Find the point 3x - 7x₂ =5. of the lines x, - 5x2 = 1 and of intersection

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:

(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’

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.

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:

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.

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).

[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.

[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]

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 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.

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:

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:

  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:

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.

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:

-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:

[tex]\sqrt2[/tex] is irrational

Step-by-step explanation:

Let us assume that [tex]\sqrt2[/tex] is rational. Thus, it can be expressed in the form of fraction [tex]\frac{x}{y}[/tex], where x and y are co-prime to each other.

[tex]\sqrt2[/tex] = [tex]\frac{x}{y}[/tex]

Squaring both sides,

[tex]2 = \frac{x^2}{y^2}[/tex]

Now, it is clear that x is an even number. So, let us substitute x = 2u

Thus,

[tex]2 = \frac{(2u)^2}{y^2}\\y^2 = 2u^2[/tex]

Thus, [tex]y^2[/tex]is even, which follows the fact that y is also an even number. But this is a contradiction as x and y have a common factor that is 2 but we assumed that the fraction [tex]\frac{x}{y}[/tex]  was in lowest form.

Hence, [tex]\sqrt2[/tex] is not a rational number. But [tex]\sqrt2[/tex] is a an irrational number.

Step-by-step explanation:

When two dies are tossed, possible outcomes are (1,1) (1,2) ,(1,3) (1,4), (1,5) (1,6),(2,1) (2,2) (2,3) (2,4) (2,5) (2,6), (3,1)(3,2)(3,3)(3,4),(3,5)(3,6)(4,1) (4,2),(4,3)(4,4)(4,5)(4,6)(5,1)(5,2)(5,3)(5,4)(5,5)(5,6)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)

so the total sample space =36

From de possible outcome, probability of getting a die that five does not appear on either side is given by total outcome of not getting five from either side/sample space

=30/36 = 5/6

And also probability of getting the difference of numbers to be 2 are (3,1)(4,2) (5,3) (6,4) =4outcomes

the probability of getting a difference of 2 is given by outcome/sample space 4/36 = 1/9

probability of getting the summation of two numbers to be equal to or greater than ten are; Outcome/sample space.

the outcome are (4,6) (5,5) (5,6)(6,4)(6,5)and (6,6)= 6outcomes

=6/36 =1/12.

The probabilities for the three events are approximately 0.6944, 0.2222, and 0.1667 respectively. These figures were achieved by comparing the number of favorable outcomes to the total number of outcomes when rolling two dice.

The subject of this question is probability which involves using numbers for calculation. When you roll two dice, there are 36 possible outcomes.

Event C: The sum of the numbers is 10 or more happens in 6 cases (5 and 5, 6 and 4, 4 and 6, 6 and 5, 5 and 6, 6 and 6). So the probability is 6/36, which simplifies to 1/6 or around 0.1667 in decimal approximation.

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The smallest of five consecutive numbers, whose sum equals 320, is 62. This is determined by setting up an equation where the sum of these numbers equals 320, then solving for the smallest number 'n'.

The subject at hand involves understanding patterns within consecutive numbers (numbers that follow each other in order, without gaps). In the given example of four consecutive even numbers 2, 4, 6 and 8, we see that the difference between them is constantly 2.

When it comes to the sum of five consecutive numbers equating to 320, let's presume the first (and smallest) number is 'n'. Therefore, the consecutive numbers would be n, (n+1), (n+2), (n+3) and (n+4). Their sum ought to total 320, so we write the expression n + (n+1) + (n+2) + (n+3) + (n+4) = 320. By simplifying, we receive 5n+10 = 320.

Further simplifying, we subtract 10 from both sides to afford: 5n = 310. Divide 310 by 5 to isolate 'n', which results in 'n' equals 62. Consequently, the smallest of the five consecutive numbers is 62.

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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.

https://brainly.com/question/33851452

Answer:

Step-by-step explanation:

A steam and leaf plot is the arrangement of numerical data into different groups with place value.  For eg, 17,20,21 is shown as

stem  leaf

1         7

2       0,1

A histogram is a bar chart that described frequency distribution.

A stem and leaf plot displays more information than a histogram.

Hence we have the correct answers are:

A stem-and-leaf display describes the individual observations.

A stem-and-leaf display has slightly more information than a histogram.

A stem-and-leaf display provides detailed individual data points and their distribution, while a histogram offers aggregated data into bins, showing the overall data distribution without individual details.

The differences between a histogram and a stem-and-leaf display are significant in how they present data. A stem-and-leaf display retains the individual data values and is beneficial for small datasets, showing the exact values and the frequency of data for each "stem" which provides a clear view of the distribution shape. On the contrary, a histogram groups data into contiguous bins, providing a visual representation of data distribution, showing the spread and most frequent values but without detailing individual data points. Therefore, a stem-and-leaf display has slightly more information than a histogram because it describes individual observations, unlike a histogram that aggregates data into bins.

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:

[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:  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:

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 number of books in the box is 60.

Step-by-step explanation:

Since it is given that the number of books in the box is divisible by 2,3,4,5 and 6.

So, the number of books in the box is multiple of these numbers.

Thus we have to find Least Common Multiple (L.C.M.) of these number

L.C.M. of (2,3,4,5,6) = 60

Thus the number of books in the box is multiples of 60 i.e. 60, 120, 180, 240,... etc.

The other factor that can we add in statement so there is only one possible solution is: "The number of books in the box is smallest number divisible by 2,3,4,5 and 6".

Answer:

The number is 60

Step-by-step explanation:

So the first way to solve this would be to multiply the greatest numbers in the sequence, you have 5 and 6, the result is 30, since 30 is not divisible by 4 you need to find the next number that is divisible by 5 and 6, that would be 60, since 60 is divisible by 4, then that is the answer, 60 is the first number that is divisible by 2, 3, 4, 5, and 6.

Answer:  [tex]11.7<\mu<15.9[/tex]

Step-by-step explanation:

Given : Significance level : [tex]\alpha:1-0.95=0.05[/tex]

Sample size : n=45

Critical value : [tex]z_{\alpha/2}=1.96[/tex]

Sample mean : [tex]\overline{x}=13.8\text{ years}[/tex]

Standard deviation : [tex]\sigma=7.3\text{ years}[/tex]

The confidence interval for population mean is given by :-

[tex]\overline{x}\pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}\\\\=13.8\pm(1.96)\dfrac{7.3}{\sqrt{45}}\\\\\approx13.8\pm2.1\\\\=(13.8-2.1,\ 13.8-2.1)=(11.7,\ 15.9)[/tex]

Hence, the 95% confidence interval estimate for the average number of years served by all Supreme Court justices is [tex]11.7<\mu<15.9[/tex]

Answer:

$12000 should be invested in a fund with a 4% return.

Step-by-step explanation:

Consider the provided information.

An amount of $15,000 is invested in a fund that has a return of  6%.

We need to calculate how much money is invested in a fund with a 4% return if the  total return on both investments is $1380.

Let $x should be invested in a fund with a 4% return.

The above information can be written as:

[tex]1380=15000\times 6\%+x\times 4\%[/tex]

[tex]1380=15000\times \frac{6}{100}+x\times \frac{4}{100}[/tex]

[tex]1380=150\times6+x\times 0.04[/tex]

[tex]1380-900=0.04x\\480=0.04x\\x=12000[/tex]

Hence, $12000 should be invested in a fund with a 4% return.