Solve the equation |x-2|-3=0 first by finding the zeros of
y=|x-2|-3 and then algebraically.

Answer:

A 99% confidence interval for the mean breaking strength of blocks produced is [tex][959.987, 1011.213][/tex]

Step-by-step explanation:

A (1 - [tex]\alpha[/tex])x100% confidence interval for the average break in these conditions It is an interval for the population mean with unknown variance and is given by:

[tex][\bar x -T_{(n-1,\frac{\alpha}{2})} \frac{S}{\sqrt{n}}, \bar x +T_{(n-1,\frac{\alpha}{2})} \frac{S}{\sqrt{n}}][/tex]

[tex]\bar X = 985.6psi[/tex]

[tex]n = 9[/tex]

[tex]\alpha = 0.01[/tex]

[tex]T_{(n-1,\frac{\alpha}{2})}=3.355[/tex]

[tex]S = 22.9[/tex]

With this information the interval is determined by:

[tex][985.6 - 3.355\frac{22.9}{\sqrt{9}}, [985.6 - 3.355\frac{22.9}{\sqrt{9}}] = [959.987, 1011.213] [/tex]

Answer with Step-by-step explanation:

We are  given that  a, b, c and x are elements in the group G.

We have to find the value of x in terms of a, b and c.

a.[tex]x^2a=bxc^{-1}[/tex]

[tex]x^2ac=bxc^{-1}c=bx[/tex]

[tex]x^{-1}x^2ac=x^{-1}bx=b[/tex] ([tex]x^{-1}bx=b[/tex])

[tex]xac=b[/tex]

[tex]xacc^{-1}=bc^{-1}[/tex]

[tex]xa=bc^{-1}[/tex]    ([tex]cc^{-1}=[/tex])

[tex]xaa^{-1}=bc^{-1}a^{-1}[/tex]

[tex]x=bc^{-1}a^{-1}[/tex]

b.[tex]acx=xac[/tex]

[tex]acxc^{-1}=xacc^{-1}=xa[/tex]  ([tex]cc^{-1}=1,cxc^{-1}=x[/tex])

[tex]axa^{-1}=xaa^{-1}[/tex]  ([tex]aa^{-1}=1,axa^{-1}=x[/tex])

[tex]x=x[/tex]

Identity equation

Hence, given equation has infinite solution and satisfied for all values of a and c.

[tex]y(t)=8\sin(bt)[/tex] has a period of [tex]\dfrac{2\pi}b[/tex], which is to say one "arch" of the curve occurs over the interval [tex]0\le t\le\dfrac\pi b[/tex].

a. Then the area under one such arch is

[tex]\displaystyle\int_0^{\pi/b}8\sin(bt)\,\mathrm dt[/tex]

b. Substitute [tex]u=bt[/tex], so that [tex]\dfrac{\mathrm du}b=\mathrm dt[/tex]. When [tex]t=0[/tex], [tex]u=0[/tex]; when [tex]t=\dfrac\pi b[/tex], [tex]u=\pi[/tex].

Then the integral is

[tex]\displaystyle\frac1b\int_0^\pi8\sin u\,\mathrm du[/tex]

The required area is [tex]\int^{\frac{\pi }{b}}_0 8sin(bt).dt\\[/tex]

The appropriate Substitution is [tex]\dfrac{1}{b} \int^\pi _08sinu.du[/tex]

Given that,

The area under one arch of the curve y(t) = 8sin(bt) for t ≥ 0 where b is a positive constant.

We have to find,

Set up the definite integral needed to find the area.

Make an appropriate substitution.

According to the question,

The area under one arch of the curve y(t) = 8sin(bt) for t ≥ 0 where b is a positive constant.

The area under one arch is given by,

[tex]Area = \int^{\frac{\pi }{b}}_0 8sin(bt).dt\\[/tex]

The required area is [tex]Area = \int^{\frac{\pi }{b}}_0 8sin(bt).dt\\[/tex]

Then,

[tex]\dfrac{du}{b} = dt \\\\when \ t=0, \ and \ u=0\\\\when\ t = \dfrac{\pi }{b}, u = \pi[/tex]

Then,

The required integral is ,

[tex]\dfrac{1}{b} \int^\pi _08sinu.du[/tex]

The appropriate Substitution is [tex]\dfrac{1}{b} \int^\pi _08sinu.du[/tex].

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Answer:

the force that must be applied by the driver to release the clutch is 21 lb

Step-by-step explanation:

Data provided:

clutch linkage on a vehicle has an overall advantage = 24:1

Applied force by the pressure plate = 504 lb

Now,

the advantage ratio is given as:

advantage ratio = [tex]\frac{\textup{Force applied by the pressure plate}}{\textup{Force applied by the driver}}[/tex]

on substituting the respective values, we get

[tex]\frac{\textup{24}}{\textup{1}}[/tex]  = [tex]\frac{\textup{504 lb}}{\textup{Force applied by the driver}}[/tex]

or

Force applied by the driver to release the clutch = [tex]\frac{\textup{504 lb}}{\textup24}}[/tex]

or

Force applied by the driver to release the clutch = 21 lb

Hence,

the force that must be applied by the driver to release the clutch is 21 lb

Using the mechanical advantage of the clutch linkage (24:1), the force the driver must apply to release the clutch is calculated to be 21 pounds.

The student has asked about the amount of force a driver must apply to release the clutch in a vehicle, given that the clutch linkage has an overall mechanical advantage of 24:1 and the pressure plate applies a force of 504lb. To find the force the driver needs to apply (Fdriver), we use the relationship provided by the mechanical advantage. Mechanical advantage (MA) is defined as the output force (Fout) divided by the input force (Fdriver). From this, we can formulate the equation MA = Fout / Fdriver, which can be rearranged to solve for the driver's force: Fdriver = Fout / MA.

Substituting the given values:

Fdriver = 504lb / 24

Fdriver = 21lb

Therefore, the driver must apply a force of 21 pounds to release the clutch.

Answer:

  Her sign is in error. The answer is -3/8.

Step-by-step explanation:

Nancy's answer has the correct magnitude. It is obtained by multiplying 1/4 by -3/2. However, the sign of that product will be negative. Nancy has reported a positive answer, so it is incorrect.

Answer:

Angle C=72º

Step-by-step explanation:

If two angles are congruent they are equals then angle A is 54º and angle B is 54º and the sum of the internal angles of a triangle is 180º then.

C=180º-54º-54º=72º

To know the value of on AC and BC we have to know the value of the other side AB, to find the values of the sides we can use the law of sines;

[tex]\dfrac{AB}{sin(72)}=\dfrac{BC}{sin(54)}=\dfrac{AC}{sin(54)}[/tex]

Answer:

The variable in this study is age.

Step-by-step explanation:

The variable in this study is Age, which has a  relationship  of cause and effect. Consequently,it is clear that  happiness does not depend on the passing of time , but on  the age of each group of people.  

Answer:

1) A.

2) No

Step-by-step explanation:

1 is A because the shaded line extends over numbers greater than -8 but less than -1. The reason it is greater than or EQUAL to -1 is because the dot above -1 is shaded in.

2 is No because in order to solve this equation, you plug in the numbers from the coordinates into the inequality. An ordered pair is always structured as (x,y), so in this case x = -1 and y = 4. To solve, the first step is to plug the numbers in, and you end up with 4< 2(-1) +5.

Then, simplify by adding and multiplying as needed. Now you will end up with 4<-2 +5. Simplify again. Finally you end up with 4<3. The reason the answer is NO, not a solution is because the statement '4<3' (four is less than three) is false. if the equation had ended up being 4>3, then it would have been true.

The height of the swimmer's feet in the air at time [tex]t[/tex] is given according to

[tex]y=1.20\,\mathrm m+\left(4.00\dfrac{\rm m}{\rm s}\right)t-\dfrac g2t^2[/tex]

where [tex]g[/tex] is the magnitude of the acceleration due to gravity (taken here to be 9.80 m/s^2).

a. Solve for [tex]t[/tex] when [tex]y=0[/tex]:

[tex]1.20\,\mathrm m+\left(4.00\dfrac{\rm m}{\rm s}\right)t-\dfrac g2t^2=0\implies\boxed{t=1.05\,\mathrm s}[/tex]

(The other solution is negative; ignore it)

b. At her highest point [tex]y_{\rm max}[/tex], the swimmer has zero velocity, so

[tex]-\left(4.00\dfrac{\rm m}{\rm s}\right)^2=-2g(y_{\rm max}-1.20\,\mathrm m)\implies\boxed{y_{\rm max}=2.02\,\mathrm m}[/tex]

c. Her velocity at time [tex]t[/tex] is

[tex]v=4.00\dfrac{\rm m}{\rm s}-gt[/tex]

After 1.05 s in the air, her velocity will be

[tex]v=4.00\dfrac{\rm m}{\rm s}-g(1.05\,\mathrm s)\implies\boxed{v=-6.29\dfrac{\rm m}{\rm s}}[/tex]

The swimmer's feet are in the air for approximately 0.816 seconds. Her highest point above the diving board is approximately .43 m. She hits the water with a velocity of approximately -8.00 m/s.

To answer these questions, we need to use physics equations that describe motion. The swimmer's motion can be broken down into two parts - the upward motion and the downward motion. Let's discuss each with respect to the provided variables.

To calculate the time, we can use the equation of motion given by: t = (v_f - v_i)/g where v_f is the final velocity (which is 0 at the highest point), v_i is the initial velocity (4.00 m/s), and g is the acceleration due to gravity (approx -9.81m/s²). The time taken for the upwards journey is: t = (0 - 4)/-9.81 ≈ 0.408 seconds. Since motion up and motion down take the same amount of time, we double this to get the total time: 2*0.408 = 0.816 seconds.

Let's use the equation h = v_i * t + 0.5*g*t², where h is the height, t is the time (0.408 seconds), g is the gravity (-9.81 m/s²), and v_i is the initial velocity (4.00 m/s). The highest point above the board is: h = 4*0.408 + 0.5*-9.81* (0.408)² = 1.63 m above the water surface or .43 m above the diving board.

Here, we can repurpose the equation v_f = v_i + g*t. Notice that the time here is the total time her feet were in the air (0.816 seconds). Using these values we get: v_f = 0 + (-9.81 * 0.816) = -8.00 m/s. She hits the water at a speed of 8.00 m/s.

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Answer:

(p ∧ q)’ ≡ p’ ∨ q’

Step-by-step explanation:

First, p and q have just four (4) possibilities, p∧q is true (t) when p and q are both t.

p ∧    q

t t t

t f f

f f t

f f f

next step is getting the opposite

(p∧q)'

    f

    t

    t

    t

Then we get p' V q', V is true (t) when the first or the second is true.

p' V  q'

f  f  f

f  t  t

t  t  f

t  t  t

Let's compare them, ≡ is true if the first is equal to the second one.

(p∧q)'   ≡     (p' V q')

    f              f

    t              t

    t              t

    t              t

Both are true, so

(p ∧ q)’ ≡ p’ ∨ q’

Final answer:

Using the given probabilities, we find that the probability is 0.965, or 96.5%.

Explanation:

To find the probability that a random person does not pass through the system and is without any security problems, we need to calculate the complement of two events: a person being a security hazard and the security system denying a person without security problems.

First, let's calculate the probability of a person being a security hazard:

Probability of a person being a security hazard = 0.02

Next, let's calculate the probability of the security system denying a person without security problems:

Probability of the security system denying a person without security problems = 1.5% = 0.015

To find the probability that a person does not pass through the system and is without any security problems, we can use the formula:

Probability = (1 - probability of being a security hazard) * (1 - probability of the security system denying a person without security problems)

Probability = (1 - 0.02) * (1 - 0.015)

Probability = 0.98 * 0.985

Probability = 0.9653

Therefore, the probability that a random person does not pass through the system and is without any security problems is 0.965, or 96.5% (rounded to 3 decimal places).

Answer:

The solution to this set of linear equations is:

[tex]x=-\frac{1}{3}\\y=-\frac{2}{3}[/tex]

Step-by-step explanation:

This is a system of two equations with two unknown variables x and y, let's call them

Equation 1: [tex]9x-3y=-1[/tex]

Equation 2: [tex]\frac{1}{5}x+\frac{2}{5}y=-\frac{1}{3}[/tex]

The first step is to solve Equation 1 for y, this means to leave the y alone on one side of the equal

[tex]y= 3x+\frac{1}{3}[/tex]

Then with this equation, you can find the value of x by replacing y in Equation 2

[tex]\frac{1}{5}x+\frac{2}{5}(3x+\frac{1}{3})=-\frac{1}{3}[/tex]

Then simplify this equation to find x

[tex]\frac{1}{5}x+\frac{6}{5}x+\frac{2}{15}=-\frac{1}{3}[/tex]

[tex]\frac{1}{5}x+\frac{6}{5}x=-\frac{1}{3}-\frac{2}{15}[/tex]

[tex]\frac{7}{5}x=-\frac{5}{15}-\frac{2}{15}[/tex]

[tex]\frac{7}{5}x=-\frac{7}{15}[/tex]

Now you solve for x

[tex]x=-\frac{1}{3}[/tex]

Now you use this value of x to find y

[tex]y=3(-\frac{1}{3})+\frac{1}{3}\\y=-\frac{2}{3}[/tex]

You can check if this answer is correct by replacing the values of x and y into Equation 1 or 2, in this case, let's take Equation 1:

[tex]9(-\frac{1}{3})-3(-\frac{2}{3})=-1\\-3+2=-1\\-1=-1\\[/tex]

Answer:

6000000 sq yd

Step-by-step explanation:

Data provided in the question:

Length of the fencing = 7000 yd

let the perpendicular sides be 'P'

and the length parallel to the river be 'L'

according to the given question

L = P + 2500  ............(1)

also,

Length to be fenced  = 2P + L

thus,

2P + L = 7000  ...........(2)

substituting L from (1), we get

2P + P + 2500 = 7000  

or

3P = 7000 - 2500

or

3P = 4500

or

P = 1500 yd

Thus,

L = 1500 + 2500 = 4000 yd

Therefore,

the area of the rectangular land = L × P = 4000 × 1500 = 6000000 sq yd

Answer:

Area of land = 6000000 sq yd

Step-by-step explanation:

Given,

length of fencing= 7000 yd

Let's assume that the length of the land parallel to the river is l and the breadth of the land perpendicular to the river is b.

Then, it is given that

    l = b +2500

Since, there is no need of fencing along the river so, we can write

   l +2b = 7000

=>b+2500 = 7000

=> b = 7000-2500

        = 4000

As the area of rectangular land can be given as

A = length x breadth

   = 4000 x 2500 sq yd

   = 6000000 sq yd

So, the area of the enclosed land will be 6000000 sq yd.

Answer:

(a) [tex]\frac{7}{8}[/tex]

(b) [tex]\frac{9}{10}[/tex]

Step-by-step explanation:

Given,

P(A) = 0.5 ⇒ [tex]P(A^c)=1-P(A) = 1 - 0.5 = 0.5[/tex]

P(B) = 0.6 ⇒ [tex]P(B^c)=1-P(B) = 1 - 0.6 = 0.4[/tex]

P(A∩B) = 0.15

∵ [tex]P(A\cap B^c)=P(A) - P(A\cap B) = 0.5 - 0.15 = 0.35[/tex]

Similarly,

[tex]P(B\cap A^c)=P(B) - P(B\cap A) = 0.6 - 0.15 = 0.45[/tex]

Now,

(a) [tex]P(\frac{A}{B^c})=\frac{P(A\cap B^c)}{P(B^c)}=\frac{0.35}{0.4}=\frac{35}{40}=\frac{7}{8}[/tex]

(b) [tex]P(\frac{B}{A^c})=\frac{P(B\cap A^c)}{P(A^c)}=\frac{0.45}{0.5}=\frac{45}{50}=\frac{9}{10}[/tex]

Answer:

The population increased by 34.69% over 20 years.

Step-by-step explanation:

It is given that the population of dolphins increases at a constant rate of 1.5% every year for 20 years.

Formula for population increase:

[tex]P=a(1+r)^t[/tex]

where, a is initial population, r is growth rate and t is time in years.

If the population of dolphins increases at a constant rate of 1.5% every year for 20 years, then the population after 20 years is

[tex]P=a(1+0.015)^{20}[/tex]

[tex]P=a(1.015)^{20}[/tex]

[tex]P=1.346855a[/tex]

Where, a is the initial population.

The total percentage increase over the 20 years is

[tex]\% change=\frac{P-a}{a}\times 100[/tex]

where, P is population after 20 years and a is initial amount.

[tex]\% change=\frac{1.346855a-a}{a}\times 100[/tex]

[tex]\% change=\frac{0.346855a}{a}\times 100[/tex]

[tex]\% change=0.346855\times 100[/tex]

[tex]\% change=34.6855[/tex]

[tex]\% change\approx 34.69[/tex]

Therefore the population increased by 34.69% over 20 years.

[tex]\underbrace{\begin{bmatrix}-2&-1&2\\-2&2&3\\-4&1&3\end{bmatrix}}_A\underbrace{\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}}_x=\underbrace{\begin{bmatrix}-1\\-1\\4\end{bmatrix}}_b[/tex]

Multiply [tex]A[/tex] on the left side with the following elimination matrix [tex]E_1[/tex]:

[tex]\underbrace{\begin{bmatrix}1&0&0\\-1&1&0\\-2&0&1\end{bmatrix}}_{E_1}A=\begin{bmatrix}-2&-1&2\\0&3&1\\0&3&-1\end{bmatrix}[/tex]

Multiply [tex]E_1A[/tex] on the left by another elimination matrix [tex]E_2[/tex]:

[tex]\underbrace{\begin{bmatrix}1&0&0\\0&1&0\\0&-1&1\end{bmatrix}}_{E_2}(E_1A)=\begin{bmatrix}-2&-1&2\\0&3&1\\0&0&-2\end{bmatrix}[/tex]

[tex]\implies\boxed{U=\begin{bmatrix}-2&-1&2\\0&3&1\\0&0&-2\end{bmatrix}}[/tex]

Multiply on the left by the inverse of [tex]E_2E_1[/tex]:

[tex](E_2E_1)^{-1}(E_2E_1)A=(E_2E_1)^{-1}U[/tex]

[tex]A=\underbrace{({E_1}^{-1}{E_2}^{-1})}_LU[/tex]

We have

[tex]{E_1}^{-1}=\begin{bmatrix}1&0&0\\1&1&0\\2&0&1\end{bmatrix}[/tex]

[tex]{E_2}^{-1}=\begin{bmatrix}1&0&0\\0&1&0\\0&1&1\end{bmatrix}[/tex]

[tex]\implies\boxed{L=\begin{bmatrix}1&0&0\\1&1&0\\3&1&1\end{bmatrix}}[/tex]

1.5x + 1.3x = - 8.4

2.8x = -8.4

x= - 8.4/2.8

x = 3

On simplification of liner equation 1.5x + 1.3x = -8, we get x = -3.

To solve the linear equation 1.5x + 1.3x = -8.4, we need to start by combining like terms.

Both terms on the left side of the equation have the variable x, so we can add them together.

1.5x + 1.3x = 2.8x.

2.8x = -8.4.

To find the value of x, we need to isolate the variable by dividing both sides of the equation by 2.8.

Divide both sides by 2.8:

x = -8.4 / 2.8.

x = -3.

Answer:

Because every “x” value has two “y” values.

Step-by-step explanation:

In the graph of a function every value of x has one and only one value of y. So, if we draw a straight line which is parallel to y-axis and it cuts the graph in only one point, this graph will correspond to a function.

Answer:  40000

Step-by-step explanation:

The formula to find the sample size is given by :-

[tex]n=p(1-p)(\dfrac{z_{\alpha/2}}{E})^2[/tex], where p is the prior estimate of the population proportion.

Here we can see that the sample size is inversely proportion withe square of margin of error.

i.e. [tex]n\ \alpha\ \dfrac{1}{E^2}[/tex]

By the equation inverse variation, we have

[tex]n_1E_1^2=n_2E_2^2[/tex]

Given : [tex]E_1=0.05[/tex]   [tex]n_1=1000[/tex]

[tex]E_2=0.025[/tex]

Then, we have

[tex](1000)(0.05)^2=n_2(0.025)^2\\\\\Rightarrow\ 2.5=0.000625n_2\\\\\Rightarrow\ n_2=\dfrac{2.5}{0.000625}=4000[/tex]

Hence, the sample size will now have to be 4000.

The new sample size will have to be approximately 4000.

The formula to calculate the sample size (n) needed to estimate a population proportion with a given maximum allowable error (E) and confidence level (usually 95% or 1.96 standard deviations for a two-tailed test) is given by:

[tex]\[ n = \left(\frac{z \times \sigma}{E}\right)^2 \][/tex]

Given that the original maximum allowable error was 0.05 and the sample size calculated was 1000, we can set up the equation:

[tex]\[ 1000 = \left(\frac{1.96 \times 0.5}{0.05}\right)^2 \][/tex]

Now, we want to find the new sample size when the maximum allowable error is reduced to 0.025. The new sample size can be calculated by:

[tex]\[ n_{new} = \left(\frac{z \times \sigma}{E_{new}}\right)^2 \][/tex]

Since \( z \) and[tex]\( \sigma \)[/tex] remain constant, and only \( E \) changes, the relationship between the original sample size and the new sample size is inversely proportional to the square of the ratio of the original error to the new error:

[tex]\[ n_{new} = n_{old} \times \left(\frac{E_{old}}{E_{new}}\right)^2 \] \[ n_{new} = 1000 \times \left(\frac{0.05}{0.025}\right)^2 \] \[ n_{new} = 1000 \times \left(\frac{0.05}{0.025}\right)^2 \] \[ n_{new} = 1000 \times \left(2\right)^2 \] \[ n_{new} = 1000 \times 4 \] \[ n_{new} = 4000 \][/tex]

Therefore, the new sample size will have to be approximately 4000 to reduce the maximum allowable error to 0.025."

Answer:

The augmented matrix is [tex]\left[\begin{array}{ccc|c}-2&-1&2&-1\\-2&2&3&-1\\-4&1&3&4\end{array}\right][/tex]

The Reduced Row Echelon Form of the augmented matrix is [tex]\left[\begin{array}{cccc}1&0&0&-3\\0&1&0&1\\0&0&1&-3\end{array}\right][/tex]

The rank of matrix (A|B) is 3

The system is consistent and the solutions are [tex]x_{1}= -3, x_{2} = 1, x_{3}= -3[/tex]

Step-by-step explanation:

We have the following information:

[tex]A=\left[\begin{array}{ccc}-2&-1&2\\-2&2&3\\-4&1&3\end{array}\right], X=\left[\begin{array}{c}x_{1}&x_{2}&x_{3}\end{array}\right] and \:B=\left[\begin{array}{c}-1&-1&4\end{array}\right][/tex]

    1. The augmented matrix is

We take the matrix A and we add the matrix B we use a vertical line to separate the coefficient entries from the constants.

[tex]\left[\begin{array}{ccc|c}-2&-1&2&-1\\-2&2&3&-1\\-4&1&3&4\end{array}\right][/tex]

    2. To transform the augmented matrix to the Reduced Row Echelon Form (RREF) you need to follow these steps:

[tex]\left[\begin{array}{cccc}1&1/2&-1&1/2\\-2&2&3&-1\\-4&1&3&4\end{array}\right][/tex]

[tex]\left[\begin{array}{cccc}1&1/2&-1&1/2\\0&3&1&0\\-4&1&3&4\end{array}\right][/tex]

[tex]\left[\begin{array}{cccc}1&1/2&-1&1/2\\0&3&1&0\\0&3&-1&6\end{array}\right][/tex]

[tex]\left[\begin{array}{cccc}1&1/2&-1&1/2\\0&1&1/3&0\\0&3&-1&6\end{array}\right][/tex]

[tex]\left[\begin{array}{cccc}1&1/2&-1&1/2\\0&1&1/3&0\\0&0&-2&6\end{array}\right][/tex]

[tex]\left[\begin{array}{cccc}1&1/2&-1&1/2\\0&1&1/3&0\\0&0&1&-3\end{array}\right][/tex]

[tex]\left[\begin{array}{cccc}1&1/2&-1&1/2\\0&1&0&1\\0&0&1&-3\end{array}\right][/tex]

[tex]\left[\begin{array}{cccc}1&1/2&0&-5/2\\0&1&0&1\\0&0&1&-3\end{array}\right][/tex]

[tex]\left[\begin{array}{cccc}1&0&0&-3\\0&1&0&1\\0&0&1&-3\end{array}\right][/tex]

    3. What is the rank of (A|B)

To find the rank of a matrix, we simply transform the matrix to its row echelon form and count the number of non-zero rows.

Because the row echelon form of the augmented matrix has three non-zero rows the rank of matrix (A|B) is 3

   4. Solutions of the system

This definition is very important: "A system of linear equations is called inconsistent if it has no solutions. A system which has a solution is called consistent"

This system is consistent because from the row echelon form of the augmented matrix we find that the solutions are (the last column of a row echelon form matrix always give you the solution of the system)

[tex]x_{1}= -3, x_{2} = 1, x_{3}= -3[/tex]

Answer:

Since,

[tex]|x|=\left\{\begin{matrix}x &\text{ if } x \geq 0 \\ -x &\text{ if } x < 0\end{matrix}\right.[/tex]

Here, the given equation is,

|a| < b

Case 1 : if a ≥ 0,

|a| < b ⇒ a < b

Case 2 : If a < 0,

|a| < b ⇒ -a < b ⇒ a > - b

( Since, when we multiply both sides of inequality by negative number then the sign of inequality is reversed. )

|a| < b ⇒ a < b or a > - b ⇒ -b < a < b

Conversely,

If -b < a < b

⇒ a < b or a > - b

⇒ a < b or -a <  b

⇒ |a| < b

Hence, proved..

Answer:

Option E - None of these answers is reasonable.

Step-by-step explanation:

To find : Convert 17.42 m to customary units ?

Solution :

The customary units is defined as the measure length and distances in the customary system are inches, feet, yards, and miles.

The options belong to feet and inches.

We have to convert meter into inches, feet.

Meter into feet,

[tex]1 \text{ feet} = 0.3048 \text{ meter}[/tex]

[tex]1 \text{ meter} = \frac{1}{0.3048}\text{ feet}[/tex]

[tex]17.42 \text{ meter} = \frac{17.42}{0.3048}\text{ feet}[/tex]

[tex]17.42 \text{ meter} = \frac{174200}{3048}\text{ feet}[/tex]

[tex]17.42 \text{ meter} =57 \frac{464}{3048}\text{ feet}[/tex]

[tex]17.42 \text{ meter} =57 \frac{58}{381}\text{ feet}[/tex]

Now, Feet into inches

[tex]1 \text{ feet} = 12\text{ inches}[/tex]

[tex] \frac{58}{381} \text{ feet} = 12\times \frac{58}{381}\text{ inches}[/tex]

[tex] \frac{58}{381} \text{ feet} =\frac{232}{381}\text{ inches}[/tex]

i.e.  [tex]17.42 \text{ meter} =57\text{ feet }\frac{232}{381}\text{ inches}[/tex]

or [tex]17.42 \text{ meter} =57'\frac{232}{381}''[/tex]

None of these answers is reasonable.

Therefore, Option E is correct.

Answer:

The answer is:  {x | x is a day of the week}

Step-by-step explanation:

The mathematical expression {x | x is a day of the week}, uses set theory notation and can be translated as: The set of all x such that x is a day of the week. Since the original set contains all days of the week, therefore it is equivalent to the expression  {x | x is a day of the week}.

Final answer:

The correct equivalent set to { Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday} is {x | x is a day of the week}, as both sets contain the days of the week.

Explanation:

The student has asked which option is equivalent to the set {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}. This set represents the seven days of the week. Therefore, the correct option that is equivalent to this set is {x | x is a day of the week}. All other options listed represent different sets with no connection to the days of the week. When comparing sets for equivalence, we look for a one-to-one correspondence between the members of each set, which is only present in the option directly referring to days of the week.

Answer:

h(t)=-1/3(x)+50/3

h intercept is (0,50/3)

t intercept is (50,0)

Step-by-step explanation:

Find the slope of the table by using the slope formula then plug in to y-y1=m(x-x1) then solve for y this gives you the formula

sub in y =0 for the x intercept

sub in x=0 for the y intercept

Answer:

The angles between the curves at the points of intersection are:

0º, 1.3º

Step-by-step explanation:

The intersections points are found by setting the equations equal to each other and solving the resulting equation:

[tex]7x^2=7x^3\\x^3-x^2=0\\x^2(x-1)=0\\x=0,x=1[/tex]

The angles of the tangent lines can be found by stating their slopes.

To find the slope we differentiate the equations:

[tex]y'_1=14x,y'_2=21x^2[/tex]

Then we plug the x-coordinates of the intersections:

For x=0 we get the slopes are both 0:

[tex]y'_1=14(0)=0,y'_2=21(0)^2=0[/tex]

So the angles of inclination of the lines are the same their difference is 0. Hence the angle  between the tangent curves is also 0º at the point of intersection at x=0

For x=1 we get the following slopes:

[tex]y'_1=14(1)=14,y'_2=21(1)^2=21[/tex]

The slopes are the tangents of the angles. Therefore, to get the angle between the lines we do:

[tex]arctan(21)-arctan(14)\approx87.2737\º-85.9144\º\approx1.3\º[/tex]

So, 1.3º is the angle between the curves at the second point of intersection at x=1.

Answer:

[tex]-5[/tex], [tex]-4.5[/tex], [tex]-\frac{5}{4}[/tex], [tex]-\frac{4}{5}[/tex], [tex]-0.54[/tex] and [tex]-0.4[/tex].

Step-by-step explanation:

We are asked to write the given numbers from least to greatest.

-4/5, -5/4, -4.5, -0.54, -5, -0.4

We know that the more negative number has least value.

Let us convert each number into decimal.

[tex]-\frac{4}{5}=-0.8[/tex]

[tex]-\frac{5}{4}=-1.25[/tex]

We can see that -5 is most negative, so it will be least.

Order from more negative to less negative:

[tex]-5[/tex], [tex]-4.5[/tex], [tex]-1.25[/tex], [tex]-0.8[/tex], [tex]-0.54[/tex] and [tex]-0.4[/tex].

Therefore, the least to greatest numbers would be [tex]-5[/tex], [tex]-4.5[/tex], [tex]-\frac{5}{4}[/tex], [tex]-\frac{4}{5}[/tex], [tex]-0.54[/tex] and [tex]-0.4[/tex].

Answer:

-7 feet

Step-by-step explanation:

To find the difference in elevation between the diver and the school fish SUBTRACT the elevation of the diver from that of the fish

i.e. difference in elevation = -12 - (-5)

= -12 + 5

= -7 feet

Final answer:

The difference in elevation between the diver at -5 feet and the school of fish at -12 feet is 7 feet, calculated by taking the absolute value of their elevations' difference.

Explanation:

The question asks for the difference in elevation between a diver and a school of fish, with the diver at -5 feet and the fish at -12 feet relative to sea level. To find the difference in elevation, you subtract the diver's elevation from the fish's elevation.

Here is the calculation:

The absolute value is used because we are interested in the positive difference in elevation, which is the distance between the two elevations regardless of direction.

Therefore, the difference in elevation between the diver and the school of fish is 7 feet.

Answer:

147

Step-by-step explanation:

Given:

Progesterone =  3.8 g

Glycerin = 7 mL

2% methylcellulose solution 50 mL

Cherry syrup ad = 90 mL

Now,

The total volume of the solution = 7 + 50 + 90 = 147 mL

Also,

2% methylcellulose solution 50 mL is concluded as:

the volume of  methylcellulose in the solution is 2% of the total volume of the solution

thus,

volume of methylcellulose = 0.02 × 50 mL = 1 mL

Therefore,

Ratio strength of methylcellulose in the finished product

=[tex]\frac{\textup{volume of methylcellulose}}{\textup{ total volume of the solution}}[/tex]

or

= [tex]\frac{\etxtup{1}}{\textup{ 147}}[/tex]

Hence, the answer according to the question is 147

Answer:

3222212.11113

Step-by-step explanation:

First, you should take care of the fractional separator (the dot) so we split the problem in two parts: one for the integer and other for the fractional part.

Since 4 is a power of 2, we can just take two digits from the orignal number and asign it to its corresponding number in base 4:

[tex]\left[\begin{array}{cc}Binary&Base 4\\00&0\\01&1\\10&2\\11&3\end{array}\right][/tex]

Start with the fractional part from the fractional point to the right:

[tex]\left[\begin{array}{ccccc}01&01&01&01&11\\1&1&1&1&3\end{array}\right][/tex]

Then do the same to the integer part starting from the fractional point to the left.

[tex]\left[\begin{array}{ccccccc}11&10&10&10&10&01&10\\3&2&2&2&2&1&2\end{array}\right][/tex]

By joining them together, we obtain the response.

Answer:

When we change a decimal into fraction, then we follow the following steps,

Step 1 : first we write the decimal number with the denominator 1,

Step 2 : Multiply numerator by 10s ( eg 10, 100, 100 etc ) for omitting decimal.

Step 3 : Multiply the denominator ( i.e 1 ) by the same number,

Step 4 : Reduce the fraction in the simplest form if possible by dividing both numerator and denominator by the HCF of numerator and denominator.

a)

[tex]0.25 =\frac{0.25}{1}=\frac{0.25\times 100}{100}=\frac{25}{100}=\frac{25\div 25}{100\div 25}=\frac{1}{4}[/tex]

b)

[tex]0.08 =\frac{0.08}{1}=\frac{0.08\times 100}{100}=\frac{8}{100}=\frac{8\div 4}{100\div 4}=\frac{2}{25}[/tex]

c)

[tex]0.400 =\frac{0.4}{1}=\frac{0.4\times 10}{10}=\frac{4}{10}=\frac{4\div 2}{10\div 2}=\frac{2}{5}[/tex]

d)

[tex]1.1 =\frac{1.1}{1}=\frac{1.1\times 10}{10}=\frac{11}{10}[/tex]

e)

[tex]3.5 =\frac{3.5}{1}=\frac{3.5\times 10}{10}=\frac{35}{10}=\frac{35\div 5}{10\div 5}=\frac{7}{2}[/tex]

Final answer:

To convert decimals into simplified fractions: 0.25 is 1/4, 0.08 is 2/25, 0.400 is 2/5, 1.1 is 11/10 and 3.5 is 7/2. Numbers in scientific notation are written in decimal form by adjusting the decimal point. When rounding to three significant figures, ensure only the first three digits after the leading non-zero digit are kept.

Explanation:

When converting decimals to fractions and simplifying them, it's important to consider the place value of the decimal. Here's how you would convert and simplify the provided decimals:

0.25 can be written as 25/100, which simplifies to 1/4.

0.08 is 8/100, which simplifies to 1/12.5 or 2/25 when expressed as a simplified fraction.

0.400 is 400/1000, which simplifies to 2/5.

1.1 is equivalent to 11/10 or 1 1/10 in mixed number form.

3.5 equals 35/10, which simplifies to 7/2 or 3 1/2 in mixed number form.

For scientific notation, numbers are converted to their decimal forms by moving the decimal point:

5.65 x 10-3 means the decimal point is moved 3 places to the left, giving 0.00565.

9.25 x 10-4 means the decimal point is moved 4 places to the left, resulting in 0.000925.

To write numbers in scientific notation:

4500 becomes 4.5 x 103.

2220000 turns into 2.22 x 106.

0.0035 is 3.5 x 10-3.

0.7 can be written as 7 x 10-1.

858.67 is expressed as 8.5867 x 102.

When rounding to three significant figures:

0.0004505 becomes 4.51 x 10-4 (count starts from the first non-zero digit).

0.00045050 also rounds to 4.51 x 10-4.

For 7.210 x 106, it remains unchanged as it already has three significant figures.

5.00 x 10-6 stays the same, with three significant figures present.