HELP PLEASE DUE IN AN HOUR

We are given the following differential equation, [tex]\frac{dx}{dt}-4x=cos(5t)[/tex], with the initial condition [tex]x(0)=1[/tex]. This is a linear DE in the form, [tex]\frac{dx}{dt}+P(t)x=Q(t)[/tex], where [tex]\mu(t)=e^{\int\ {P(t)} \, dt }[/tex].

First, finding [tex]\mu(t)[/tex].

[tex]\Longrightarrow\mu(t)=e^{\int\ {P(t)} \, dt }\Longrightarrow\mu(t)=e^{\int\ {-4} \, dt }[/tex]

[tex]\mu(t)=e^{-4t }[/tex]

Multiply the DE by [tex]\mu(t)[/tex].

[tex]\Longrightarrow\frac{dx}{dt}-4x=cos(5t)\Longrightarrow e^{-4t} [\frac{dx}{dt}-4x=cos(5t)][/tex]

[tex]\Longrightarrow (e^{-4t}) \frac{dx}{dt}-(e^{-4t})4x=(e^{-4t})cos(5t)[/tex]

[tex]\Longrightarrow e^{-4t} \frac{dx}{dt}-4e^{-4t}x=e^{-4t}cos(5t)[/tex]

Verify that the L.H.S is a product rule of [tex]e^{-4t}x[/tex].

[tex]\Longrightarrow \frac{d}{dt}[e^{-4t}x ] =(e^{-4t}*-4)x+(e^{-4t})(\frac{dx}{dt} )[/tex]

[tex]\Longrightarrow \frac{d}{dt}[e^{-4t}x ] =-4e^{-4t}x+e^{-4t}\frac{dx}{dt}[/tex]

This matches, so we can move forward by integrating both sides of the DE.

[tex]\int\ {[e^{-4t}x ]'} \, =\int\ {e^{-4t}cos(5t)} \, dt[/tex]

The integral on the R.H.S is pretty nasty to do using integration by parts and u-sub. I will use a table of integrals.

[tex]\int\ {e^{ax}cos(bx) } \, dx=\frac{e^{ax}(acos(bx)+bsin(bx))}{a^2+b^2}[/tex]

Let [tex]a=-4[/tex] and [tex]b=5[/tex]

[tex]\int\ {[e^{-4t}x ]'} \, =\int\ {e^{-4t}cos(5t)} \, dt[/tex]

[tex]\Longrightarrow e^{-4t}x = \frac{e^{-4t}(-4cos(5t)+5sin(5t))}{(-4)^2+(5)^2}+C[/tex]

[tex]\Longrightarrow e^{-4t}x = \frac{e^{-4t}(-4cos(5t)+5sin(5t))}{41} +C[/tex]

Now for the initial condition, [tex]x(0)=1[/tex].

[tex]\Longrightarrow e^{-4(0)}(1) = \frac{e^{-4(0)}(-4cos(5(0))+5sin(5(0)))}{41} +C[/tex]

[tex]\Longrightarrow (1)(1) = \frac{(1)(-4(1)+(0))}{41} +C[/tex]

[tex]\Longrightarrow 1 = -\frac{4}{41} +C[/tex]

[tex]\Longrightarrow 1+\frac{4}{41} = C[/tex]

[tex]\Longrightarrow C=\frac{45}{41}[/tex]

Now we have, [tex]e^{-4t}x = \frac{e^{-4t}(-4cos(5t)+5sin(5t))}{41} +\frac{45}{41}[/tex], solve for x.

[tex]\Longrightarrow x = \frac{\frac{e^{-4t}(-4cos(5t)+5sin(5t))}{41}}{ e^{-4t}} +\frac{\frac{45}{41}}{e^{-4t}}[/tex]

[tex]\Longrightarrow x = \frac{(-4cos(5t)+5sin(5t))}{41}} +\frac{45e^{4t}}{41}[/tex]

Thus, the answer to the given differential equation with an initial condition is, [tex]x = \frac{(-4cos(5t)+5sin(5t))}{41}} +\frac{45e^{4t}}{41}[/tex].

Answer:

1 - [tex]e^{inx}[/tex]

2 - [tex]e^{5in}[/tex]

3 - [tex]e^{in^{3} x}[/tex]

Step-by-step explanation:

The correct answer is to Reject the null hypothesis. This is strong evidence that the mean amount of sleep is less than 8 hours.

The test statistic of t = -2.03 and P-value of approximately 0.025 suggest that the mean amount of sleep for students is significantly less than 8 hours.

The P-value of 0.025 is less than the significance level of 0.05, which means that there is enough evidence to reject the null hypothesis that the mean amount of sleep is 8 hours.

Therefore, the appropriate conclusion at the a = 0.05 significance level is that the mean amount of sleep for students is less than 8 hours.

Hence, the correct answer is C.

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The area of the tabletop in square inches is 1071 square inches.

An object's area is how much space it takes up in two dimensions. It is the measurement of the quantity of unit squares that completely cover the surface of a closed figure. The square unit, which is frequently expressed as square inches, square feet, etc., is the accepted unit of area.

Given that, there are 3 inches between each side of the aquarium and each side of the tabletop.

The length of the tabletop will be:

l = 48 + 3 = 51 inches.

The width of the tabletop will be:

w = 18 + 3 = 21 inches.

The area of the table top will be:

A = (l)(b)

A = (51)(21)

A = 1071 square inches.

The area of the tabletop in square inches is 1071 square inches.

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The new arc length function for the curve after reparametrization is as follows

[tex]\\\\r(s) = ( 4 - \frac{s}{\sqrt{26}})i + (\frac{4s}{\sqrt{26} } + 4)j+(\frac{3s}{\sqrt{26} } + 3)k[/tex]

What is meant by the arc length of a curve?

The distance between two places along a segment of a curve is known as the arc length.

The formula for arc length is used to determine how far something is along the curved line that makes up an arc. The arc length is, to put it simply, the distance that passes through the curved line of the circle that forms the arc.

Given,

Vector equation of curve  r(t) = (5 - t) i + (4t - 3) j + 3t k

    Given P = (4, 1, 3)

5 - t = 4

So, t = 1

Now differentiate r(t)

 r'(t) = -i + 4j + 3k

Magnitude of r'(t) = |r'(t)| = √[(-1)² + 4² + 3²] = √26

The arc length function for the curve

 [tex]s = \int\limits^t_1 { |r'(t)| } \, dt = \int\limits^t_1 { \sqrt{26} } \, dt = \sqrt{26} (t-1)[/tex]

  [tex]t = \frac{s}{\sqrt{26} } + 1[/tex]

Now to reparametrize the curve with respect to arc length, substitute t with (s/√26) + 1 in r(t).

[tex]r(s) = (5 - ( \frac{s}{\sqrt{26} } + 1)) i + (4( \frac{s}{\sqrt{26} } + 1) - 3) j + 3( \frac{s}{\sqrt{26} } + 1) k\\\\r(s) = ( 4 - \frac{s}{\sqrt{26}})i + (\frac{4s}{\sqrt{26} } + 4)j+(\frac{3s}{\sqrt{26} } + 3)k[/tex]

Therefore the new arc length equation of the curve when  [tex]t = \frac{s}{\sqrt{26} } + 1[/tex] is as follows

[tex]\\\\r(s) = ( 4 - \frac{s}{\sqrt{26}})i + (\frac{4s}{\sqrt{26} } + 4)j+(\frac{3s}{\sqrt{26} } + 3)k[/tex].

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I think that answer is yes

Step-by-step explanation:

The answer should be -11

Answer:  -11 (negative eleven)

An integer is a real number. Which includes 0, positive numbers, and negative numbers. Since the temperature outside went down by 11 degrees rather than increased, it is negative. Which means our integer is -11.

I hope this helps & Good Luck <3 !

The point where the net electric field is equal to zero is located approximately 0.48936m away from q1 and 24m - 0.48936m = 23.51064m away from q2.

Charge is a fundamental property of matter that describes its electrical state. It is a scalar quantity that can be either positive, negative, or neutral, and is measured in Coulombs (C). The concept of charge is used to explain many of the phenomena that occur in electricity and magnetism, such as electric fields, currents, and voltages. Positive and negative charges are thought to be due to the presence or absence of electrons, which are the smallest negatively charged particles in matter. Neutral objects have an equal number of positive and negative charges, resulting in a net charge of zero. The interaction of charges with each other results in many of the electrical and magnetic phenomena observed in the world around us.

The net electric field at a point is equal to zero if the electric field due to one charge is exactly canceled by the electric field due to the other charge.

Since q2 is 49 times greater than q1, the electric field due to q2 will be 49 times greater than the electric field due to q1 at any given point. In order to cancel the electric field due to q1, we need to place q2 at a point that is 49 times closer to q1 than the original separation of 24m.

Let's call the new separation between the charges x. Then, x = 24m / 49, which is approximately equal to 0.48936m.

So, the point where the net electric field is equal to zero is located approximately 0.48936m away from q1 and 24m - 0.48936m = 23.51064m away from q2.

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Answer:

28.93 gallons of paint

Step-by-step explanation:

The volume of a sphere can be found using the formula:

V = (4/3)πr^3

We are given that the water tower holds 6,684 cubic feet of water, so we can use this to find the radius:

6684 = (4/3)πr^3

(6684 * 3) / (4 * π) = r^3

r = (6684 * 3 / (4 * π))^(1/3) = 35.73 feet

Next, we need to find the surface area of the sphere. The surface area of a sphere can be found using the formula:

A = 4πr^2

So, the surface area of this water tower can be found as:

A = 4π * 35.73^2 = 4π * 1286.13 = 5071.57 square feet

Finally, we need to find the amount of paint needed to paint the surface area twice. One gallon of paint can cover 350 square feet, so we can divide the surface area by 350 to find the number of gallons of paint needed:

Gallons of paint = (5071.57 * 2) / 350 = (10143.14 / 350) = 28.93 gallons

So, 28.93 gallons of paint are needed to paint the water tower twice.

Note that the proof related to the above parallelogram is given as follows:

ΔSVX ≅ ΔUTX                      Given

SV || TU                                 Given

SV ≅ TU                                Corresponding part of congruent triangle

VUTS parallelogram             Definition of parallelogram.

A quadrilateral having two pairs of parallel sides is known as a parallelogram. It is possible to demonstrate that the opposing sides of a parallelogram are congruent.

A parallelogram is any quadrilateral having two pairs of congruent opposing sides. Parallelograms contain two congruent opposing angles, thus a quadrilateral with two congruent opposite angles must be a parallelogram.

A parallelogram's diagonals intersect each other, therefore a quadrilateral with diagonals that bisect each other must be a parallelogram. Finally, if a quadrilateral contains two parallel and congruent sides, it must be a parallelogram.

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The function at the given value of the independent variable, f(x) = 1/√(x-4)f(x) - f(5) / x - 5 is (x-5).

Each element of X is given precisely one element of Y by the function from a set X to a set Y. The sets X and Y are collectively referred to as the function's domain and codomain, respectively. Initially, functions represented the idealised relationship between two changing quantities.

We have given function as ,

f(x) = 1/√(x-4)

We have to find f(x) - f(5)/x-5

f(x) = f(5)/x-5 = 1/√(5-4)/(x-5)

= 1/√(5-4)/(x-5)

= (x-5)/√(5-4)

= (x-5)

Therefore, The function at the given value of the independent variable, f(x) = 1/√(x-4)f(x) - f(5) / x - 5 is (x-5).

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It is true that if points F and G are contained in a plane, then the line segment FG that connects them is entirely contained in that plane.

A two-dimensional plane, also known as a plane in two dimensions, is a flat surface that has length and width, but no thickness. It is an abstract mathematical concept that can be represented graphically.

Here,
In geometry, if two points are contained in a plane, then any line segment connecting those two points is also contained in that plane. So, if points F and G are contained in a plane, then the line segment FG that connects them is entirely contained in that plane.

Thus, it is true that if points F and G are contained in a plane, then the line segment FG that connects them is entirely contained in that plane.

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The standard form of the polynomial function y = x(x + 5)(x - 3) is

y = x³ + 2x² - 15x

Function is a relation between a set of inputs and a set of outputs which are permissible. In a function, for particular values of x we will get only a single image in y. It is denoted by f(x).

Vertical line test:-

Whenever we want to check whether a given expression is a function or not we can apply a vertical line test, in this test we check for a single image of x , we are getting a single image or more.

If we get more images then it will not be a function.

For example, let us take, y² = 4ax

y = ±√4ax

For single value of x we get two values of y

Hence, it will not be a function.

Given that,

The polynomial function,

y = x(x+5)(x−3)

To obtain the standard form, we need to multiply out the expression and then arrange the terms in decreasing order of the degree of the monomials. The degree of a polynomial is the highest degree of its monomials.

x(x + 5)(x - 3) = x³ + (5x²) - (3x²) + (5x²) + (-15x) = x³ + 2x² - 15x

In the standard form, the coefficient of the monomial with the highest degree is 1, and the rest of the terms are arranged in decreasing order of the degree.

Thus, the standard form is x³ + 2x² - 15x

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The triangular table that Mrs. Gomez is buying for her classroom is a right triangle table because 3² +4² = 5². That is option C.

A right angle triangle is the type of triangle that has one side of its angle perpendicular to another and the value of the three sides can be calculated through the use of the Pythagorean formula.

The Pythagorean formula is expressed as, c2 = a2 + b2, where 'c' = hypotenuse of the right triangle and 'a' and 'b' are the other two legs.

Since, 5= c

b= 3

a = 4

Therefore, 3² +4² = 5² shows that the table is a right angle triangle.

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The subsets are {}, {0}.

What is the subset in mathematics?

In mathematics, a subset is a set consisting of elements that are also elements of another set. A set A is said to be a subset of set B if and only if every element of A is also an element of B. In other words, if you have a set A and another set B, and all the elements of set A are present in set B, then set A is considered a subset of set B.

In mathematics, a subset is a collection of elements that belong to a larger set. The elements in a subset can be any type of mathematical objects, such as numbers, vectors, or functions. A set is a subset of itself, and the empty set is a subset of every set.

The subsets of the given set {0} are the empty set {} and the set itself, {0}. So, the subsets of the given set 0 are {}, {0}.

Hence, the subsets are {}, {0}.

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About 0.04% percent of quokkas weigh over 15.5 pounds.

Normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

Given that, the weights of quokkas can be described by the Normal model N(10, 2.7).

Let Y the weights of quokkas

Given that, Y~N(μ= 10, σ²=2.7)

So, σ=1.643

a) The probability, P(Y＞15.5) denotes chances that quokkas weight over 9.3 pounds, and it is,

P(Y>9.3) = P(Z> (15.5-μ)/σ)

= P(Z> (15.5-10)/1.643)

= P(Z>3.347)

= 0.0004

= 0.0004×100 =0.04%

b) The probability, P(Y＜11.9) denotes chances that quokkas weight under 11.9 pounds, and it is,

P(Y>9.3) = P(Z> (11.9-μ)/σ)

= P(Z> (11.9-10)/1.643)

= P(Z>1.156)

= 0.1250

= 0.1250×100 =12.5%

c) The probability, P(11.9≤Y≤15.5) denotes chances that quokkas weight will be between 11.9 and 15.5 pounds, and it is,

P(11.9≤Y≤15.5) = P((15.5-μ)/σ ≤ Z ≤ (11.9-μ)/σ)

= P((15.5-10)/1.643) ≤ Z ≤ (11.9-10)/1.643)

= P(3.347≤ Z ≤1.156)

= P(Z≤3.347)-PZ ≤1.156)

= 0.0004-0.1250

= 0.1246

= 0.1246×100

= 12.46%

Therefore, about 0.04% percent of quokkas weigh over 15.5 pounds.

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Eric’s gross pay for one year if he earns a salary of $3,500 per month is $42,000.

The total amount of money an employee receives before taxes is referred to as gross pay.

Given  that;

Eric’s gross pay for one year = Eric's salary per month × 12 months

Eric’s gross pay for one year = $3,500 × 12

Eric’s gross pay for one year = $42,000

Therefore, the gross pay is $42,000.

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Answer:

Hi there! The answer is 3/5

Step-by-step explanation:

The rise is 3 and the run is 5.

Brainiest, please? Can you click on the crown on my answer?

Answer: 3/5

Step-by-step explanation:

Looks at the points (0,1) and (5,4).

Formula: y2-y1/x2-x1

Substitute: 4-1/5-0 = 3/5

m = 3/5

Using Compound Interest, The initially invested amount was $5400.

The interest you earn on interest is known as compound interest. Simple math may be used to demonstrate this: if you have $100 and it generates 5% interest annually, you will have $105 at the end of the first year. You'll have $110.25 after the second year is over.

The formula for compound interest is

[tex]C = P(1 + \frac{r}{n})^{nt}[/tex]

Where C is the compound interest value after n years.

P is the principal amount.

I is interest.

n is the number of years.

It is given that percentage of interest is 5% so I=0.05. C=$5676.27.

n=7.

According to the formula of compound interest,

5676.27 = P(1+0.05)7 P = 5676.27 1.051084

P = 5400

Therefore the amount that was invested initially was $5400.

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Answer:

Can you please post the picture of the question, then I can help.

Based on the given system of equations, the value of x and y are undefined.

In order to solve the given system of equations, we would apply the substitution method. From the information provided in the image attached above, we have the following system of equations:

x = 5 - 2y      .......equation 1.

6y = 5 - 3x            .......equation 2.

By using the substitution method to substitute equation 1 into equation 2, we have the following:

6y = 5 - 3(5 - 2y)

6y = 5 - 15 + 6y

6y = -10 + 6y

6y - 6y = -10 (no solution).

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According to the model, the current population of Algeria is 38 million.

The model for the population in t years from now is given by the model P = 38 ( 1 + r ) ^ t.

This means that when given the model, the current population of Algeria is 38 million because the 38 in the model represents the value of population when no years have elapsed.

We can solve for this to get :

= 38 ( 1 + r ) ^ t

= 38 ( 1 + r ) ⁰

= 38 x 1

= 38

= 38 million

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The number of ways in which a committee of four Democrats and three Republicans be formed from a group of ten Democrats and eleven Republicans is 34650 ways.

By choosing some items from a set and creating subsets, permutation and combination are two approaches to represent a group of objects. It outlines the numerous configurations for a particular set of data. Permutations are the selection of data or objects from a set, whereas combinations are the order in which they are represented. Both ideas are critical to mathematics.

Here, the differences between permutation and combination are described in detail.

The number of ways in which we can select a committee of 4 democrats from ten democrats is:

P = 10C 4 = 10! / (4)! (10 - 4)!

P = 210

The number of ways in which we can select a committee of 3 Republicans from 11 Republicans is:

P = 11C 3 = 11! / 3! (11 - 3)!

P = 165

The number of ways in which a committee of four Democrats and three Republicans be formed from a group of ten Democrats and eleven Republicans is:

P = (210) (165) = 34650

The number of ways in which a committee of four Democrats and three Republicans be formed from a group of ten Democrats and eleven Republicans is 34650 ways.

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Answer: JD's markup price is $80 * 120% = $96 per bat.

So, the retail price of each bat is $80 + $96 = $176.

The total revenue from the sale of 16 bats is 16 * $176 = $2,816.

He still has 30 - 16 = 14 bats left.

The discount price of each bat is $176 - ($176 * 40%) = $106.40.

The total revenue from the sale of 14 bats at the discount price is 14 * $106.40 = $1,493.60.

The total revenue from all the bats is $2,816 + $1,493.60 = $4,309.60.

The profit JD makes from selling all the bats is $4,309.60 - ($80 * 30) = $1,039.60.

Step-by-step explanation:

The profit which will he make after selling all of the bats is $1,039.60.

This is referred to as the amount of money that you gain when you are paid more for something than it cost you to make, get, or do it.

JD's markup price = $80 ×120% = $96 per bat.

Retail price = $80 + $96 = $176.

Revenue of 16 bats is 16 * $176 = $2,816.

Since 30 - 16 = 14 bats which is what we have left.

The discount price of each bat is $176 - ($176 * 40%) = $106.40.

The total revenue from the sale of 14 bats at the discount price is 14 * $106.40 = $1,493.60.

Therefore the total revenue from all the bats is $2,816 + $1,493.60 = $4,309.60.

The total profit = $4,309.60 - ($80 * 30) = $1,039.60.

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a) The domain of the function is 0 to 15 miles

b) The range of the function is 0 to 270 miles

The domain of a function is the set of values that we are allowed to plug into our function. This set is the x values in a function such as f(x). The range of a function is the set of values that the function assumes. This set is the values that the function shoots out after we plug an x value in.

The range is the set of outputs of a relation or function. In other words, it's the set of possible y values. Recall that ordered pairs are of the form (x,y) so the y coordinate is listed after the x. The output is listed after the input.

Given data ,

Let the mileage of the car = 18 miles per gallon

The amount of gasoline the car can hold = 15 gallons

Let the distance to empty be a function of mileage of the car × amount of gas in gallons

Now , the domain of a function is the set of values that we are allowed to plug into our function which ranges from 0 to 15

So , distance to empty = 18 x 15 = 270

So , the range of the function is the set of output values which ranges from 0 to 270

Hence , the domain of the function is 0 to 15 miles and the range of the function is 0 to 270 miles

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The value of a that makes u parallel to v is a = 4 and  the value of a that makes u perpendicular to v is -7.

a. For two vectors to be parallel, their direction ratios must be equal. In other words, their components must have the same ratio. For vectors u = (a, 7) and v = (4, 4), we have:

u1/u2 = a/7 = v1/v2 = 4/4

So the value of a that makes u parallel to v is a = 4.

b. For two nonzero vectors to be perpendicular, their dot product must be equal to zero. The dot product of u = (a, 7) and v = (4, 4) is given by:

u · v = a * 4 + 7 * 4 = 4a + 28

To make this equal to zero, we can solve for a:

4a + 28 = 0

4a = -28

a = -7

So the value of a that makes u perpendicular to v is -7.

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Answer:

745.57

Step-by-step explanation:

[tex]691.95 \times \frac{7.75}{100} = 53.62[/tex]

[tex]53.62 + 691.95 = 745.57[/tex]

The correct option regarding the solution of the system of equations is given as follows:

One solution at (-1, 0).

We want to obtain the graphic solution for the system of equations, hence we obtain the point of intersection of the two equations.

For this problem, we have a single equation, hence the solution is obtained when the graph crosses the x-axis.

The graph crosses the x-axis when x = -1 and y = 0, hence the solution is given as follows:

(-1,0).

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