is a three-sided closed-plane figure formed by joining three noncolinear points. Based on the side property triangles are of three types they are Equilateral triangle, Scalene triangle, and Isosceles triangle.
We know the area of a given ΔABC can be obtained by (1/2)×a×c×sin(∅/2)
where sin(Ф/2) is the included angle.
Given, ΔHIJ, h = 730 cm, i = 390 cm and ∠J=35°.
∴ The area of the ΔHIJ is (1/2)×730×390×sin(35/2) square centimeter.
= 142,350×sin(17.5) square centimeter.
= 42805 square centimeters.
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suppose that the ages of medical residents are normally distributed with a mean of 27 years and standard deviation 2 years. what percent of medical residents are less than 28 years old?
The percent of medical residents less than 28 years old is 69.15%.
For a normally distributed set of data, given the mean and standard deviation, the probability can be determined by solving the z-score and using the z-table.
First, solve for the z-score using the formula below.
z-score = (x – μ) / σ
where x = individual data value = 28
μ = mean = 27
σ = standard deviation = 2
z-score = (28 - 27) / 2
z-score = (1) / 2
z-score = 0.5
Find the probability that corresponds to the z-score in the z-table. (see attached images)
at z = 0.5, p = 0.6915
Multiply the probability by 100 to get the percentage.
% = p x 100
% = 0.6915 x 100
% = 69.15
Hence, the percent of medical residents that are less than 28 years old is 69.15%.
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Determine whether AB and CD are parallel, perpendicular, or neither for A (-1,-1), B (1,5), C (1,2), D (5,4). Graph each line to verify your answer.
Answer:
you can do the favour of marking the answer brainlest pls
the quotient of Y and 9 is greater than 4
help please and thankyou 4
The ratios of the right triangle to solve for x is as follows;
sin x = 6 / 10cos x = 5 / 10tan x = 4 / 6How to find the angles of a right triangle?A right angle triangle is a triangle that has one of its angles as 90 degrees.
The sides of a right angle triangle can be named base on the position of the angles.
The sides of a right angle triangles are hypotenuse sides, opposite and adjacent side. The hypotenuse side is the longest side of the right triangle.
Therefore, let's find the ratio that can be used to find the angle x.
Triangle PNM
sin x = opposite / hypotenuse
sin x = 6 / 10
Triangle ABC
cos x = adjacent / hypotenuse
cos x = 5 / 10
Triangle DEF
tan x = opposite / adjacent
tan x = 4 / 6
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11) Chloe sells her "secret blend" of coffee, which sells for $21.84 per pound. To blend she uses "special dark"
which is $67 per pound and "lightly roasted" which is $14 per pound. She needs 100 pounds of "secret blend"
to sell. How much of each type does she need?
Answer:
ya poop in the coffee and then ya see how many pounds there is find the the type urself
Step-by-step explanation:
X=11 find X if y is 2
Answer:
it is 3 0 o 90 not up okbg 56
Which inequality represents all possible values for x in -2x>-48?
A X <-24
BX<24
CX-24
DX> 24
Answer:
Step-by-step explanation:
An inequality which represents all possible values for x in -2x>-48 is: B. x < 24.
What is an inequality?In Mathematics, an inequality can be defined as a mathematical relation that is typically used for comparing two (2) or more numerical data such as integers, and variables in an equation based on any of the following inequality symbols:
Greater than (>).Less than (<).Greater than or equal to (≥).Less than or equal to (≤).Next, we would find the solution and possible values to this given inequality as follows:
-2x > -48
Dividing both sides of the inequality by -2, we have:
-2x/-2 > -48/-2
x < 48/2
x < 24.
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Deepak is a landscaper who charges $30 for each job he does plus an additional $15 for each hour he works. He only accepts jobs if he will earn at least $90 per job. He writes this inequality to determine x, the number of hours he must work during each job in order to accomplish this. 30 + 15 x greater-than-or-equal-to 90 which best describes the restrictions on the jobs deepak will accept?.
Deepak will work for at-least 4 hours in each job
Linear Inequality in one variable:
A linear inequality in one variable that can be written either in the form of,
a+bx<0
a+bx≤0
a+bx>0
a+bx≥0
where,
where a and b are real numbers and a≠0 a ≠ 0
For the given question,
Deepak charges $30 for each job he does plus an additional $15 for each hour he works.
He only accepts jobs if he will earn at least $90 per job.
If,
x is number of hours Deepak works in each job then,
required inequality can be written as,
30 + 15x [tex]\geq[/tex] 90
15x ≥ 90 - 30
15x ≥ 60
x ≥ 60/15
x ≥ 4
thus,
we conclude that Deepak will work for at-least 4 hours in each job.
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Solve the equation for y
2.5y = 5x -5
Answer:
y=2x-2
Step-by-step explanation:
2.5y=5x-5
/2.5. /2.5
y=2x-2
Hopes this helps please mark brainliest
Answer:
y=2x−2
Step-by-step explanation:
The slope-intercept form is
y=mx+b, where m
is the slope and b
is the y-intercept.y =mx+b
Divide each term in 2.5y=5−5 by 2.5
and simplify. y=2x−2
-2 = 3x + 8/2 Solve for x.
Simplify your answer as much as possible.
Answer:
x= -2
Step-by-step explanation:
-2 = 3x + 8/2
-2+2=3x+4+2 (add 2 to both sides)
0=3x+6
-6=3x+6-6
3x=-6
x=-6/3
x= -2
a cylinder is inscribed in a right circular cone of height 2.5 and radius (at the base) equal to 7.5. what are the dimensions of such a cylinder which has maximum volume? radius
At that particular critical point, the second derivative needs to be negative in order to obtain the dimensions of the cylinder with the maximum volume. The values of the function that contain both variables are then substituted to obtain the dimensions.
A right circular cone with a height of 2.5 and a base radius of 7.5 inscribes the given cylinder.
Assume that h is the height and r is the radius of the cylinder. Using triangles that are similar, express h in terms of r as follows:
2.5/7.5 = 2.5-h/r
2.5r = 18.75 x 7.5 x 2.5 h = 2.5 x 1/3r
V= πr²h is the formula for the cylinder's volume.
As a result, only write the volume in terms of r.
V= r2h V= r2(2.5-1/3r)
V= 2.5πr² - 1/3πr³
As a result, the cylinder must have dimensions of r=5 and h=0.83.
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1. a car manufacturer is concerned about poor customer satisfaction at one of its dealerships. the management decides to evaluate the satisfaction surveys of its next 62 customers. the dealership will be fined if the number of customers who report favorably is between 45 and 50. the dealership will be dissolved if fewer than 45 customers report favorably. it is known that 84% of the dealership's customers report favorably on satisfaction surveys. a. what is the probability that the dealership will be fined? b. what is the probability that the dealership will be dissolved?
Probability that the dealership will be fined is 0.157
Probability that the dealership will be dissolved is 0.158
the management decides to evaluate the satisfaction surveys = 62 customers
the sample mean is 0.84
we solve this using sampling distribution of the proportion
standard deviation of the sample is [tex]\sqrt{\frac{PQ}{n} }[/tex] P = 0.84 Q= 0.16 n= 62
[tex]\sqrt{\frac{0.84*0.16}{62} }[/tex] = 0.04
mean(P) = 0.84
standard deviation = 0.04
45/62 = 0.72
50/62 = 0.80
(a) the probability that the dealership will be fined
p(45 < x < 50) = p(x< 50) - p(x<45)
Z-value for 50 is (0.80- 0.84)/0.04 = -1
p(x<50)= 0.1586
Z-value for 45 is (0.72-0.84)/0.04 = -3
p(x<45) = 0.0013
p(45 < x < 50) = 0.1586-0.0013 = 0.1573
(b) the probability that the dealership will be dissolved
p(x<45) = 0.1586
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A rectangular playground is to be fenced off and divided into two parts by a fence parallel to one side of the playground. 1200 feet of fencing is used. Find the dimensions of the playground that will enclose the greatest total area.
The greatest total area, length will be 300 ft. and width will be 200 ft.
Given,
In the question:
A rectangular playground is to be fenced off and divided into two parts by a fence parallel to one side of the playground. 1200 feet of fencing is used.
Now, According to the question:
Lets assume, the length of the playground is l ft. and width is w ft.
Suppose, the playground is divided into two parts by a fence parallel to width. That means the length of the divider fence will be w ft.
So, the total length of the fence needed = (2l+ 3w)ft.
It is given in the question that 1200 feet of fencing is used.
2l + 3w = 1200
2l = 1200 - 3w
l = 600 - [tex]\frac{3}{2}w[/tex] ......(1)
Now, the area of the playground....
A = l × w
A = ( 600 - [tex]\frac{3}{2}w[/tex] ) × w
A = 600w - [tex]\frac{3}{2}w^2[/tex]
Taking derivative on both side in respect of "w" ,
we will get:
dA/dw = 600 - [tex]\frac{3}{2} (2w)[/tex]
dA/dw = 600 - 3w
A will be maximum when dA/dw = 0
600 - 3w = 0
600 = 3w
w = 600/3
w = 200
Now plugging this w = 200 into equation (1)...
l = 600 - [tex]\frac{3}{2}[/tex] × 200
l = 600 - 300
l = 300
Hence, for the greatest total area, length will be 300 ft. and width will be 200 ft.
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