Let R be a communtative ring and a, b elements in R. Prove that if a and b are units, then so is ab. What can we say about ab when a is a unit and b is a zero divisor? Prove your claim.

There are more than one zero i.e. three zeros of the given polynomial.

            The zeros are:

                        [tex]x=0,-2,-3[/tex]

We are given a function f(x) by:

             [tex]f(x)=x^3+5x^2+6x[/tex]

We know that the zeros of the function f(x) are the possible values of x at which the function is equal to zero.

Hence, when f(x)=0 we have:

[tex]x^3+5x^2+6x=0\\\\i.e.\\\\x^3+3x^2+2x^2+6x=0\\\\i.e.\\\\x^2(x+3)+2x(x+3)=0\\\\i.e.\\\\(x^2+2x)(x+3)=0\\\\x(x+2)(x+3)=0\\\\i.e.\\\\x=0\ or\ x=-2\ or\ x=-3[/tex]

The zeros of the function f(x) = x^3 + 5x^2 + 6x are x = 0, x = -3, x = -2.

To find the zeros of the function f(x) = x^3 + 5x^2 + 6x, we first need to rewrite it in a factorized form in order to identify the roots. This is done by factoring out the common factor, which is 'x' in this equation, giving us x(x^2 + 5x + 6) = 0.

Therefore, x = 0 gives the first zero of this function. Now we are left with the quadratic equation x^2 + 5x + 6 = 0. Using the quadratic formula (-b ± sqrt[b^2 - 4ac]) / 2a, we can figure out the other roots of the equation. For this equation, a=1, b=5 and c=6.

Plugging these values into the quadratic formula, we get: x = [ -5 ± sqrt( (5)^2 - 4*1*6) ] / 2*1 = [ -5 ± sqrt(25 - 24) ] / 2 = -5 ± 1 / 2. Therefore, x = -3 and x = -2 are the other roots of this equation.

So, the zeros of the function f(x) = x^3 + 5x^2 + 6x are: x = 0, x = -3, x = -2.

https://brainly.com/question/29078812

#SPJ3

Answer:

a. 1 m = 100 cm

b. 1 m = 1.000.000.000 nm

c. 1 mm = 0,001 m

d. 1 L = 1.000 mL

Step-by-step explanation:

1 m means 1 meter, that will be our reference word.  

a. A cm is a centimeter, “centi” is the prefix that means “one hundred”. So, 1 meter equals 100 centimeters.

b. It happens the same when we have nanometers. “nano” means “one billion” (nine ceros behind the number one), so 1 meter equals 1.000.000.000 nanometers.

c. In this case, it goes the other way around. If we follow the logic that we use before, 1 meter equals 1.000 millimeters (“milli” means "one million"). But the problem says we have 1 mm, so we have to do a direct rule of 3. If 1000 millimeters equals 1 meter, 1 millimeter equals 1/1000 (or 0,01) meters.

d. Now we have liters instead of meters but is the same logic. 1 liter equals 1000 milliliters.  

Answer:

None

Step-by-step explanation:

The slopes are the same. The y intercepts are 2 and - 6. These lines are parallel which means they never intersect. No intersection means no solution. Just to show you what this means, I graphed this on Desmos for you.

Red: y = 3x - 6

Blue: y = 3x + 2

These two never meet.

Answer:

Step-by-step explanation:

Explanation:

Graph both the equations and read the coordinates of the point of intersection as shown in the graph below:

Final answer:

2 to the power of three halves is equivalent to the square root of 2 cubed, which is approximately 2.83.

Explanation:

In mathematics, when we raise a number to a fraction exponent, we are essentially taking the root of that number. In this case, 2 to the power of three halves is equivalent to the square root of 2 cubed.

2 to the power of three halves = [tex]\sqrt{(2^3)}[/tex] = [tex]\sqrt{8}[/tex] = 2.83

Answer:

100%

Step-by-step explanation:

If there is a 40% response rate and there are 400 numbers, that means that 160 people will respond.

Answer:

  105/252 = 0.41666...

Step-by-step explanation:

There are (7C4)(3C1) = (35)(3) = 105 ways to choose exactly 4 boys. There are 10C5 = 252 ways to choose 5 youths, so the probability that a randomly chosen team will consist of exactly 4 boys is ...

  105/252

_____

nCk = n!/(k!(n-k!))

Answer:

There is a 41.67% probability that exactly 4 boys are picked in this team of 5.

Step-by-step explanation:

The order is not important, so we use the combinations formula.

[tex]C_{n,x}[/tex] is the number of different combinatios of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

Number of desired outcomes.

Four boys and one girl: So

[tex]C_{7,4}*C_{3,1} = \frac{7!}{4!(7-4)!}*\frac{3!}{1!(3-1)!} = 35*3 = 105[/tex]

Number of total outcomes:

Combination of five from a set of 10.

So

[tex]C_{10,5} = \frac{10!}{5!(10-5)!} = 252[/tex]

What is the probability that exactly 4 boys are picked in this team of 5?

[tex]P = \frac{105}{252} = 0.4167[/tex]

There is a 41.67% probability that exactly 4 boys are picked in this team of 5.

The probability that the average age of the 100 residents selected is less than 68.5 years is approximately 0.1949 or 19.49%.

The subject of this problem refers to statistics, specifically the concept of the sampling distribution of sample means. It is related to the central limit theorem, which states that if you take sufficiently large random samples from a population, the distribution of the sample means will approximate a normal distribution, regardless of the shape of the population's distribution. We know that the population average (mean) is 69 and the standard deviation is 5.8.

We are given a sample size of 100, and hence we calculate its standard deviation as 5.8/√100 = 0.58. With this value, we can use the z-score formula (Z = (X - μ) / σ), where X is the sample mean, μ is the population mean, and σ is the standard deviation of the sample mean, to find the z-score for a sample mean of 68.5 years: Z = (68.5 - 69) / 0.58 ≈ -0.86.

Finally, this Z score is used to find the probability that the sample mean age is less than 68.5 years, by referring to a standard normal distribution table, also known as the Z-table. It should be taken into account that this table provides the probability that a value is less than the given Z score, which is exactly what we need in this case. Consulting the Z table with Z=-0.86, we find that the probability is approximately 0.1949 or 19.49%.

https://brainly.com/question/33780340

#SPJ3

Answer:

We have a normal distribution with a mean of 19 minutes and a standard deviation of 3 minutes. To solve the problem we're going to need the help of a calculator:

P(z>20) = 0.3694

Therefore, the percentage of costumbers that will receive the service for half-price is: 36.94%.

Also, we've found that p(z>25.16) = 0.02. Therefore, if they only want to offer half-price discount to only 2% of its costumber, the time limit should be 25.16 minutes.

Answer:

Step-by-step explanation:

Great question, it is always good to ask away and get rid of any doubts that you may be having.

This process is called Rule-Making. It is basically when the Federal Government makes regulations on certain topics. These Regulations help advance and set strict boundaries on projects so that people know what they can and cannot do, as well as protect others from being scammed or robbed by these projects. This is all done by the administrative process known as Rule Making.

I hope this answered your question. If you have any more questions feel free to ask away at Brainly.

Answer:

h(-4) = -3

h(-2) = -3

h(0) = -2

Step-by-step explanation:

We can see that the function's value is -3 for all numbers less than or equals to -2 and the function is (x-1)^2-3 between -1 and 1

So,

h(-4) = -3

As -4 is less than -2, so the value of function will be equal to -3.

h(-2) = -3

Similarly, on x=-2 the value of function will be -3.

And for h(0)

h(0) = (0-1)^2-3

= (-1)^2-3

=1-3

=-2

..

Answer: The next three terms are [tex]\dfrac{7}{23},\dfrac{8}{26},\dfrac{9}{29}[/tex]

Step-by-step explanation:

Since we have given that

[tex]\dfrac{1}{5},\dfrac{2}{8},\dfrac{3}{11},\dfrac{4}{14},\dfrac{5}{17},\dfrac{6}{20}[/tex]

After analyzing the above pattern, we get that

In numerator , numbers are run in consecutive manner.

In denominator, number is added to 3 to get the next term.

So, the general form would be [tex]\dfrac{n}{3n+2}[/tex]

So, the next three terms would be

[tex]\dfrac{7}{3\times 7+2}=\dfrac{7}{23}\\\\\dfrac{8}{3\times 8+2}=\dfrac{8}{26}\\\\\dfrac{9}{3\times 9+2}=\dfrac{9}{29}[/tex]

Hence, the next three terms are

[tex]\dfrac{7}{23},\dfrac{8}{26},\dfrac{9}{29}[/tex]

Answer:

(8x-1) (x-1)

Step-by-step explanation:

8x² - 9x + 1

(8x -   ) (x - )

We know it is minus because we have -9x

We have +1 so both have to be -

To fill in the blanks we put 1

The only combination is 1*1 =1

(8x-1) (x-1)

Lets check

8x^2 -x -8x +1

(8x^2 -9x+1

Answer:

Step-by-step explanation:

Given that there are 3 events as

rolling 3 dice, are the events A: sum divisible by 3 and B: sum divisible 5

The sample space will have

(1,1,1)...(6,6,6)

Sum will start with 3 and end with 18

Sum divisible by 3 are 3,6,9..18

Sum divisible by 5 are 5,10,....15

The common element is 15

Hence these two events are not mutually exclusive.

Answer:

  28 inches high by 7 inches wide

Step-by-step explanation:

Let x represent the width of the poster with margins. Then the printable width is (x -2). The printable height will be 100/(x-2), so the overall poster height is ...

  height = 100/(x -2) +8 = (8x +84)/(x -2)

The poster's overall area is the product of its width and height, so is ...

  A = x(8x +84)/(x -2)

The derivative of this with respect to x is ...

  A' = ((16x +84)(x -2) -(8x^2 +84x)(1))/(x -2)^2

This is zero when the numerator is zero, so ...

  8x^2 -32x -168 = 0

  x^2 -4x -21 = 0 . . . . . . divide by 8

  (x +3)(x -7) = 0 . . . . . . . factor

The values of x that make these factors be zero are -3 and +7. The height corresponding to a width of 7 is ...

  height = 100/(7 -2) +8 = 28

The amount of paper is minimized when the poster is 7 inches wide by 28 inches tall.

_____

Comment on the problem and solution

You will notice that the poster is 4 times as high as it is wide. It is no accident that this ratio is the ratio of the vertical margin to the horizontal margin. That is, the fraction of the poster devoted to margin is the same in each direction. This is the generic solution to this sort of problem.

Knowing that the margins have a ratio of 4:1 tells you the printable area will have a ratio of 4:1, hence is equivalent to 4 squares, each with an area of 100/4 = 25 square inches. That means the printable area is √25 = 5 inches wide by 4×5 = 20 inches high, so the overall poster area is 28 inches high by 7 inches wide. This arithmetic can be all mental and does not involve derivatives.

To minimize the amount of paper used, the overall dimensions of the rectangular poster should be 102 inches in width and 108 inches in height.

To minimize the amount of paper used, we need to find the dimensions of the rectangle that will enclose 100 in2 of printing. Since there is a 4-inch margin at the top and bottom, the height of the rectangle will be the printing area plus the margins, which is 100 + 4 + 4 = 108 inches. Similarly, there is a 1-inch margin on each side, so the width of the rectangle will be the printing area plus the margins, which is 100 + 1 + 1 = 102 inches. Therefore, the overall dimensions of the rectangle that will minimize the amount of paper used are 102 inches in width and 108 inches in height.

https://brainly.com/question/35514186

#SPJ11

Answer with Step-by-step explanation:

We are given S be any set which is countable and nonempty.

We have to prove that their exist a surjection g:N[tex]\rightarrow S[/tex]

Surjection: It is also called onto function .When cardinality of domain set is greater than or equal to cardinality of range set then the function is onto

Cardinality of natural numbers set =[tex]\chi_0[/tex]( Aleph naught)

There are two cases

1.S is finite nonempty set

2.S is countably infinite set

1.When S is finite set and nonempty set

Then cardinality of set S is any constant number which is less than the cardinality of set of natura number

Therefore, their exist a surjection from N to S.

2.When S is countably infinite set and cardinality with aleph naught

Then cardinality of set S is equal to cardinality of set of natural .Therefore, their exist a surjection from N to S.

Hence, proved

Answer:

x₃=1.599997=1.6

Step-by-step explanation:

f(x)=cos x, x₁=1.6

[tex]f'(x)=\frac{d}{dx}cos\ x\\=-sin\ x[/tex]

First iteration

At x₁=1.6

f(x₁)=f(1.6)

=cos(1.6)

=-0.0292

f'(x₁)=f'(1.6)

=-sin(1.6)

=-0.9996

[tex]\frac{f(x_1)}{f'(x_1)}=\frac{-0.0292}{-0.9996}\\=0.0292\\x_2=x_1-\frac{f(x_1)}{f'(x_1)}=1.6-(0.0292)\\\therefore x_2=1.5708[/tex]

[tex]f(x_2)=f(1.5708)\\=cos 1.5708\\=0.000003\\f'(x_2)=-sin1.5708\\=-1\\x_3=x_2-\frac{f(x_2)}{f'(x_2)}=1.6-\frac{0.000003}{-1}\\\therefore x_3=1.599997=1.6[/tex]

Answer: 1.205

Step-by-step explanation:

Given : Significance level : [tex]\alpha=0.025[/tex]

[tex]H_0:\mu=50.6\\\\H_1:\mu>50.6[/tex]

We assume that population is normally distributed.

The sample size : [tex]n=10[/tex], which is less than 30 , so we apply t-test.

Mean : [tex]\overline{x}=54.6[/tex]

Standard deviation : [tex]\sigma=10.5[/tex]

The test statistic for population mean is given by :-

[tex]t=\dfrac{\overline{x}-\mu_0}{\dfrac{\sigma}{\sqrt{n}}}\\\\=\dfrac{54.6-50.6}{\dfrac{10.5}{\sqrt{10}}}=1.20467720387\approx1.205[/tex]

Hence, the critical value = 1.205

Substitute [tex]v(x)=-2x+y(x)[/tex], so that [tex]\dfrac{\mathrm dv}{\mathrm dx}=-2+\dfrac{\mathrm dy}{\mathrm dx}[/tex]. Then the ODE is equivalent to

[tex]\dfrac{\mathrm dv}{\mathrm dx}+2=v^2-7[/tex]

which is separable as

[tex]\dfrac{\mathrm dv}{v^2-9}=\mathrm dx[/tex]

Split the left side into partial fractions,

[tex]\dfrac1{v^2-9}=\dfrac16\left(\dfrac1{v-3}-\dfrac1{v+3}\right)[/tex]

so that integrating both sides is trivial and we get

[tex]\dfrac{\ln|v-3|-\ln|v+3|}6=x+C[/tex]

[tex]\ln\left|\dfrac{v-3}{v+3}\right|=6x+C[/tex]

[tex]\dfrac{v-3}{v+3}=Ce^{6x}[/tex]

[tex]\dfrac{v+3-6}{v+3}=1-\dfrac6{v+3}=Ce^{6x}[/tex]

[tex]\dfrac6{v+3}=1-Ce^{6x}[/tex]

[tex]v=\dfrac6{1-Ce^{6x}}-3[/tex]

[tex]-2x+y=\dfrac6{1-Ce^{6x}}-3[/tex]

[tex]y=2x+\dfrac6{1-Ce^{6x}}-3[/tex]

Given the initial condition [tex]y(0)=0[/tex], we find

[tex]0=\dfrac6{1-C}-3\implies C=-1[/tex]

so that the ODE has the particular solution,

[tex]\boxed{y=2x+\dfrac6{1+e^{6x}}-3}[/tex]

Answer:

The maximum volume of the open box is 24.26 cm³

Step-by-step explanation:

The volume of the box is given as [tex]V=g(x)[/tex], where [tex]g(x)=x(6-2x)(8-2x)[/tex] and [tex]0\le x\le3[/tex].

Expand the function to obtain:

[tex]g(x)=4x^3-28x^2+48x[/tex]

Differentiate  wrt  x to obtain:

[tex]g'(x)=12x^2-56x+48[/tex]

To find the point where the maximum value occurs, we solve

[tex]g'(x)=0[/tex]

[tex]\implies 12x^2-56x+48=0[/tex]

[tex]\implies x=1.13,x=3.54[/tex]

Discard x=3.54 because it is not within the given domain.

Apply the second derivative test to confirm the maximum critical point.

[tex]g''(x)=24x-56[/tex], [tex]g''(1.13)=24(1.13)-56=-28.88\:<\:0[/tex]

This means the maximum volume occurs at [tex]x=1.13[/tex].

Substitute [tex]x=1.13[/tex] into [tex]g(x)=x(6-2x)(8-2x)[/tex] to get the maximum volume.

[tex]g(1.13)=1.13(6-2\times1.13)(8-2\times1.13)=24.26[/tex]

The maximum volume of the open box is 24.26 cm³

See attachment for graph.

Answer:  The required solution of the given differential equation is

[tex]x+y\log y=Cy.[/tex]

Step-by-step explanation:  We are given to solve the following ordinary differential equation :

[tex]ydx+(y-x)dy=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]

We will be using the following formulas for integration and differentiation :

[tex](i)~d\left(\dfrac{x}{y}\right)=\dfrac{ydx-xdy}{y^2},\\\\\\(ii)~\int\dfrac{1}{y}dy=\log y.[/tex]

From equation (i), we have

[tex]ydx+(y-x)dy=0\\\\\Rightarrow ydx+ydy-xdy=0\\\\\\\Rightarrow \dfrac{ydx+ydy-xdy}{y^2}=\dfrac{0}{y^2}~~~~~~~~~~~~~~~~~~~~[\textup{dividing both sides by }y^2]\\\\\\\Rightarrow \dfrac{ydx-xdy}{y^2}+\dfrac{1}{y}dy=0\\\\\\\Rightarrow d\left(\dfrac{x}{y}\right)+d(\log y)=0.[/tex]

Integrating the above equation on both sides, we get

[tex]\int d\left(\dfrac{x}{y}\right)+\int d(\log y)=C~~~~~~~[\textup{where C is the constant of integration}]\\\\\\\Rightarrow \dfrac{x}{y}+\log y=C\\\\\Rightarrow x+y\log y=Cy.[/tex].

Thus, the required solution of the given differential equation is

[tex]x+y\log y=Cy.[/tex].

Answer:

Step-by-step explanation:

Given : The readings on thermometers are normally distributed with

Mean : [tex]\mu=\ 0[/tex]

Standard deviation : [tex]\sigma= 1[/tex]

The formula to calculate the z-score :-

[tex]z=\dfrac{x-\mu}{\sigma}[/tex]

For x = -1.52

[tex]z=\dfrac{-1.52-0}{1}=-1.52[/tex]

For x = -0.81

[tex]z=\dfrac{-0.81-0}{1}=-0.81[/tex]

The p-value = [tex]P(-1.52<z<-0.81)=P(z<-0.81)-P(z<-1.52)[/tex]

[tex]0.2089701-0.0642555=0.1447146\approx0.1447[/tex]

Hence, the probability that a randomly selected thermometer reads between negative 1.52 and negative 0.81 = 0.1447

To find the probability, standardize the values using z-scores and find the area under the normal curve between the z-scores.

To find the probability that a randomly selected thermometer reads between -1.52 and -0.81, we need to find the area under the normal curve between these two values. First, we need to standardize the values by finding the z-scores for these values using the formula z = (x - μ) / σ, where x is the given value, μ is the mean, and σ is the standard deviation. After finding the z-scores, we can then use the normal distribution table or a calculator to find the area between these z-scores.

The z-score for -1.52 is z = (-1.52 - 0) / 1.00 = -1.52 and the z-score for -0.81 is z = (-0.81 - 0) / 1.00 = -0.81. Using a normal distribution table or a calculator, we can find the area to the left of -1.52 and the area to the left of -0.81. The probability that a randomly selected thermometer reads between -1.52 and -0.81 is the difference between these two areas: P(-1.52 < X < -0.81) = P(X < -0.81) - P(X < -1.52).

Using the normal distribution table or a calculator, we can find that P(X < -0.81) is approximately 0.2123 and P(X < -1.52) is approximately 0.0655. Therefore, the probability that a randomly selected thermometer reads between -1.52 and -0.81 is approximately 0.2123 - 0.0655 = 0.1468, or 14.68%. The sketch of the region would be a shaded area under the standard normal curve between -1.52 and -0.81.

https://brainly.com/question/34741155

#SPJ3

Answer:

  1/16

Step-by-step explanation:

Substituting the given value of x into the equation, we get ...

  [tex]y = 4\times 2^x=4\times 2^{-6}=\dfrac{4}{2^6}=\dfrac{4}{64}=\dfrac{1}{16}[/tex]

Answer: 0.0136

Step-by-step explanation:

Given : Mean : [tex]\mu=25[/tex]

Standard deviation : [tex]\sigma=3.2[/tex]

Sample size : [tex]n=50[/tex]

The formula to calculate the z-score :-

[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

For x = 24

[tex]z=\dfrac{24-25}{\dfrac{3.2}{\sqrt{50}}}=-2.21[/tex]

The p-value = [tex]P(z\leq-2.21)= 0.0135526\approx0.0136[/tex]

Hence, the probability that the sample mean age for 50 randomly selected women to marry is at most 24 years = 0.0136

Answer:

The given expression is divisible by 3 for all natural values of x.

Step-by-step explanation:

The given expression is

[tex]2^{2x+1}+1[/tex]

For x=1,

[tex]2^{2(1)+1}+1=2^{3}+18+1=9[/tex]

9 is divisible by 3. So, the given statement is true for x=1.

Assumed that the given statement is true for n=k.

[tex]2^{2k+1}+1[/tex]

This expression is divisible by 3. So,

[tex]2^{2k+1}+1=3n[/tex]              .... (1)

For x=k+1

[tex]2^{2(k+1)+1}+1[/tex]

[tex]2^{2k+2+1}+1[/tex]

[tex]2^{(2k+1)+2}+1[/tex]

[tex]2^{2k+1}2^2+1[/tex]

Using equation (1), we get

[tex](3n-1)2^2+1[/tex]

[tex](3n)2^2-2^2+1[/tex]

[tex](3n)2^2-4+1[/tex]

[tex](3n)4-3[/tex]

[tex]3(4n-1)[/tex]

This expression is also divisible by 3.

Therefore the given expression is divisible by 3 for all natural values of x.

[tex]f(t)=\begin{cases}9&\text{for }0\le t<2\\(t-5)^2&\text{for }2\le t<5\\2te^{6t}&\text{for }t\ge5\end{cases}[/tex]

and presumably 0 for [tex]t<0[/tex]. We can express [tex]f(t)[/tex] in terms of the unit step function,

[tex]u(t-c)=\begin{cases}1&\text{for }t\ge c\\0&\text{for }t<c\end{cases}[/tex]

[tex]f(t)=9(u(t)-u(t-2))+(t-5)^2(u(t-2)-u(t-5))+2te^{6t}u(t-5)[/tex]

Quick explanation: [tex]9u(t)=9[/tex] for [tex]t\ge0[/tex], and [tex]9u(t-2)=9[/tex] for [tex]t\ge2[/tex]. So subtracting these will cancel the value of 9 for all [tex]t\ge2[/tex] and leave us with the value of 9 over the interval we want, [tex]0\le t<2[/tex]. The same reasoning applies for the other 3 terms.

Recall the time displacement theorem:

[tex]\mathcal L_s\{f(t-c)u(t-c)\}=e^{-sc}\mathcal L_s\{f(t)\}[/tex]

By this property, we have

[tex]\mathcal L_s\{9u(t)\}=\mathcal L_s\{9\}=\dfrac9s[/tex]

[tex]\mathcal L_s\{9u(t-2)\}=e^{-2s}\mathcal L_s\{9\}=\dfrac{9e^{-2s}}s[/tex]

[tex]\mathcal L_s\{(t-5)^2u(t-2)\}=\mathcal L_s\{((t-2)-3)^2u(t-2)\}[/tex]

[tex]=e^{-2s}\mathcal L_s\{(t-3)^2\}=\left(\dfrac2{s^3}-\dfrac6{s^2}+\dfrac9s\right)e^{-2s}[/tex]

[tex]\mathcal L_s\{(t-5)^2u(t-5)\}=e^{-5s}\mathcal L_s\{t^2\}=\dfrac{2e^{-5s}}{s^3}[/tex]

[tex]\mathcal L_s\{2te^{6t}u(t-5)\}=\mathcal L_s\{2e^{30}(t-5)e^{6(t-5)}+10e^{30}e^{6(t-5)}\}[/tex]

[tex]=2e^{30-5s}\mathcal L_s\{te^{6t}+5e^{6t}\}=2e^{30-5s}\left(\dfrac1{(s-6)^2}+\dfrac5{s-6}\right)[/tex]

Putting everything together, we end up with

[tex]\boxed{\mathcal L_s\{f(t)\}=\dfrac{(2-6s)e^{-2s}-2e^{-5s}}{s^3}+\dfrac9s-\dfrac{2e^{30-5s}(29-5s)}{(s-6)^2}}[/tex]

Let [tex]\vec h[/tex] and [tex]\vec\eta[/tex] be two vectors in [tex]H[/tex].

[tex]H[/tex] is a subspace of [tex]V[/tex] if (1) [tex]\vec h+\vec\eta\in H[/tex] and (2) for any scalar [tex]k[/tex], we have [tex]k\vec h\in H[/tex].

(1) True;

[tex]\mathrm{tr}(\vec h+\vec\eta)=\mathrm{tr}(\vec h)+\mathrm{tr}(\vec eta)=0[/tex]

so [tex]\vec h+\vec\eta\in H[/tex].

(2) Also true, since

[tex]\mathrm{tr}(k\vec h)=0k=k[/tex]

Therefore [tex]H[/tex] is a subspace of [tex]V[/tex].

Answer: Yes, H is a subspace of V

Step-by-step explanation:

We know that V is the space of all the 2x2 matrices with real entries.

H is the set of all 2x2 matrices with real entries that have trace equal to 0.

Obviusly the matrices that are in the space H also belong in the space V (because in H you have some selected matrices and in V you have all of them). The thing we need to prove is if H is an actual subspace.

Suppose we have two matrices that belong to H, A and B.

We must see that:

1) if A and B ∈ H, then (A + B)∈H

2) for a scalar number k, k*A ∈ H

lets write this as:

[tex]A = \left[\begin{array}{ccc}a1&a2\\a3&a4\\\end{array}\right] B = \left[\begin{array}{ccc}b1&b2\\b3&b4\\\end{array}\right][/tex]

where a1 + a4 = 0 = b1 + b4

then:

[tex]A + B = \left[\begin{array}{ccc}a1 + b1&a2 + b2\\a3 + b3&a4 + b4\\\end{array}\right][/tex]

the trace is:

a1 + b1 - (a4 + b4) = (a1 - a4) + (b1 - b4) = 0

then the trace is nule, and (A + B) ∈ H

and:

[tex]kA = \left[\begin{array}{ccc}k*a1&k*a2\\k*a3&k*a4\end{array}\right][/tex]

the trace is:

k*a1 - k*a4 = k(a1 - a4) = 0

so kA ∈ H

then H is a subspace of V

Answer:

$1850.

Step-by-step explanation:

We are asked to find amount of interest earned in one year on an amount of $5000.

We will use simple interest formula to solve our given problem.

[tex]I=Prt[/tex], where,

I = Amount of interest earned,

r = Annual interest rate in decimal form.

t = Time in years.

Let us convert our given interest rate in decimal form.

[tex]37\%=\frac{37}{100}=0.37[/tex]

Upon substituting our given value in simple interest formula, we will get:

[tex]I=\$5000\times 0.37\times 1[/tex]

[tex]I=\$5000\times 0.37[/tex]

[tex]I=\$1850[/tex]

Therefore, an amount of $1850 is earned as interest in one year.

Answer:

Step-by-step explanation:

By definition of Laplace transform we have

L{f(t)} = [tex]L{{f(t)}}=\int_{0}^{\infty }e^{-st}f(t)dt\\\\Given\\f(t)=7t^{3}\\\\\therefore L[7t^{3}]=\int_{0}^{\infty }e^{-st}7t^{3}dt\\\\[/tex]

Now to solve the integral on the right hand side we shall use Integration by parts Taking [tex]7t^{3}[/tex] as first function thus we have

[tex]\int_{0}^{\infty }e^{-st}7t^{3}dt=7\int_{0}^{\infty }e^{-st}t^{3}dt\\\\= [t^3\int e^{-st} ]_{0}^{\infty}-\int_{0}^{\infty }[(3t^2)\int e^{-st}dt]dt\\\\=0-\int_{0}^{\infty }\frac{3t^{2}}{-s}e^{-st}dt\\\\=\int_{0}^{\infty }\frac{3t^{2}}{s}e^{-st}dt\\\\[/tex]

Again repeating the same procedure we get

[tex]=0-\int_{0}^{\infty }\frac{3t^{2}}{-s}e^{-st}dt\\\\=\int_{0}^{\infty }\frac{3t^{2}}{s}e^{-st}dt\\\\\int_{0}^{\infty }\frac{3t^{2}}{s}e^{-st}dt= \frac{3}{s}[t^2\int e^{-st} ]_{0}^{\infty}-\int_{0}^{\infty }[(t^2)\int e^{-st}dt]dt\\\\=\frac{3}{s}[0-\int_{0}^{\infty }\frac{2t^{1}}{-s}e^{-st}dt]\\\\=\frac{3\times 2}{s^{2}}[\int_{0}^{\infty }te^{-st}dt]\\\\[/tex]

Again repeating the same procedure we get

[tex]\frac{3\times 2}{s^2}[\int_{0}^{\infty }te^{-st}dt]= \frac{3\times 2}{s^{2}}[t\int e^{-st} ]_{0}^{\infty}-\int_{0}^{\infty }[(t)\int e^{-st}dt]dt\\\\=\frac{3\times 2}{s^2}[0-\int_{0}^{\infty }\frac{1}{-s}e^{-st}dt]\\\\=\frac{3\times 2}{s^{3}}[\int_{0}^{\infty }e^{-st}dt]\\\\[/tex]

Now solving this integral we have

[tex]\int_{0}^{\infty }e^{-st}dt=\frac{1}{-s}[\frac{1}{e^\infty }-\frac{1}{1}]\\\\\int_{0}^{\infty }e^{-st}dt=\frac{1}{s}[/tex]

Thus we have

[tex]L[7t^{3}]=\frac{7\times 3\times 2}{s^4}[/tex]

where s is any complex parameter

Answer:

276 feet

Step-by-step explanation:

The best way to do this is to complete the square, which puts the quadratic in vertex form. The vertex of a negative parabola, which is what this is, is the highest point of the function...the max value.  The k coordinate of the vertex will tell us that highest value.  To complete the square, we will first set the quadratic equal to 0, then move the constant over to the other side:

[tex]-16t^2+112t=-80[/tex]

The rule for completing the square is that the leading coefficient must be a positive 1.  Ours is a negative 16, so we have to factor out -16:

[tex]-16(t^2-7t)=-80[/tex]

Now the next thing is to take half the linear term, square it, and then add it to both sides.  Our linear term is 7, half of that is 7/2.  Squaring 7/2 gives you 49/4.  So we add 49/9 into the parenthesis on the left.  However, we can't forget that there is a -16 out front there that refuses to be ignored.  We have to add then (-16)(49/4) onto the right:

[tex]-16(t^2-7t+\frac{49}{4})=-80-196[/tex]

The purpose of this is to create a perfect square binomial that serves as the h value of the vertex (h, k).  Stating that perfect square on the left and doing the addition on the right:

[tex]-16(t-\frac{7}{2})^2=-276[/tex]

Now we finalize by moving the constant back over and setting it back equal to y:[tex]y=-16(t-\frac{7}{2})^2+276[/tex]

The vertex is [tex](\frac{7}{2},276)[/tex]

That translates to "at 3.5 seconds the particle is at its max height of 276 feet".

The correct answer is b)276 feet.

The standard form of a quadratic equation is[tex]H(t) = at^2 + bt + c.[/tex]For the given equation, a = -16, b = 112, and c = 80. The vertex (h, k) of the parabola can be found using the formula h = -b/(2a).

Let's calculate h:

h = -b/(2a) = -112 / (2 * -16) = -112 / -32 = 3.

Now we substitute h back into the original equation to find k, the maximum height:

k = H(h) = -16(3.5)^2 + 112(3.5) + 80

Calculating k:

k = -16(12.25) + 392 + 80

k = -196 + 392 + 80

k = 196 + 80

k = 276

So the maximum height is 276 feet, which corresponds to option 2.

Answer:

The length of a side of the square is 22 inches.

Step-by-step explanation:

Let each side of square be = s

Let each side of triangle be = s - 8

Perimeter of square, p₁ = 4s

Perimeter of triangle = p₂ = 3s

                                          = 3(s-8)

                                          = 3s - 24

Therefore, according to the question

p₁ - p₂ = 46

4s - (3s - 24) = 46

4s - 3s + 24 = 46

s = 46 - 24

s = 22

The length of a side of the square is 22 inches.