Answer:
the distance between (4,-2) and (0,4) is √(52) or approximately 7.2 units.
Step-by-step explanation:
To find the distance between two points in a Cartesian coordinate plane, we can use the distance formula:
distance = √((x2-x1)² + (y2-y1)²)
In this case, the two points are (4,-2) and (0,4). So,
x1 = 4, y1 = -2 and x2 = 0, y2 = 4
By substituting these values in the distance formula, we get:
distance = √((0-4)² + (4-(-2))²)
distance = √((-4)² + 6²)
distance = √(16 + 36)
distance = √(52)
So the distance between (4,-2) and (0,4) is √(52) or approximately 7.2 units.
let e be the amount in dollars that jacques spends on salad and pizza. if jacques buys s bowls of salad and p slices of pizza, then the total amount of money he spends (e) can be represented by the equation
The total amount of money (e) spent can be represented by the equation E = ($3.00S) + ($5.00P).
The total amount of money spent will be the sum of money sent of salad and pizza.
The total amount of salad = number of bowls of salad × cost of each bowl of salad
Similarly, the total amount of pizza = number of slices of pizza × cost of each slice of pizza
So, the combined equation will be -
Total money spent = number of bowls of salad × cost of each bowl of salad + number of slices of pizza × cost of each slice of pizza
Keep the values in above mentioned equation -
E = (S × 5) + (P × 3)
Performing multiplication of constant and variable on Right Hand Side of the equation
E = 5S + 3P
Thus, third option is correct.
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A student decides to finance a used car over a 5-yr (60-month) period. After making a down
payment of $2000, the remaining cost of the car including tax and interest is $14,820. The
amount owed y = A(t) (in $) is given by A(t) = 14,820-247t, where t is the number of
months after purchase and 0 ≤t≤ 60. Determine the t-intercept and y-intercept and interpret
their meanings in context.
Answer:The t-intercept of a function is the point at which the function crosses the t-axis. To find the t-intercept of the function A(t) = 14,820 - 247t, we need to find the value of t when A(t) = 0. Setting A(t) = 0 and solving for t, we get:
14,820 - 247t = 0
t = (14820/247)
The t-intercept of the function is 60 months, which means that after 60 months (or 5 years) the amount owed on the car will be $0.
The y-interval of a function is the point at which the function crosses the y-axis. To find the y-intercept of the function A(t) = 14,820 - 247t, we need to find the value of A(t) when t = 0.
At t = 0, A(t) = 14,820 - 247(0) = 14,820.
The y-intercept of the function is $14,820 which means that before making the first payment, the cost of the car including tax and interest is $14,820.
Step-by-step explanation:
If a coin is flipped 60 times and a head comes up 42 times, what is the
relative frequency of a head coming up?
A. 0.60
B. 0.55
C. 0.65
D. 0.70
Answer: D. 0.70
Step-by-step explanation:
42/60= 0.7
If You write all numbers from 30 to 65 including 30 and 65 how many numbers did you write
Numbers written from 30 to 65 including 30 and 65 = 36 numbers
What is number?A number is a basic component of mathematics. Numbers are used for counting, measuring, keeping things in order, indexing, etc. We have different types of numbers based on their properties such as natural numbers, whole numbers, rational and irrational numbers, integers, real numbers, complex numbers, even and odd numbers, etc.
Given,
Numbers 30 to 65
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65
Total count of numbers 30 to 65 = 36
Hence, 36 numbers are written from 30 to 65 including 30 and 65.
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Please help me please
Taking w as the unknown number, the equation for the given statement is 2(w-8) = 5.
To solve this problem, we need to convert the given statement in mathematical form.
We assume that "w" is the unknown number in the equation.
First, we have to take a closer look at the given statement. It was stated that "twice the different of... is equal to 5". It shows that the result of equation 5 is the double value of another equation. We assume the unknown equation as y. Then:
2y = 5
Next, the rest of the statement stated that "a number and 8". From this part of statement, we know that the unknown equation "y" we assume above should be replace with "w-8". Hence:
2(w - 8) = 5
If we want to solve this question, we can solve it by dividing both sides with 2:
2(w - 8) = 5
------------------- : 2
w - 8 = 5/2
w = 5/2+ 8
w = 10.5
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What is the Probability.
The value of the conditional Probability is P(A|B) is 1/3.
What is probability ?The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a statement is true. An event's probability is a number between 0 and 1, where, broadly speaking, 0 denotes the event's impossibility and 1 denotes certainty.
The main distinction between a probability and a conditional probability is that a probability is the likelihood of an event, such as A, occurring, whereas a conditional probability assumes that another event, such as B, has already happened in order to define the probability of an event, such as in the conditional probability of A given B.
Conditional probability is the likelihood that any event A will occur in the presence of another event B that is related to A. P(A|B) illustrates it.
By, Conditional Probability :
P (A∩B) =P (A∩B') - P(B)= 4/9 - 1/3 = 1/9
P(A|B) = P (A∩B)/ P(B)
= (1/9)/(1/3) = 1/3
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Drag the tiles to the correct boxes to complete the pairs.
Match each data set with the value that best describes its center.
6
2
4
9
Data set A: 2, 3, 1, 4, 2, 0, 0
arrowRight
Data set B: 6, 4, 5, 6, 5, 7, 8, 7
arrowRight
Data set C: 1, 4, 0, 2, 4, 4, 6, 7
arrowRight
Data set D: 7, 8, 10, 11, 8, 9, 10, 9, 11
arrowRight
The measure of center that best describes each data-set is given as follows:
Data set A: 2.Data set B: 6.Data set C: 4.Data set D: 9.How to obtain the median of a data-set?The median of a data-set is the middle value of a data-set, the value of which 50% of the measures are less than and 50% of the measures are greater than.
The median is usually considered the best measure of center of a data-set, as it is not affected by outliers.
Each of the data-sets in this problem have 7 elements, which is an even cardinality, hence the median is given by the fourth element of the ordered data-set.
The ordered data-set A is given as follows:
0, 0, 1, 2, 2, 3, 4.
The fourth element is of 2, which is the median, and the same procedure is applied for all the other data-sets.
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Find the least common denominator of 1/2x-6 and 2x/7x-21.
The least common denominator of the fractions 1/2x-6 and 2x/7x-21 is (2x-6)(7x-21)
What is a fraction?A fraction can be described as the part of a whole variable or number.
Some types of fractions in mathematics are;
Proper fractionsImproper fractionsMixed fractionsComplex fractionsSimple fractionsWhat is the least common denominator?The least common denominator of a fraction can be defined as the smallest number of all the common multiples of the denominators given.
From the information given, we have the fractions as;
1/2x-6 and 2x/7x-21
Note that the denominator of both fractions are
(2x-6)(7x-21)
Since they are in form of expression, the least common denominator is their product.
Hence, the LCD is (2x-6)(7x-21)
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what is the mid point of (3.5,2,2) (1.5,-4.8)
The above equation's midpoint is represented by the expression Midpoint = (3.5,2,2) (1.5,-4.8). ( 2.5,-1.3 )
What do you meant by mid point ?Midpoint =( 2.5,-1.3 )
The formula for determining the midpoint is
[tex]$\left(x_m, y_m\right)=\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$[/tex]
[tex]$\left(x_m, y_m\right)=$[/tex]the midpoint's coordinates
[tex]$\left(x_1, y_1\right)=$[/tex] the first point's coordinates
[tex]$\left(x_2, y_2\right)=$[/tex] the second point's coordinates
Divide the distance between the two endpoints by 2, then multiply it by 3. The middle of that line is at these distances from each end. As an alternative, multiply the sum of the two endpoints' x coordinates by 2. For the y coordinates, repeat the process. the location on a line when the distances to both ends are equal. a period of time midway between the start and finish of an event.
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2+2x[2+2x(2+2:2)]:8=
=
Answer:
2+2×2+2×2+2:2
Step-by-step explanation:
well the answer would be 2×2×2+2+2+2
4×2+2+2
8+2+2
10+2
12:2=8
that's the best that I could solve
please i need help with steps
The total distance ran by the wide receiver is given as follows:
195.6 yd.
How to obtain the distance between two points?Suppose that we have two points with coordinates given as follows:
[tex](x_1, y_1)[/tex] and [tex](x_2, y_2)[/tex]
Then the formula for the distance between these two points is given as follows:
[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
For the out route, the distance is given as follows:
[tex]D = \sqrt{(60 - 30)^2+(50 - 20)^2} = 42.43[/tex]
To the end zone, the distance is given as follows:
[tex]D = \sqrt{(60 - 80)^2+(50 + 30)^2} = 82.46[/tex]
From the end zone back to the huddle, the distance is given as follows:
[tex]D = \sqrt{(80 - 30)^2+(-30 - 20)^2} = 70.71[/tex]
Hence the total distance is of:
42.43 + 82.46 + 70.71 = 195.6 yd.
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Robert enters data for weight (in pounds) and calories burned per minute into a statistics software package and finds a regression equation of ŷ = 2.2 + 0.05x, where weight is the explanatory variable. Based on this information, select the conclusion about weight and calories burned per minute that is TRUE.
Based on this information, the conclusion about weight and calories burned per minute that is true is: D. For each additional pound of weight, calories burned per minute increases by 0.05 calories.
What is the slope-intercept form?In Mathematics, the slope-intercept form of a line can be represented or modeled by using this linear equation:
y = mx + c
Where:
m represents the slope.c represents the y-intercept.x and y are the data points.Based on the information provided about the data for weight (in pounds) and calories burned per minute, a regression equation that models the situation is given by:
ŷ = 2.2 + 0.05x
Where:
ŷ is the calories.x is the pounds of weight.In conclusion, we can logically deduce that the amount of calories burned per minute increases by 0.05 calories for each additional pounds of weight.
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Complete Question:
a.) For each additional pounds of weight, calories burned per minute increases by 2.2 calories.
b.) For each additional pounds of weight, calories burned per minute stays relatively the same.
c.) For each additional pounds of weight, calories burned per minute decreases by 0.05 calories.
d.) For each additional pounds of weight, calories burned per minute increases by 0.05 calories.
Majka Company was started on January 1, Year 1. During Year 1, the company experienced the following accounting events: (1) Issued 52,500 worth of common stock (2) earned cash revenues of $33,400, (3) paid cash expenses of $14,800, and (4) paid a $3,100 cash dividend to its stockholders. These were the only events that affected the company during Year 1.
The cash from financing activities for Majka Company, can be found to be
How to find the cash from financing activities ?The net amount of financing a business generates during a specific time period is called cash flow from financing activities. The issuing and repayment of equities, the payment of dividends, the issuance and repayment of debt, and capital lease obligations are all examples of financial activity.
The cash from Financing activities for Majka Company is therefore:
= Common stock issued - Cash dividends
= 52, 500 - 3, 100
= $ 49, 400
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The full question is:
Majka Company was started on January 1, Year 1. During Year 1, the company experienced the following accounting events: (1) Issued 52,500 worth of common stock (2) earned cash revenues of $33,400, (3) paid cash expenses of $14,800, and (4) paid a $3,100 cash dividend to its stockholders. These were the only events that affected the company during Year 1.
What were the cash flows from financing for the year?
Mariah made a cylinder out of clay that had a radius of 6 cm and a volume of 72π cm^3. She painted the entire surface of the cylinder purple. How many square centimeters of the cylinder
did Mariah paint in terms of π?
Enter the correct answer in the box in terms of π.
answer: _______ cm squared
The number of square centimeters of the cylinder that Mariah painted is given as follows:
36π cm².
How to obtain the volume and the surface area of a cylinder?The volume of a cylinder of radius r and height h is given by the equation presented as follows:
V = πr²h.
Mariah made a cylinder out of clay that had a radius of 6 cm and a volume of 72π cm^3, hence the height of the cylinder is obtained as follows:
36πh = 72π
h = 2.
The surface area of a cylinder is obtained as follows:
S = 2π r h + 2π r
Hence it's value is of:
S = 2π x (2 x 6 + 6)
S = 36π cm².
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One measure of the accuracy of a forecasting model is the:
a. trend component
b. mean absolute deviation
c. seasonal index
d. smoothing constant
Mean Absolute Deviation (MAD) is a measure of accuracy for a forecasting model.
It measures the average distance between the actual value and the predicted value, and can be calculated using the following formula: MAD = 1/n * Σ |Ai-F i |, where n is the number of data points, Ai is the actual value, and Fi is the forecasted value. MAD is often used to compare the accuracy of different forecasting models. A lower MAD value indicates that the model has a better accuracy in predicting future values. Additionally, MAD can also be used to measure the accuracy of a single forecasting model over time. If MAD increases, then the accuracy of the forecasting model is decreasing.
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when the fraction 1/70000000 is written as a decimal, which digit occurs i the 2023 place after the decimal point
The decimal representation of 1/70000000 is 0.0000000142857142857....
In math, what is a fraction?
An element of a whole is a fraction. The number is represented mathematically as a quotient, where the numerator and denominator are split. Both are integers in a simple fraction. A fraction appears in the numerator or denominator of a complex fraction. The numerator of a proper fraction is less than the denominator.
To find the digit that occurs in the 2023rd place after the decimal point, we can use long division to divide 1 by 70000000 and track the remainders.
Since the decimal representation of 1/70000000 is repeating every 6 digits, we can find the digit at the 2023rd place by taking the remainder when 2023 is divided by 6.
2023 % 6 = 1
So the digit in the 2023 place is the first digit after the repeating decimal, which is 4.
Therefore the digit in the 2023rd place after the decimal point is 4.
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-25 POINTS-
What is the equation?
[?]n+[?]
An equation is a mathematical statement that proves two mathematical expressions are equal in algebra, and this is how it is most commonly used. Answer is y = -2x + 9.
What is the equation?An equation is a mathematical statement that proves two mathematical expressions are equal in algebra, and this is how it is most commonly used.
Fundamental Equations and Ideas:
Operational hierarchy: BPEMDAS
Form of the Slope-Intercept: y = mx + b
meter slope
y-intercept of b
Define variables first
Degree (Slope m) = -2
Random thought (1, 7)
Step 2: Locate b's y-intercept
Equation to be written down is y = -2x + b.
Substitute: 7 = -2(1) + b
Add the following factors together to multiply: 7 = -2 + b
2 more on either side:
9 = b
Write a linear equation in step 3
Replace with our freshly discovered b.
y = -2x + 9
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Prove divisibility
thanks!
The prove that the given expression is divisible by 43 is explained below.
Divisibility of numbers.A number is said to be divisible by a given number when it leaves no remainder after the division. Example; show that 7^7 is divisible by 49.
7^7 = 823543
So that;
823543/ 49 = 16807
To prove that the expression is divisible by 43, we have;
7^8 = 5764801
7^7 = 823543
7^6 = 117649
Then,
7^8 - 7^7 + 7^6 = 5764801 - 823543 + 117649
= 4941258 + 117649
= 5058907
Then,
5058907/ 43 = 117649
Thus the expression is divisible by 43.
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Each marble bag sold by Lena's Marble Company contains 5 red marbles for every 8 blue marbles. If a bag has 40 blue marbles, how many red marbles does it contain?
Answer:25
Step-by-step explanation:
40 multiple by 5 over 8 cross multiplication
I need help Asap!!!!!!!!!
Solve: 9log9(4)=
Answer: 34.3527303398
Step-by-step explanation: hope it is right
Fill in the P(X=x) values in the table below give a legitimate probability distribution for the discrete random variable X whose possible values are 0, 3, 4, 5 and 6.
The legitimate probability distribution for the discrete random variable X whose possible values are 0, 3, 4, 5 and 6 is: P(X=0) = 0.2, P(X=3) = 0.2, P(X=4) = 0.3, P(X=5) = 0.2, and P(X=6) = 0.1.
For a probability distribution to be legitimate, the probabilities of all the possible values of a random variable need to add up to 1. So, in this case, the sum of all the probabilities must be equal to 1.
For the given values, we can assign the probability of 0.2 to the values 0 and 3, 0.3 to the value 4 and 0.1 to the value 6, so that the sum of all the probabilities is 1.
Therefore, the legitimate probability distribution for the given discrete random variable X is as follows:
| Value (x) | P(X=x) |
|---|---|
| 0 | 0.2 |
| 3 | 0.2 |
| 4 | 0.3 |
| 5 | 0.2 |
| 6 | 0.1 |
The legitimate probability distribution for the discrete random variable X whose possible values are 0, 3, 4, 5 and 6 is: P(X=0) = 0.2, P(X=3) = 0.2, P(X=4) = 0.3, P(X=5) = 0.2, and P(X=6) = 0.1.
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Find the average/mean of following numbers:
6,11,7,13,3
Answer:
8
Step-by-step explanation:
[tex]mean = \frac{sum \: of \: numbers}{how \: many \: numbers \: there \: are} [/tex]
[tex]mean = \frac{6 + 11 + 7 + 13 + 3}{5} [/tex]
[tex] mean = \frac{40}{5} = 8[/tex]
A bookstore had 84 copies of a magazine. Yesterday it sold 1/6 of them today it sold 4/7 of what remained. How many copies does the bookstore have left? Answer?
The bookstore has 30 copies of books left after they sold (4/7)th today.
What is a fraction?A fraction is written in the form of p/q, where q ≠ 0.
Fractions are of two types they are proper fractions in which the numerator is smaller than the denominator and improper fractions where the numerator is greater than the denominator.
Given, A bookstore had 84 copies of a magazine.
Yesterday it sold 1/6 of them which is, (84 - (1/6)×84).
= 84 - 14 books remaining.
= 70 books remaining.
Today it sold 4/7 of what remaining which is,
= (70 - (4/7)×70).
= 70 - 40.
= 30 books.
So, Now the bookstore has 30 books left.
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Pls answer ASAP.
Select all the true statements.
24 ÷ 8 > 24 ÷ 6
12 ÷ 12 = 7 ÷ 7
6 ÷ 1 < 9 ÷ 1
12 ÷ 3 > 12 ÷ 6
10 ÷ 5 < 10 ÷ 1
The true statements are;
a) 12 ÷ 12 = 7 ÷ 7 (both side are equal to 1)
b) 6 ÷ 1 < 9 ÷ 1 ( The left quotient is less than the right)
c) 12 ÷ 3 > 12 ÷ 6 (The left quotient is greater than the right)
d) 10 ÷ 5 < 10 ÷ 1 ( The left quotient is less than the right)
What are the true statements?We have to note that the true statements have to be the statements that satisfy the in equality that have been written in the question. We would have to look at the inequality in the question and then decide whether or not it is true.
Let us note that the in equality must take into account the expressions that are on both sides and then know if it holds true under all the circumstances that are possible mathematically.
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The logarithmic functions, f(x) and g(x), are shown on the graph.
What is the equation that represents g(x)? Explain your reasoning.
The image of the function after the translation is g(x) = log(x + 1) + 4
What is graph?In mathematics, the graph of a function f is the set of ordered pairs, where {\displaystyle f(x)=y.} In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in two-dimensional space and thus form a subset of this plane.
here, we have,
f(x)=logx
How to determine the equation of g(x)?
From the graph, the given parameters are:
f(x) = log(x)
From the graph, we can see that:
The graph of f(x) is translated left by 1 unit
The graph of f(x) is translated up by 4 units
This transformation is given as
g(x) = f(x + 1) + 4
Mathematically, this transformation can be represented as
(x, y) = (x + 1, y + 4)
When represented as a function, we have
g(x) = f(x + 1) + 4
Substitute the equation f(x) = log(x)
f(x + 1) = log(x + 1) + 4
So, we have the following equation
g(x) = log(x + 1) + 4
Hence, the equation of g(x) is g(x) = log(x + 1) + 4
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two triangles triangle abc and triangle abd sit inside a circle with a diameter, represented by the line ab, equal to 8. which of the following statements regarding angle 1 (angle acb) and angle 2 (angle abd) is correct?
A line segment is a piece of a line that can connect two places. Hence ar (ABC) = ar (ABD)
What is line-segment?A line segment is a piece of a line that can connect two places. The following diagrams can help us grasp the line segment: A line like this! It stretches eternally in both directions and has no ends. It becomes a line segment when you mark the two points A and B on it and choose this segment independently. It is described as a long, continuous, straight line that is illustrated by arrowheads pointing in both directions. Both directions are covered by its reach. Line Segment: A line segment is a section of a straight line that runs between two locations.Given
[tex]$\triangle \mathrm{ABC}$[/tex] and [tex]$\triangle \mathrm{ABD}$[/tex] are two triangles on the same base [tex]$\mathrm{AB}$[/tex].
To show :
ar(ABC)=ar(ABD)
Proof :
Since the line segment [tex]$\mathrm{CD}$[/tex] is bisected by [tex]$\mathrm{AB}$[/tex] at [tex]$\mathrm{O}[/tex] . [tex]\mathrm{OC}=\mathrm{OD}$[/tex].
In [tex]$\triangle \mathrm{ACD}$[/tex], We have [tex]$\mathrm{OC}=\mathrm{OD}$[/tex].
So, [tex]$\mathrm{AO}$[/tex] is the median of[tex]$\triangle \mathrm{ACD}$[/tex]
Also we know that median divides a triangle into two triangles of equal areas.
∴ar(ΔAOC)=ar(ΔAOD) _______ (1)
Similarly , In[tex]$\triangle \mathrm{BCD}$[/tex],
[tex]$\mathrm{BO}$[/tex]is the median. ([tex]$\mathrm{CD}$[/tex] bisected by [tex]$\mathrm{AB}$[/tex] at [tex]$\mathrm{O}$[/tex])
∴ar(ΔBOC)=ar(ΔBOD) _______ (2)
On adding equation (1) and (2) we get,
ar(ΔAOC)+ar(ΔBOC)=ar(ΔAOD)+ar(ΔBOD)
∴ar(ΔABC)=ar(ΔABD)
The complete question is,
In figure, ABC and ABD are two triangles on the same base AB. If line-segment CD is bisected by AB at O, show that ar (ABC) = ar (ABD).
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Write the expression in exponential form. 9×9×9×9x9x9x9=|
its 9 of the 7th power
its also 9^7
Answer:
(9)^7
Step-by-step explanation:
All you have to do is count how many 9's you have, and the number you get will be the (power)
Put (9) as the base and then add the power
Which gives us (9)^7
[tex] {9}^{7} [/tex]
If ‘a’ varies jointly with ‘b’ and ‘c’, and inversely as the square of ‘d’, how would ‘a’ be affected if ‘b’ is tripled and both ‘c’ and ‘d’ are doubled.
When two or more variables vary jointly, their product is always constant. In this case, if 'a' varies jointly with 'b' and 'c' it means that their product is always constant, so abc = k, where k is a constant value.
When a variable 'a' varies inversely as the square of another variable 'd', it means that a*1/d^2 = k, where k is a constant value.
So, if 'b' is tripled, 'c' is doubled and 'd' is doubled, we can see the effect on 'a' by substituting the new values into the equations.
abc = k => a3b2c = k
a1/d^2 = k => a1/(2d)^2 = k
So the effect on 'a' if 'b' is tripled, 'c' is doubled, and 'd' is doubled is that it will be divided by 4.
a3b2c = a1/(2d)^2 => a = k / (3b2c*(2d)^2) = (k/(12bcd^2))
So a = k/(12bcd^2) = a/4.
Therefore, the value of 'a' is decreased by a factor of 4.
The plane with normal vector <-6, 3, -3> containing the point (7, -4, 5) has equation Ax + By + Cz = D. If A= - 6 find what B, C, and D equals.
Step-by-step explanation:
for a plane defined as
Ax + By + Cz + D = 0
(A, B, C) is the normal vector (90° with the plane).
so, we know with the given normal vector (-6, 3, -3) that
A = -6
B = 3
C = -3
we are only missing D.
for that we use the given point (7, -4, 5) :
-6×7 + 3×-4 + -3×5 = D
-42 - 12 - 15 = D
-69 = D
Consider the system y' + 4y = f(t), where f(t) = 4e^-t a. Solve the ODE with y(0) = 0 using the technique of integrating factors: (Do not use Laplace transforms ) y(t) = ...?b. Find the transfer function of the system: H(s) = ...?c. Find the impulse response of the system: h(t) = L^-1 [H](t) d. Evaluate the convolution integral (h*f)(t) , and compare the resulting function with the solution obtained in part (a): (h*f)(t) = ∫ dw =
The result of the convolution integral is the same as the solution obtained in part (a).
a. Solving the ODE with y(0) = 0 using integrating factors,
multiplying both sides of the equation by e^4t, we get
e^4t y' + 4e^4t y = 4e^-te^4t
Integrating both sides,
∫ (e^4t y' + 4e^4t y) dt = ∫ 4e^-te^4t dt
Therefore,
e^4t y = ∫ 4e^-te^4t dt + C
Since y(0) = 0, C = 0,
y = 1/4 ∫ 4e^-te^4t dt
Using integration by parts,
y = 1/4 [te^-t + 4/3 e^-t]
Therefore,
y(t) = 1/4 [te^-t + 4/3 e^-t]
b. The transfer function of the system is given by
H(s) = Y(s)/F(s) = 1/4s/(s+4)
c. The impulse response of the system is given by
h(t) = L^-1[H(s)] = 1/4e^-t
d. The convolution integral (h*f)(t) can be evaluated as follows,
(h*f)(t) = ∫ h(t-w)f(w)dw
= ∫ 1/4e^-(t-w) 4e^-w dw
= 1/4 ∫ 4e^-(t+w) dw
= 1/4 [te^-t + 4/3 e^-t]
The result of the convolution integral is the same as the solution obtained in part (a).
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