A motorboat takes 5 hours to travel 150km going upstream. The return trip takes 3 hours going downstream. What is the rate of the boat in still water and what is the rate of the current?

Answer:

Step-by-step explanation:

The factor-of-ten relationship between the different units means we can combine the numbers in decimal fashion. If 1 unit is 1 zhang, then 1 chi is 0.1 zhang and 1 cun is 0.01 zhang. Hence 6 chi 8 cun is 0.68 zhang.

Let x and y represent the width and height, respectively. In terms of zhang, we have ...

  y - x = 0.68

  x^2 +y^2 = 1^2

Substituting y = 0.68 +x into the second equation gives ...

  x^2 + (x +0.68)^2 = 1

  2x^2 +1.36x - 0.5376 = 0 . . . . . eliminate parentheses, subtract 1

Using the quadratic formula, we have ...

  x = (-1.36 ±√(1.36^2 -4(2)(-0.5376)))/(2·2) = (-1.36 ±√6.1504)/4

  x = 0.28 . . . . . the negative root is of no interest

  y = 0.28 +0.68 = 0.96

In units of chi and cun, the dimensions are ...

  height: 9 chi 6 cun

  width: 2 chi 8 cun

Answer:

Height: 9 chi 6 cun

Width: 2 chi 8 cun

Step-by-step explanation:

Now there is a door whose height is more than its width by 6 chi 8 cun. The distance between the [opposite] corners is 1 zhang. The height of the door is  9 chi 6 cun  and the width of the door is 2 chi 8 cun.

1 zhang = 10 chi = 100 cun

The rate at which the area of the square is increasing when the area of the square is [tex]\( 81 \, \text{cm}^2 \) is \( 90 \, \text{cm}^2/\text{s} \)[/tex].

Let x represent the length of a side of the square, and A represent the area of the square.

We know that the area A of a square is given by [tex]\( A = x^2 \).[/tex]

Differentiating both sides with respect to time t, we get:

[tex]\[ \frac{dA}{dt} = 2x \frac{dx}{dt} \][/tex]

Given that each side of the square is increasing at a rate of [tex]\( 5 \, \text{cm/s} \), we have \( \frac{dx}{dt} = 5 \, \text{cm/s} \).[/tex]

We are asked to find the rate at which the area of the square is increasing when the area of the square is [tex]\( 81 \, \text{cm}^2 \)[/tex], so we need to find [tex]\( \frac{dA}{dt} \) when \( A = 81 \)[/tex].

We can find x when A=81:

[tex]\[ A = x^2 \]\[ 81 = x^2 \]\[ x = 9 \][/tex]

Now, plug in x=9 and [tex]\( \frac{dx}{dt} = 5 \)[/tex] into the equation for  [tex]\[ \frac{dA}{dt} = 2x \frac{dx}{dt} \][/tex]

[tex]\[ \frac{dA}{dt} = 2(9)(5) \][/tex]

[tex]\[ \frac{dA}{dt} = 90 \, \text{cm}^2/\text{s} \][/tex]

So, the rate at which the area of the square is increasing when the area of the square is [tex]\( 81 \, \text{cm}^2 \) is \( 90 \, \text{cm}^2/\text{s} \)[/tex].

Answer:

Expected Grade=2 i.e., C

Variance=1.2

Step-by-step explanation:

[tex]Expected\ value=E\left [ x \right ]=\sum _{i=1}^{k} x_{i}p_{i}[/tex]

The x values are

A = 4

B = 3

C = 2

D = 1

F = 0

Probability of each of the events

P(4)=0.1

P(3)=0.2

P(2)=0.4

P(1)=0.2

P(0)=0.1

[tex]E\left [ x \right ]=4\times 0.1+3\times 0.2+2\times 0.4+1\times 0.2+0\times 0.1\\\therefore E\left [ x \right ]=2[/tex]

Variance

[tex]Var\left ( x\right)=E\left [ x^2 \right ]-E\left [ x \right ]^2[/tex]

[tex]E\left [ x^2 \right ]=4^2 \times 0.1+3^2 \times 0.2+2^2 \times 0.4+1^2 \times 0.2+0^2 \times 0.1\\\Rightarrow E\left [ x^2 \right ]=5.2\\E\left [ x \right ]^2=2^2=4\\\therefore Var\left ( x\right)=5.2-4=1.2\\[/tex]

Final answer:

To have the same number of plates, napkins, and cups, Whitney will need at least 80 of each.

Explanation:

To find the least number of packages of plates, napkins, and cups so that Whitney has the same number of each, we need to find the least common multiple (LCM) of 10, 16, and 8. The LCM is the smallest multiple that all three numbers have in common.

10 = 2 x 5, 16 = 2 x 2 x 2 x 2, and 8 = 2 x 2 x 2

We can identify the prime factors of each number and then multiply the highest factor from each number. In this case, the LCM is 2 x 2 x 2 x 2 x 5 = 80.

Therefore, Whitney will need at least 80 plates, 80 napkins, and 80 cups to have the same number of each.

Answer:

sin Ф=3/√13

Cos Ф=2/√13

Tan Ф=3/2

Step-by-step explanation:

Let x=2

Let y=3

Let r be the length of line segment drawn from origin to the point

[tex]r=\sqrt{x^2+y^2}[/tex]

Find r

[tex]r=\sqrt{2^2+3^2} =\sqrt{4+9} =\sqrt{13}[/tex]

Apply the relationship for sine, cosine and tan of Ф where

r=hypotenuse

Sine Ф=length of opposite side÷hypotenuse

Sin Ф=O/H where o=3, hypotenuse =√13

sin Ф=3/√13

CosineФ=length of adjacent side÷hypotenuse

Cos Ф=A/H

Cos Ф=2/√13

Tan Ф=opposite length÷adjacent length

TanФ=O/A

Tan Ф=3/2

Answer:

Yes, because the experiment satisfies all the criteria for a binomial experiment.

Step-by-step explanation:

A binomial experiment has the following criterias,

1. There must be a fixed number of trials

2. Each trial is independent of the others

3. There must be only two outcomes  ( success and failure )

4. The probability of each outcome is same.

Given,

An experimental drug is administered to 80 randomly selected​ individuals, with the number of individuals responding favorably recorded,

The number of trials = 80 ( fixed )

Each individuals is independent,

Total outcomes = 2 ( yes or no ),

Also, the probability of each individual is same,

Hence, the given probability experiment represent a binomial​ experiment.

Answer:

The acceleration is 1.0416 m/[tex]s^{2}[/tex]

Step-by-step explanation:

In order to solve this problem we first need to know the formula for acceleration which is the following.

[tex]acceleration = \frac{final.velocity - initial.velocity}{final.time - initial.time}[/tex]

Since the time acceleration is calculated as [tex]m/s^{2}[/tex] we need to convert the km/h into m/s. Since 1km = 1000m and 1 hour = 3600 seconds, then

[tex]\frac{20*1000 }{3600s} = \frac{20,000m}{3600s} = \frac{20m}{3.6s}[/tex]

**Dividing numerator and denominator by 1000 to simplify**

[tex]\frac{65*1000 }{3600s} = \frac{65,000m}{3600s} = \frac{65m}{3.6s}[/tex]

**Dividing numerator and denominator by 1000 to simplify**

Now we can plug in the values into the acceleration formula to calculate the acceleration.

[tex]acceleration = \frac{\frac{65m}{3.6s}-\frac{20m}{3.6s} }{12s-0s}[/tex]

[tex]acceleration = \frac{\frac{45m}{3.6s}}{12s}[/tex]

[tex]acceleration = \frac{\frac{12.5m}{s}}{12s}[/tex]

[tex]acceleration = \frac{\frac{1.0416m}{s}}{s}[/tex]

Finally we can see that the acceleration is 1.0416 m/[tex]s^{2}[/tex]

In order to find the acceleration, we first need to convert the speed from km/h to m/s. Then we use the formula for acceleration which is the change in velocity divided by the change in time. The final acceleration is approximately 1.042 m/s².

The subject of the question pertains to the concept of acceleration in Physics. Acceleration, measured in meters per second per second (m/s²), is the rate at which an object changes its velocity. To determine this, we first have to convert the velocities from kilometers per hour (km/h) to meters per second (m/s). We know that 1 km = 1,000 m and 1 hour = 3,600 s. Therefore:

Next we use the formula for acceleration: a = Δv / Δt. The change in velocity (Δv) is the final velocity minus the initial velocity. Thus, Δv = 18.06 m/s - 5.56 m/s = 12.5 m/s. The change in time (Δt) is given as 12 seconds.

Substituting these values into the formula, we get: a = Δv / Δt = 12.5 m/s ÷ 12 s = 1.042 m/s². Therefore, the car's acceleration, assuming it's constant, is approximately 1.042 m/s².

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Answer:

The height is 28.57 cm.

The surface area is 9,628 cm^2.

Step-by-step explanation:

I assume the cooler is shaped like a rectangular prism with length and width of the base given, and with an unknown height.

volume = length * width * height

First, we convert the volume from liters to cubic centimeters.

60 liters * 1000 mL/L * 1 cm^3/mL = 60,000 cm^3

Now we substitute every dimension we have in the formula and solve for height, h.

60,000 cm^3 = 60 cm * 35 cm * h

60,000 cm^3 = 2,100 cm^2 * h

h = (60,000 cm^3)/(2,100 cm^2)

h = 28.57 cm

The height is 28.57 cm.

Now we calculate the internal surface area.

total surface area = area of the bases + area of the 4 sides

SA = 2 * 60 cm * 35 cm + (60 cm + 35 cm + 60 cm + 35 cm) * 28.57 cm

SA = 9,628 cm^2

The surface area is 9,628 cm^2.

Answer:

Step-by-step explanation:

Consider the 3x3 matrices in row echelon form:

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

and

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

a) The augmented matrix

[tex]\left[\begin{array}{ccc|c}1&2&0&1\\0&1&2&1\\0&0&0&2\end{array}\right][/tex]

corresponds to an inconsistent system.

b) The augmented matrix

[tex]\left[\begin{array}{ccc|c}1&2&0&1\\0&1&2&1\\0&0&0&0\end{array}\right][/tex]

corresponds to a consistent system with infinite solutions.

(c) The augmented matrix

[tex]\left[\begin{array}{ccc|c}1&2&0&1\\0&1&2&1\\0&0&1&1\end{array}\right][/tex]

corresponds to a consistent system with infinite solutions.

Final answer:

The 95% confidence interval for the mean mathematics ACT score for all calculus students at this college is calculated to be between 24.693 and 27.307.

Explanation:

To calculate a 95% confidence interval for the mean mathematics ACT score for all calculus students at the college, we use the sample mean and standard deviation along with the z-score for a 95% confidence level. Since the sample size is large (n=81), we can use the z-distribution as an approximation for the t-distribution.

The formula for a confidence interval is:

Mean ± (z-score * (Standard Deviation / √n))
Here, the sample mean (μ) is 26, the standard deviation (s) is 6, and the sample size (n) is 81. For a 95% confidence level, the z-score is approximately 1.96.

Now we calculate the margin of error (ME):
ME = 1.96 * (6 / √81) = 1.96 * (6 / 9) = 1.96 * 0.6667 = 1.307

Therefore, the 95% confidence interval is
26 ± 1.307

Lower Limit = 26 - 1.307 = 24.693
Upper Limit = 26 + 1.307 = 27.307
The confidence interval is (24.693, 27.307).

We estimate with 95 percent confidence that the true population mean for the mathematics ACT score for all calculus students at the college is between 24.693 and 27.307.

Answer:

y=3/5x+13/5

Step-by-step explanation:

Finding the slope:

m=6/10

m=3/5

The y intercept is 13/5

Thus, the equation is y=3/5x+13/5

Answer:

Equation of the given line is, y = (3/5)x +13/5

Step-by-step explanation:

Points to remember

Equation of the line passing through the poits (x1, y1) and (x2, y2)  and slope m is given by

(y - y1)/(x - x1) = m    where slope m = (y2 - y1)/(x2 - x1)

To find the slope of line

Here (x1, y1) = (-6, -1) and (x2, y2) = (4, 5)

Slope = (y2 - y1)/(x2 - x1)

 = (5 - -1)/(4 - -6)

 = 6/10 = 3/5

To find the equation

(y - y1)/(x - x1) = m

(y - -1)/(x - -6) = 3/5

(y + 1)/(x + 6) = 3/5

5(y + 1) = 3(x + 6)

5y + 5 = 3x + 18

5y = 3x + 13

y = (3/5)x +13/5

Explanation:

This can be explained by thinking numbers on the number line as:

Lets take we have to multiply a positive number (say, 2) with a negative number say (-3)

2×(-3)

Suppose someone is standing at 0 on the number line and to go to cover -3 , the person moves 3 units in the left hand side. Since, we have to compute for 2×(-3), The person has to cover the same distance twice. At last, he will be standing at -6, which is a negative number.

A image is shown below to represent the same.

Thus, a positive times a negative is a negative number.

Answer:

a) $ 1465.418

b) $ 214.582

Step-by-step explanation:

Since, the monthly payment formula of a loan is,

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

Where, PV is the principal amount of the loan,

r is the monthly rate,

n is the total number of months,

Here, P = $ 30, r = 1.25 % = 0.0125, n = 36 ( since, time is 3 years also 1 year = 12 months )

Substituting the values,

[tex]30=\frac{PV(0.0125)}{1-(1+0.0125)^{-36}}[/tex]

By the graphing calculator,

[tex]PV=865.418[/tex]

a) Thus, the cost of the TV = Down Payment + Principal value of the loan

= $ 600 + $ 865.418

= $ 1465.418

b) Now, the total payment = Monthly payment × total months

= 30 × 36

= $ 1080

Hence, the total amount of interest paid = total payment - principal value of the loan

=  $ 1080 - 865.418

=  $ 214.582.

Answer:

Option D "If n is odd or n is 2 then n is prime"

Step-by-step explanation:

we know that

To form the converse of the conditional statement, interchange the hypothesis and the conclusion.

In this problem

the conditional statement is "If n is prime then n is odd or n is 2."

The hypothesis is "If n is prime"

The conclusion is "n is odd or n is 2."

therefore

interchange the hypothesis and the conclusion

The converse is "If n is odd or n is 2 then n is prime"

Final answer:

The converse of 'If n is prime then n is odd or n is 2' is 'If n is odd or n is 2 then n is prime', therefore, the correct answer from the given options is D.

Explanation:

The converse of the statement 'If n is prime then n is odd or n is 2' is constructed by reversing the hypothesis and the conclusion. The converse would state 'If n is odd or n is 2 then n is prime'. Hence, the correct answer is D. We establish the converse by suggesting that the given properties (being odd or being the number 2) necessarily imply that a number is prime; however, it's important to note that while all prime numbers other than 2 are indeed odd, not all odd numbers are prime. Therefore, the converse is not logically equivalent to the original statement.

Answer:

  90 adults; 160 students

Step-by-step explanation:

Let "a" represent the number of adults who went. The number of students can be represented by (250-a). Then the total cost of tickets is ...

  4.50a +2.50(250-a) = 805

  2a + 625 = 805 . . . . . . simplify

  2a = 180 . . . . . . . . . . . . subtract 625

  a = 90 . . . . . . . . . . . . . . divide by 2

  # of students = 250 -90 = 160

160 students and 90 adults went to the movies.

Answer:

the answer is 90 adults , and 160 students

Answer:[tex]\frac{1}{1000}[/tex]

Step-by-step explanation:

For the lottery game three numbers must match in exact order

From 0 to 9 total 10 numbers are there

Therefore selecting exactly same numbers as of winner is

=[tex]^{10}C_1\times ^{10}C_1\times ^{10}C_1 [/tex]

Since numbers are repeatable therefore each time we have a choice of choosing 1 number out o 10

=[tex]10\times 10\times 10[/tex]

Probability of winning=[tex]\frac{1}{1000}[/tex]

In a lottery game where the player must match a repeatable sequence of three numbers ranging from 0-9, the probability of a single ticket having the winning sequence is 1/1000, or 0.1%.

The subject of this problem is probability in Mathematics, specifically in a lottery context. Each digit in the sequence can be any number from 0 to 9. Because these numbers are repeatable, this means there are 10 possible numbers for each of the three digits in the sequence. Therefore, to find the total number of possible sequences, you multiply the ten options for the first digit by the ten for the second and the ten for the third, which comes to 10 * 10 * 10 = 1000 total possible sequences.

Since you are only looking for one specific sequence being the winning number, that means there’s only 1 favorable outcome out of 1000. Therefore, the probability of your ticket having the winning number sequence is 1/1000, or 0.001 in decimal form or 0.1% in percentage form.

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Answer:

[tex]a_{n}=15 + (-1)^n * 5[/tex]

Step-by-step explanation:

First, we notice that the when n is odd, [tex]a_{n}[/tex] = 10. And when n is even, [tex]a_{n}[/tex] = 20.

The average of 10 and 20 is [tex](10+20)/2 = 15[/tex]. So, the distance between 15 and 10 is the same that between 15 and 20.

That distance is 5.  

From 15, we need to subtract 5 to get 10 when n is odd and we need to add 5 to get 20 when n is even.

The easiest way to express that oscilation is using [tex](-1)^n[/tex], because it is (-1) when n is odd and 1 when is even. And when multiplied by 5, it will add or subtract 5 as we wanted.

Answer:

$1.710

Step-by-step explanation:

Mike deposited $850 into the bank in July.

In August his balance will be: $850×1.15 = $977.5

In September his balance will be: $977.5×1.15 = $1124.125

In October his balance will be: $1124.125×1.15 = $1292.74375

In November his balance will be: $1292.74375×1.15 = $1.486,6553125

In December his balance will be: $1.486,6553125×1.15 = $1.709,653609375

Therefore, the amount of money he will have after december will be $1.710

Answer:

1234.08 weeks.

Step-by-step explanation:

Since, the amount formula in compound interest is,

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

Where, P is the principal amount,

r is the rate per period,

t is the number of periods,

Given,

P = $ 150,000,

Annual rate = 8 % = 0.08,

So, the weekly rate, r = [tex]\frac{0.08}{52}[/tex]  ( 1 year = 52 weeks )

A = 1,000,000

By substituting the values,

[tex]1000000=150000(1+\frac{0.08}{52})^t[/tex]

[tex]\frac{1000000}{150000}=(\frac{52+0.08}{52})^t[/tex]

[tex]\frac{20}{3}=(\frac{52.08}{52})^t[/tex]

Taking log both sides,

[tex]log(\frac{20}{3})=t log(\frac{52.08}{52})[/tex]

[tex]\implies t = \frac{log(\frac{20}{3})}{ log(\frac{52.08}{52})}=1234.07630713\approx 1234.08[/tex]

Hence, for become a millionaire we should wait 1234.08 weeks.

Answer:

Mortality rate is 0.5 per 1000 habitant

the number of deaths have increased (from 210 to 250)

Step-by-step explanation:

Hello

Mortality rate is the number of deaths in a specific population,usually expressed in units of deaths per 1,000 individuals per year

[tex]Mortality\ rate\ (S)= \frac{(D)}{(P)} *10^{n} \\\\\\\\[/tex]

where

S=Mortality rate

D=deaths

P= population

n= is a conversion form ,such as multiplying by  10^{3}=1000, to  get mortality rate per 1,000 individuals( we will use n=3)

step 1

asign

S=unknown

D=250

P=500000

n=3, (for each  1000 people)

step 2

Replace

[tex]S=\frac{250}{500000}*1000\\S=\ 0.5\ per\ 1000[/tex]

Mortality rate is 0.5 per 1000 habitant

step 3

what if S=0.42

S=0.42(assuming it is 0.42 per 1000,n=3)

D=unknown

P=500000

[tex](S)= \frac{(D)}{(P)} *10^{n}\\\\isolatin D\\\\S*P=D*10^{n}\\ D=\frac{S*P}{10^{n} } \\\\and\ replacing\\\\D=\frac{(0.42)*(500000)}{10^{3} }\\\\ Deaths=210\\[/tex]

the number of deaths have increased (from 210 to 250)

Have a great day

Answer:

Suppose,

(C - A) ∩ (B - A) ≠ ∅

Let x is an element of (C - A) ∩ (B - A),

That is, x ∈ (C - A) ∩ (B - A),

⇒ x ∈ C - A and x ∈ B - A

⇒ x ∈ C, x ∉ A and x ∈ B, x ∉ A

⇒ x ∈ B ∩ C and x ∉ A

⇒ B ∩ C ⊄ A

But we have given,

B ∩ C ⊂ A

Therefore, our assumption is wrong,

And, there is no common elements in (C - A) and (B-A),

That is,  (C - A) ∩ (B - A) = ∅

Hence proved...

Answer:

(141)₁₀→(241)₇

Step-by-step explanation:

(141)₁₀→(?)₇

for conversion of number from decimal to base 7 value we have to

factor 141 by 7

which is shown in the figure attached below.

from the attached figure we can clearly see that the colored digit will

give the conversion

we will write the digit from the bottom as shown in figure

(141)₁₀→(241)₇

Each calculation has been evaluated, taken all significant figures into consideration, and results have been presented in scientific notation. Special attention was given to rules related to multiplying numbers in scientific notation.

The given calculations require us to use scientific notation and proper treatment of significant figures. We are using the fundamentals of arithmetic with scientific notation, which is based on the rules of exponents. Each calculation is treated as follows:

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Answer:82,368

Step-by-step explanation:

Final answer:

The total number of eight-card hands formed from a standard 52-card deck where the hand consists of three cards of one denomination, three cards of another, and two cards of a third is 27,456. This is calculated by finding combinations of denominations and the specific cards within those denominations.

Explanation:

To answer the question of how many eight-card hands can be formed from a standard 52-card deck, where the hand consists of three cards of one denomination, three cards of another denomination, and two cards of a third denomination, we have to use combinations. This is a problem of combinatorics in which we are finding the number of ways to select items from a group without regard to order.

First, we choose the denominations. There are 13 denominations and we want to choose 3 of them for our hand. Using the combination formula, this can be done in C(13, 3) ways:

C(13, 3) = 13! / (3! × (13-3)!) = 286

After choosing the denominations, we need to select the specific cards. For each of the first two denominations, we select 3 out of the 4 available suits, and for the third denomination, we select 2 out of the 4 suits. This can be done as follows:

C(4, 3) for the first denomination: C(4, 3) = 4 ways
C(4, 3) for the second denomination: C(4, 3) = 4 ways
C(4, 2) for the third denomination: C(4, 2) = 6 ways

Multiplying the ways to choose the denominations by the ways to choose the cards for each denomination gives us the total number of distinct hands.

Total hands = C(13, 3) × C(4, 3) × C(4, 3) × C(4, 2) = 286 × 4 × 4 × 6 = 27,456

Therefore, there are 27,456 different eight-card hands that meet the given criteria.

Answer:

The smallest fraction is b 11/60

To find the smallest of the unknown fractions given that their average with 1/5 is 1/4, we denoted the smallest fraction as x, leading to a simple algebraic equation solution. After fixing an arithmetic error, we found that the smallest fraction is 1/6.

The question is asking for the smallest unknown fraction when the average of this smallest fraction, another fraction that is twice its size, and a known fraction 1/5 is 1/4. Let's denote the smallest fraction as x, which implies that the other fraction is 2x. Since the average of three numbers is the sum of those numbers divided by three, our equation is (1/5 + x + 2x) / 3 = 1/4. Simplifying the equation by combining like terms and solving for x will reveal the smallest fraction.

First, we sum the unknowns and the known fraction: 1/5 + x + 2x = 3x + 1/5. Now we multiply both sides of the equation by 3 (to eliminate the division by 3 on the left side), we get: 3x + 3/5 = 1/4. To solve for x, we must have like denominators, therefore we convert all fractions to have a common denominator of 20. The equation then becomes 60x + 12 = 5. Subtracting 12 from both sides gives us 60x = -7, thus x = -7/60.

However, since fractions cannot be negative in this context, we made an error in our calculations. We correctly need to equate 3x + 1/5 to 3/4 (because the average is 1/4 which is the same as 3/4 for three numbers), and now the solution proceeds without error. Solving from here, we find that x = 1/6, which is the smallest fraction and is the correct answer to the question.

Answer:

Given:  [tex]\begin{bmatrix}2&5&-2&1&2\\1&1&-2&1&4\end{bmatrix}[/tex]

[tex]2x_{1}+5 x_{2}  - 2_{3} + x_{4} + 2x_{5}\\[tex]

[tex]x_{1}+ x_{2}  - 2_{3} + x_{4} + 4x_{5}\\[/tex]

The system of linear equations in matrix form may be written as:

AX=B,

where,

A is coefficient matrix of order [tex]2\times 4[/tex] and is given by:

A = [tex]\begin{bmatrix}2&5&-2&1&2\\1&1&-2&1&4\end{bmatrix}[/tex]

X is variable matrix of order  [tex]5\times 1[/tex] and is given by:

X=  [tex]\begin{bmatrix}x_{_{1}}\\x_{_{2}}\\x_{3}\\x_{4}\\x_{5}\end{bmatrix}[/tex]

and B is the contant matrix of order [tex]2\times 1[/tex] and is given by:

B = [tex]\begin{bmatrix}1\\5\end{bmatrix}[/tex]

Now, AX=B

[tex]\begin{bmatrix}2&5&-2&1&2\\1&1&-2&1&4\end{bmatrix}[/tex]. [tex]\begin{bmatrix}x_{_{1}}\\x_{_{2}}\\x_{3}\\x_{4}\\x_{5}\end{bmatrix}[/tex] = [tex]\begin{bmatrix}1\\5\end{bmatrix}[/tex]

Answer with explanation:

The given differential equation

  y'y''=2--------(1)

We have to apply the following substitution

  u=y'

 u'=y"

Applying these substitution in equation (1)

u u'=2

[tex]u \frac{du}{dx}=2\\\\ u du=2 dx\\\\ \int u du=\int 2 dx\\\\\frac{u^2}{2}=2 x+K\\\\\frac{y'^2}{2}=2 x+K\\\\y'^2=4 x+2 K\\\\y'=(4 x+2 K)^{\frac{1}{2}}\\\\ dy=(4 x+2 K)^{\frac{1}{2}} d x\\\\\int dy=\int(4 x+2 K)^{\frac{1}{2}} d x\\\\y=\frac{(4 x+2 K)^{\frac{3}{2}}}{4 \times \frac{3}{2}}+J\\\\y=\frac{(4 x+2 K)^{\frac{3}{2}}}{6}+J[/tex]

Where , J and K are constant of Integration.

Answer:

Bob performed better in mathematics part of the SAT than the ACT

Step-by-step explanation:

We need to calculate the z-scores for both parts and compare them.

Z-score for the SAT is calculated using the formula:

[tex]Z=\frac{X-\mu}{\sigma}[/tex]

where [tex]\mu=514[/tex] is the mean and [tex]\sigma=113[/tex] is the standard deviation, and [tex]X=660[/tex] is the SAT test score.

We plug in these values to obtain:

[tex]Z=\frac{660-514}{113}[/tex]

[tex]Z=\frac{146}{113}=1.29[/tex] to the nearest hundredth.

We use the same formula to calculate the z-score for the ACT too.

Where [tex]\mu=20.6[/tex] is the mean and [tex]\sigma=5.1[/tex] is the standard deviation, and [tex]X=27[/tex] is the ACT test score.

We substitute the values to get:

[tex]Z=\frac{27-20.6}{5.1}=1.25[/tex] to the nearest hundredth.

Since 1.29 > 1.25, Bob performed better in mathematics part of the SAT

Using z-scores to determine where Bob's performance stands compared to others, we find that he performed slightly better on the SAT with a z-score of 1.29, than on the ACT with a z-score of 1.25.

To determine on which test Bob performed better, we have to calculate his z-scores on both the SAT and ACT. Z-score is a statistical measurement that describes a score's relationship to the mean of a group of scores. It indicates how many standard deviations an element is from the mean.

Here is how you calculate the z-score: z = (X - μ) / σ, where X is the individual score, μ is the mean, and σ is the standard deviation.

We can apply this formula to both of Bob's scores. For the SAT: z = (660 - 514) / 113 ≈ 1.29 For the ACT: z = (27 - 20.6) / 5.1 ≈ 1.25

Comparing the z-scores, Bob's score is above the mean by 1.29 standard deviations on the SAT and by 1.25 standard deviations on the ACT. Therefore, Bob performed slightly better on the SAT than on the ACT when comparing his scores to other test takers.

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Final answer:

To investigate if African Americans are underrepresented among teachers in a St. Louis suburb, a hypothesis test can be set up comparing the proportion of African American teachers to the African American population in the city. The null hypothesis is that the proportions are equal, while the alternative hypothesis is that the teacher proportion is less. A statistical test, such as a chi-square test, can then be used to assess representation.

Explanation:

To determine whether African Americans are underrepresented in the schoolteacher hiring in a St. Louis suburb, we can set up a hypothesis test. According to the data provided, African Americans make up 5.7% of the city's population. Out of 405 teachers in the school system, only 15 are African American. To establish the hypotheses, we consider the null hypothesis (H0) that African Americans are represented in teaching positions at the same rate as they are in the general population, and the alternative hypothesis (H1) that African Americans are underrepresented in teaching positions compared to their proportion in the general population.

For the null hypothesis (H0): The proportion of African American teachers is equal to the proportion of the African American population in the city (5.7%). For the alternative hypothesis (H1): The proportion of African American teachers is less than the proportion of the African American population in the city (5.7%). The next step is to perform a statistical test, such as a chi-square test of independence, to determine if the observed data (15 out of 405 teachers) significantly deviates from what would be expected under the null hypothesis, based on the proportion of the African American population in the city.

If the p-value obtained from the statistical test is below a predetermined significance level, usually 0.05, we would reject the null hypothesis in favor of the alternative hypothesis, suggesting that African Americans are indeed underrepresented. This statistical approach does not consider other factors that may influence hiring, but it does provide a starting point to understand representation.

Answer:

84.38, 73.74

Step-by-step explanation:

score given 81.6, 72.0, 81.1, 86.4, 70.2, 83.1

sample size (n) = 6

[tex]mean = \dfrac{81.6+ 72.0+ 81.1+ 86.4+ 70.2+ 83.1}{6}[/tex]

mean = 79.06

standard deviation

[tex]\sigma =\sqrt{\frac{\sum (x-\bar{x})^2}{n-1}}[/tex]

[tex]\sigma =\sqrt{\frac{ (81.6-79.06)^2+(72-79.06)^2+(86.4-79.06)^2+(70.2-79.06)^2+(81.1-79.06)^2+(83.1-79.06)^2}{6-1}}[/tex]

σ = 6.47

level of significance (α) = 1 - 90% = 10%

confidence interval

[tex]\bar{x} \pm t_{\alpha}(\frac{S}{\sqrt{n}})\\79.06 \pm 2.015(\frac{6.47}{\sqrt{6} })[/tex]

=79.06 ± 5.32

= 84.38, 73.74

The 90% confidence interval for the mean score is calculated using the sample mean, standard deviation, and the Z-value from a Z-distribution table. The resulting range gives us a 90% confidence that the real population mean lies within this interval.

To compute a 90% confidence interval for the mean score, we use the sample data we have, our sample mean, and our standard deviation. The confidence interval gives us a range of values that likely includes the true population mean. We use the standard statistical formula for the 90% confidence interval: X ± (Z(α/2) * (σ/√n))

First, calculate the sample mean (X) and the standard deviation (σ). The scores are: 81.6, 72.0, 81.1, 86.4, 70.2, 83.1. The X will be the total of all the scores divided by 6, and σ will be a measure of how much they deviate from the mean.

Next, insert these values into the determining a confidence interval formula. You will need the Z(α/2) value for a 90% confidence interval, which is 1.645 in a Z-distribution table. Finally, take your X ± the calculated result to achieve the 90% confidence interval.

Remember: a 90% confidence interval means that you can be 90% confident that the true mean for all students lies within this interval.

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