Use the Laplace transform to solve the given initial value problem. y' + 6y = e^4t ; y(0)=2 ...?
The Laplace transform method is applied to solve the differential equation y' + 6y = e^4t with the initial condition y(0)=2. After transforming, simplifying, and solving for Y(s), we use inverse Laplace transform to find the solution y(t) in the time domain.
Explanation:Laplace transform is a powerful tool in the field of mathematics used for solving differential equations. To solve the given initial value problem y' + 6y = e^4t ; y(0)=2, we can start by taking the Laplace transform of both sides of the equation.
The Laplace transform of y' is sY(s) - y(0) and the Laplace transform of y is Y(s). Therefore, the Laplace transform of y' + 6y gives sY(s) - y(0) + 6Y(s). Given that y(0)=2, this simplifies to sY(s) + 6Y(s) - 2.
On the right-hand side, the Laplace transform of e^4t is 1/(s-4). Thus, we have the equation sY(s) + 6Y(s) - 2 = 1/(s-4).
By solving for Y(s), we can find the inverse Laplace transform to get the solution y(t) in the time domain.
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Write9^3 *p without exponents
Which values of m and b will create a system of equations with no solution? Check all that apply.
y = mx + b
y = –2x + 3/2
[ ] m = –3 and b = -2/3
[ ] m = –2 and b = -1/3
[ ] m = 2 and b = -2/3
[ ] m = –3/2 and b = -2/3
[ ] m = –2 and b = -2/3
[ ] m = 3 and b = -2/3
Answer: Answer is second and fifth option.
Step-by-step explanation:
m = –2 and b =
m = –2 and b =
The height of the Empire State Building is 318.00 meters. If a stone is dropped from the top of the building, what is the stone's velocity just before it strikes the ground?
Find the lim [(ln(x+5) - ln5) / x ] as x approaches 0
...?
To find the limit of the given expression [(ln(x+5) - ln5) / x ] as x approaches 0, we can apply L'Hospital's Rule.
Explanation:To find the limit of the given expression, we can use L'Hospital's Rule. Let's first rewrite the expression:
[ln(x+5) - ln(5)] / x
As x approaches 0, when plugging in 0, we get an indeterminate form of 0/0. So, we can apply L'Hospital's Rule:
lim [(ln(x+5) - ln5) / x ] as x approaches 0 = lim [(1/(x+5)) / 1]
= lim (1 / (x+5)) as x approaches 0
Since x approaches 0, the expression approaches 1 / (0+5) = 1/5.
Which of the following is not a combination?
The number of ways a President and Vice President can be chosen from a group of 10 people.
The number of 2 person subcommittees that can be chosen from a 10 person committee.
The number of 2 person teams that can be picked form 10 people.
When there is the no of ways where the president and the vice president should be selected from the grouping of 10 people does not represent the combination.
The following information should be considered:
In the case when the people are picked out for particular role so here the order should be matter. It should be the situation of permutation, not the combination situation. Based on this, the last 2 options are considered to be the combinationTherefore we can conclude that when there is the no of ways where the president and the vice president should be selected from the grouping of 10 people does not represent the combination.
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What is the middle term of the product of (5a - 7)(2a - 1)?
-4a
-19a
-24a
The sum of three consecutive even integers is 66. Which equation represents the scenario?
The correct option is:
d. x + (x+2) + (x+4) = 66
The scenario involves three consecutive even integers. Let's denote the first even integer as x, the second one as x+2 (since it is the next consecutive even integer), and the third one as x+4 (as it is two more than the second one).
The sum of these three consecutive even integers is given to be 66. Therefore, the equation that represents this scenario is:
x + (x+2) + (x+4) = 66
So, the correct option is:
d. x + (x+2) + (x+4) = 66
The probable question may be:
The sum of three consecutive even integers is 66. Which equation represents the scenario?
a. x=66
b. x+(x+1)+(x+2)=66
c. x+x+x=66
d. x+(x+2)+(x+4)=66
The amount of time Ricardo spends brushing his teeth follows a Normal Distribution with unknown mean and standard deviation. Ricardo spends less than one minute brushing his teeth about 40% of the time. He spends more than two minutes brushing his teeth 2% of the time. Use this information to determine the mean and standard deviation of this distribution.
The graph of the function f(x) = (x – 4)(x + 1) is shown below. Which statement about the function is true? The function is increasing for all real values of x where x < 0. The function is increasing for all real values of x where x < –1 and where x > 4. The function is decreasing for all real values of x where –1 < x < 4. The function is decreasing for all real values of x where x < 1.5.
we have
[tex]f(x)=(x-4)(x+1)[/tex]
using a graph tool
see the attached figure
we know that
The function is increasing for all real values of x where [tex]x > 1.5[/tex]
and
The function is decreasing for all real values of x where [tex]x < 1.5[/tex]
therefore
Statements
case a) The function is increasing for all real values of x where x < 0
The statement is false
case b) The function is increasing for all real values of x where x < –1 and where x > 4
The statement is false
case c) The function is decreasing for all real values of x where –1 < x < 4
The statement is false
case d) The function is decreasing for all real values of x where x < 1.5
The statement is true
Answer:
The correct option is:
The function is decreasing for all real values of x where x < 1.5.
Step-by-step explanation:
We are given a function f(x) which is a product of two linear polynomial as:
[tex]f(x)=(x-4)(x+1)[/tex]
Now, we know that on multiplying the two linear polynomial we will obtain a quadratic polynomial.
So, the function f(x) will be represented as:
[tex]f(x)=x(x+1)-4(x+1)\\\\f(x)=x^2+x-4x-4\\\\f(x)=x^2-3x-4[/tex]
So, we will plot the graph of the function and check which statements about the function hold true.
1)
The function is increasing for all real values of x where x < 0.
This statement is false.
Since we get from the graph that function f(x) is decreasing for x<0.
2)
The function is increasing for all real values of x where x < –1 and where x > 4.
This option is incorrect as the function is decreasing for x<-1
whereas it is increasing for x>4.
3)
The function is decreasing for all real values of x where –1 < x < 4.
This option is incorrect.
Since, the function is both decreasing as well as increasing in the interval (-1,4).
4)
The function is decreasing for all real values of x where x < 1.5.
This option is correct.
Since it could be observed from the graph that the function is decreasing for x<1.5.
wholesale price $3.50 per item and is sold at a retail value of $5.00 what is the gross percentage of the item?
A triangle with a base of 9 units and a height of 22 units ?
If f'(x)=g'(x) for all x in an interval I, what is the relationship between f and g ...?
lim cos(pi+h)+1 / h
h->0
evaluate the limit
60% of what number is 13.5?
The required, 60 percent of 22.5 is equal to 13.5, as of the given condition.
What is percentage?The percentage is a way of expressing a fraction or a decimal number as a portion of 100. It is denoted by the symbol '%'. For example, the percentage of 0.75 can be calculated as 0.75 x 100% = 75%. In other words, we are expressing 0.75 as a fraction of 100.
Here,
To find out what number 60% of is equal to 13.5, we can use algebra. Let x be the number we are looking for. Then we can write:
60% of x = 13.5
We can convert 60% to a decimal by dividing by 100:
0.6x = 13.5
Finally, we can solve for x by dividing both sides of the equation by 0.6:
x = 13.5 ÷ 0.6
x = 22.5
Therefore, 60% of 22.5 is equal to 13.5.
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The matrix C=[1,-2_-3,7] was used to encode a phrase to [7,-28,-25,-35,-2_-21,107,90,123,17]. Find C^-1 and use it to decode the matrix.
The total distance an object travels varies directly with the length of time it travels. If an object travels 32 meters in 18 seconds, how long will it take to travel 80 meters? A. 40 seconds B. 42 seconds C. 45 seconds D. 48 seconds
Step-by-step explanation:
It is given that, the total distance an object travels varies directly with the length of time it travels.
[tex]d\propto t[/tex]
If an object travels 32 meters in 18 seconds so,
[tex]32= 18k[/tex].....(1) (k is a constant)
If distance travelled is 80 meters, then the time taken is t say.
[tex]80= kt[/tex].....(2)
On solving equation (1) and (2), we get the value of t as,
t = 45 seconds
Hence, this is the required solution.
Which of the following is the solution set of -2|x| < -8
{x | -4 > x > 4} {x | -4 < x < 4} {x | x < -4 or x > 4}
What is 8.24 written as a percent?
Fernando purchased 1 pound of dried cranberries and 4 pounds of almonds to make a trail mix. Cranberries cost $4 per pound and almonds cost $5 per pound. What is the price per pound of the trail mix?
Select the graph of the solution. Click until the correct graph appears.
x ≥ 4
Answer:
The answer is the option B
see the attached figure
Step-by-step explanation:
we have
[tex]x\geq4[/tex]
The solution is the interval ------> [4,∞)
All real numbers greater than or equal to [tex]4[/tex]
therefore
the answer in the attached figure
How many meters are equivalent to 375 centimeters
Select Is a Function or Is not a Function to correctly classify each relation.
Title
Is a Function
Is not a Function
{(3, 7),(3, 6),(5, 4),(4, 7)}
{(1, 5),(3, 5),(4, 6),(6, 4)}
{(2, 3),(4, 2),(4, 6),(5, 8
{(0, 4),(3, 2),(4, 2),(6, 5)}
Answer:
{(3, 7),(3, 6),(5, 4),(4, 7)} → is not a Function{(1, 5),(3, 5),(4, 6),(6, 4)} → is a Function{(2, 3),(4, 2),(4, 6),(5, 8)} → is not a Function{(0, 4),(3, 2),(4, 2),(6, 5)} → is a FunctionStep-by-step explanation:
A relation from a set X to a set Y is called a function, if each element of X is related to one and exactly one element in Y. Though many elements of X can be related to one element in Y (many to one).
{(3, 7),(3, 6),(5, 4),(4, 7)} → is not a Function (as (3, 7), (3, 6) share same x)
{(1, 5),(3, 5),(4, 6),(6, 4)} → is a Function
{(2, 3),(4, 2),(4, 6),(5, 8)} → is not a Function (as (4, 2), (4, 6) share same x)
{(0, 4),(3, 2),(4, 2),(6, 5)} → is a Function
Answer: ^^^ that person was correct
Step-by-step explanation:
DEFG Is a rectangle. DF=5x-5 and EG=x+11. find the value of x and the length of each diagonal. (Draw and label the sketch below) ...?
What is the solution set of the following equation?
4/5x^2=2x-4/5
1. {1, 2}
2. {1, 1/2}
3. {2, 1\2}
Answer:
{2, 1\2}
Explanation:
What is the slope of the line? y = 3x + 9
A. 9
B. 3
C. 12
D. 6
If 12 is added to the square of a positive integer, the result is 133. Find the positive integer.
The value of the positive integer is,
⇒ 11
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.
We have to given that;
Here, 12 is added to the square of a positive integer, the result is 133.
Let a positive integer = x
Hence, We can formulate;
⇒ 12 + x² = 133
⇒ x² = 133 - 12
⇒ x² = 121
⇒ x = √121
⇒ x = 11
Thus, The value of the positive integer is,
⇒ 11
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What is the simplified form of 48 over 192??
Answer: The simplified form of 48 over 192 is 1 over 4.
Step-by-step explanation:
Since, A fraction is in its simplest form when the numerator and denominator cannot be any smaller, while still being whole numbers.
Also, for find the simplified form of a fraction,
We divide both numerator and denominator by their H.C.F,
Here, the given fraction is,
[tex]\frac{48}{192}[/tex]
Since, 48 = 2 × 2 × 2 × 2 × 3
192 = 2 × 2 × 2 × 2 × 2 × 2 × 3
H.C.F(48,192) = 2 × 2 × 2 × 2 × 3 = 48
[tex]\implies \frac{48/48}{192/48}=\frac{1}{4}[/tex]
Hence, The simplified form of 48/192 is 1/4.
The simplified form of 48/192 is 1/4.
First, let's find the GCD of 48 and 192.
The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
The factors of 192 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 192.
The common factors are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
So, the greatest common divisor (GCD) of 48 and 192 is 48.
Now, we divide both the numerator and denominator by 48:
48/48 = 1
192/48 = 4
Therefore, the simplified form of 48/192 is 1/4.
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What is the value of x in the diagram?
(Diagram in the following post)
A. 1/2
B. 4/21
C. 2
D. 4
Answer:
The answer is 2 so C
Step-by-step explanation:
The correct answer is D. 4, The value of x is 4.
To find the value of [tex]\( x \)[/tex] in the diagram, we need to apply the properties of similar triangles. Assuming the diagram shows two similar triangles with corresponding sides labeled, we can set up a proportion between the sides of the similar triangles.
Let's denote the sides of the smaller triangle as [tex]\( a \), \( b \), and \( c \)[/tex], and the corresponding sides of the larger triangle as [tex]\( ka \), \( kb \), and \( kc \),[/tex] where [tex]\( k \)[/tex] is the scale factor between the two similar triangles.
Given that one side of the larger triangle is 4 times the corresponding side of the smaller triangle, we have [tex]\( k = 4 \)[/tex].
Now, if we let [tex]\( a = 21 \)[/tex] (as per the options given), then the corresponding side of the larger triangle is [tex]\( ka = 4 \times 21 = 84 \).[/tex]
The problem states that the side labeled [tex]\( x \)[/tex] in the larger triangle corresponds to the side labeled [tex]\( 1 \)[/tex] in the smaller triangle. Therefore, we have the proportion:
[tex]\[ \frac{x}{ka} = \frac{1}{a} \][/tex]
Substituting the known values, we get:
[tex]\[ \frac{x}{84} = \frac{1}{21} \][/tex]
To solve for [tex]\( x \)[/tex], we cross-multiply:
[tex]\[ x \times 21 = 84 \times 1 \][/tex]
[tex]\[ 21x = 84 \][/tex]
Now, we divide both sides by 21 to isolate [tex]\( x \)[/tex]:
[tex]\[ x = \frac{84}{21} \][/tex]
[tex]\[ x = 4 \][/tex]
what is the purpose of a questionnaire
Answer:
It enables you to explore and know more
Step-by-step explanation:
Purpose Of Questionnaire
The main purpose of a questionnaire is to extract data from the respondents. It's a relatively inexpensive, quick, and efficient way of collecting large amount data even when the researcher isn't present to collect those responses first hand