32a³b²
_____

8ab²

Simplify the following expression.

[tex]|\Omega|=14^2=196\\|A|=2\cdot8=16\\\\P(A)=\dfrac{16}{196}=\dfrac{4}{49}\approx8.2\%[/tex]

Step-by-step explanation:

[tex] \sqrt{25} = \pm \: 5 \\ \\ \sqrt{100} = \pm \: 10 \\ \\ \sqrt{36} = \pm \: 6 \\ \\ \sqrt{84} = \pm \: 9.165\\ \\ \sqrt{4} = \pm \: 2 \\ \\ [/tex]

Not entirely sure what the question is supposed to say, so here's my best guess.

First, find the partial fraction decomposition of

[tex]\dfrac{x-9}{x^2-3x-18}[/tex]

This is equal to

[tex]\dfrac{x-9}{(x-6)(x+3)}=\dfrac a{x-6}+\dfrac b{x+3}[/tex]

Multiply both sides by [tex](x-6)(x+3)[/tex], so that

[tex]x-9=a(x+3)+b(x-6)[/tex]

Notice that if [tex]x=6[/tex], the term involving [tex]b[/tex] vanishes, so that

[tex]6-9=a(6+3)\implies a=-\dfrac13[/tex]

Then if [tex]x=-3[/tex], the term with [tex]a[/tex] vanishes and we get

[tex]-3-9=b(-3-6)\implies b=\dfrac43[/tex]

So we have

[tex]\dfrac{x-9}{x^2-3x-18}=-\dfrac1{3(x-6)}+\dfrac4{3(x+3)}[/tex]

I think the final answer is supposed to be [tex]a+b[/tex], so you end up with 1.

Answer:

First city: $5,500

Second city: $5,000

Step-by-step explanation:

Let's define x as the hotel price in the first city and y the hotel price in the second city.  We can start with this equation:

y = x - 500 (The hotel before tax in the 2nd city was $500 lower than in the 1st.)

Then we can say

0.065x + 0.045y = 582.50  (the sum of the tax amounts were $582.50)

We place the value of y from the first equation in the second equation:

0.065x + 0.045 (x - 500) = 582.50

0.065x + 0.045x - 22.50 = 582.50 (simplifying and adding 22.5 on each side)...

0.11x = 605

x = 5,500

The cost of the first hotel was $5,500

Thus, the cost of the second hotel was $5,000 (x - 500)

Answer: Probability that students who did not attend the class on Fridays given that they passed the course is 0.043.

Step-by-step explanation:

Since we have given that

Probability that students attend class on Fridays = 70% = 0.7

Probability that who went to class on Fridays would pass the course = 95% = 0.95

Probability that who did not go to class on Fridays would passed the course = 10% = 0.10

Let A be the event students passed the course.

Let E be the event that students attend the class on Fridays.

Let F be the event that students who did not attend the class on Fridays.

Here, P(E) = 0.70 and P(F) = 1-0.70 = 0.30

P(A|E) = 0.95,  P(A|F) = 0.10

We need to find the probability that they did not attend on Fridays.

We would use "Bayes theorem":

[tex]P(F\mid A)=\dfrac{P(F).P(A\mid F)}{P(E).P(A\mid E)+P(F).P(A\mid F)}\\\\P(F\mid A)=\dfrac{0.30\times 0.10}{0.70\times 0.95+0.30\times 0.10}\\\\P(F\mid A)=\dfrac{0.03}{0.695}=0.043[/tex]

Hence, probability that students who did not attend the class on Fridays given that they passed the course is 0.043.

Answer:

[tex]\boxed{\text{197 m}}[/tex]

Step-by-step explanation:

The formula relating distance (d), speed (s), and time (t) is

d = st

1. Calculate the distance

d = 269 s × 4.6 m·s⁻¹ = 1240 m

2.Calculate the track radius

The distance travelled is the circumference of a circle

[tex]\begin{array}{rcl}C & = & 2 \pi r\\1240 & = & 2 \pi r\\\\r & = & \dfrac{1240}{2 \pi }\\\\& = & 197\\\end{array}\\\text{The radius of the track is }\boxed{\textbf{197 m}}[/tex]

The radius in meters is 196.9 meters.

The runner ran around the track in 269 seconds at a speed of 4.6 m/s. This will enable us to find the distance around the track which is the circumference of the track.

Distance = Speed × time

= 4.6 × 269

= 1,237.4 meters

The distance here is the circumference which can also be found by the formula:

Circumference = π × diameter

1,237.4 = 22/7 × Diameter

Diameter = 1,237.4 ÷ 22/7

= 393.7 meters

Now that we have the diameter, the radius is:

= Diameter / 2

= 196.9 meters

In conclusion, the radius is 196.3 meters

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Ok, the student needs 40 points and each question is worth 5, so 40/5 = 8 questions are needed.

Each question has 4 possibilities, 1 is right, so the chances to guess it correctly is one in 4, or 1/4, or 25%.

[tex]\frac{8}{20} = \frac{2}{5}[/tex]

To know the probability to pass the exam we can do:

[tex]\frac{25}{100}*\frac{2}{5} = 10%[/tex]

Answer: 0.102 or  10.2%.

Step-by-step explanation:

Given : Number of multiple-choice questions = 20

Number of options in any question=4

Each question is worth 5 points and only one response per question is correct.

Probability of getting a correct answer =  [tex]\dfrac{1}{4}=0.25[/tex]

If the student needs at least 40 points to pass the test, that mean he needs at-least [tex]\dfrac{40}{5}=8[/tex] questions correct.

Let x denotes the number of correct questions .

By using binomial distribution , we find

[tex]P(x\geq8)=1-P(x<8)\\\\ =1-P(x\leq7)\\\\=1-0.898\ \ \text{[By using binomial table for n= 20 , p=0.25 and x=7]}\\\\=0.102[/tex]

[Binomial table gives the probability [tex]P(X\leq x)=\sum_{x=0}^c^nC_xp^x(1-p)^{n-x}[/tex] ]

Hence, the probability the student passes is closest to 0.102 or 10.2%.

Answer:

$804,680.814 ( approx )

Step-by-step explanation:

The amount formula in compound interest is,

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

Where, P is the principal amount,

r is the annual rate of interest,

n is the compounding periods in a year,

t is the time in years,

Given, P =  $ 400,000,

r = 3.5 %=0.035,

n = 12,    ( 1 year = 12 months )

t = 20 years,

Thus, the amount would be,

[tex]A=400000(1+\frac{0.035}{12})^{240}[/tex]

[tex]=400000(\frac{12.035}{12})^{240}[/tex]

[tex]=\$ 804680.813963[/tex]

[tex]\approx \$ 804680.814[/tex]

The amount owed on the mortgage on January 1, 2030, would be approximately [tex]\$765,320.99.[/tex]

To solve this problem, we need to calculate the compound interest on the mortgage over a period of 20 years (from January 1, 2010, to January 1, 2030). The formula for compound interest is given by:

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

where:

-  A is the amount of money accumulated after n years, including interest.

-  P is the principal amount (the initial amount of money).

- r  is the annual interest rate (decimal).

-  n is the number of times that interest is compounded per year.

-  t is the time the money is invested for, in years.

Given:

- [tex]\( P = \$400,000 \)[/tex]

-[tex]\( r = 3.5\% = 0.035 \)[/tex] (as a decimal)

-  n = 12  (since the interest is compounded monthly)

-  t = 20  years

Plugging these values into the compound interest formula, we get:

[tex]\[ A = 400,000 \left(1 + \frac{0.035}{12}\right)^{12 \times 20} \][/tex]

[tex]\[ A = 400,000 \left(1 + \frac{0.035}{12}\right)^{240} \][/tex]

[tex]\[ A = 400,000 \left(1 + 0.002916667\right)^{240} \][/tex]

[tex]\[ A = 400,000 \left(1.002916667\right)^{240} \][/tex]

Now, we calculate the value inside the parentheses:

[tex]\[ 1.002916667^{240} \ = 1.913209802 \][/tex]

Finally, we multiply this by the principal amount to find out how much would be owed:

[tex]\[ A \ = 400,000 \times 1.913209802 \][/tex]

[tex]\[ A \ = \$765,320.99 \][/tex]

Final answer:

The probability of at least 16 adults requiring eyesight correction out of 17 is found by summing up individual probabilities of exactly 16 and exactly 17 adults in need, calculated using the binomial probability formula. A comparison of this probability to a typical significance threshold will determine if 16 is a significantly high number.

Explanation:

The probability that at least 16 out of 17 randomly selected adults need correction for their eyesight, given that 84% of adults need such correction, is calculated using the binomial probability formula. The binomial probability of exactly k successes in n trials, where p is the probability of success on a single trial, is:

P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)

Here, we calculate the probability for exactly 16 adults and exactly 17 adults needing vision correction and then sum these to get at least 16:

Calculate P(X = 16), where n = 17, k = 16, and p = 0.84.

Calculate P(X = 17), where n = 17, k = 17, and p = 0.84.

Sum P(X = 16) and P(X = 17) for the total probability.

To answer if 16 is a significantly high number requiring correction, it is important to compare the probability with a threshold of significance, such as 0.05 commonly used in statistics. If the probability is less than this threshold, then 16 can be considered a significantly high number.

Answer:

x² + 9x + 8 = (x + 1)(x + 8)

x² + 9x + 8 = (x + 8)(x + 1)

Step-by-step explanation:

* Lets explain how to factorize the polynomial using the pattern

- The form of the quadratic polynomial is x² + px + q, where p is the

  coefficient of x and q is the numerical term

∵ x² + (a + b)x + (ab) = (x + a)(x + b)

- From the formula above the coefficient of x is the sum of the two

 factors a and b

∴ p = a + b and q = ab

- That means p is the sum of two numbers and q is the product of

  the same numbers

* Lets solve the problem

∵ x² + 9x + 8 is a quadratic polynomial

∵ x² + px + q is the form of quadratic polynomial

∴ p = 9 and q = 8

∵ p = a + b and q = ab

∴ a + b = 9 ⇒ (1)

∴ ab = 8 ⇒ (2)

- We must to find two numbers their product is 8 and their sum is 9

∵ The possibility of 8 as a product of two numbers is:

   2 × 4 OR 1 × 8

∵ The sum of 1 + 8 = 9

∴ The value of a and b are 1 and 8

- It does't matter which of them = 1 or which of them = 8

∴ x² + (a + b)x + ab = x² + (1 + 8)x + (1)(9)

∵ x² + (a + b)x + (ab) = (x + a)(x + b)

∴ x² + (1 + 8)x + (1)(9) = (x + 1)(x + 8)

∴ x² + 9x + 8 = (x + 1)(x + 8)

- OR

∴ x² + 9x + 8 = (x + 8)(x + 1)

Step-by-step explanation:

Let n = number of years since 1995

Let C = cost of gym membership in any particular year

Initial cost in 1995 = $25

Additional cost each year = $6

We can say the following:

Cost at any given year = cost in 1995 + ($6 x number of years after 1995)

Or expressed as the following :  C = 25 + 6n   (Ans)

In 2009, the number of years since 1995,

= 2009 - 1995

= 14 years

Hence, cost in 2009,

= $25 + ($6 x 14 years)

= $109 (Ans)

 $109   is the Cost of A GYM Membership in 2009.

What does "cost" mean to you?

given,

Let n = number of years since 1995

Let C = cost of gym membership in any particular year

Initial cost in 1995 = $25

Additional cost each year = $6

formula ,

Cost at any given year = cost in 1995 + ($6 x number of years after 1995)

put value in formula

             C = 25 + 6n  

now,

In 2009, the no of member in  years since 1995,

= 2009 - 1995

= 14 years

Hence, cost of gym membership in 2009,

= $25 + ($6 x 14 years)

= $109

Therefore, cost of gym membership in 2009 = $109

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Final answer:

Explanation of quartiles, deciles, and percentiles in a dataset of gas-powered generator invoice payment times. Lastly, the 67th percentile in the data set is approximately 46 days.

Explanation:

The first quartile (Q1) is found by locating the median of the lower half of the data set, which results in a value of 34 days. The third quartile (Q3) is the median of the upper half of the data set, yielding a value of 47 days. The second decile corresponds to the 20th percentile, which is approximately 31 days. The eighth decile corresponds to the 80th percentile, which is approximately 47 days. Lastly, the 67th percentile in the data set is approximately 46 days.

Answer: Hence, the probability that the whole shipment would be accepted is 0.371.

Many would be rejected.

Step-by-step explanation:

Since we have given that

Number of tablets to be tested = 42

Probability of getting a defect = 5% = 0.05

We need to find the probability that this whole shipment will be accepted.

As we have mentioned that if there is only one or none defect, then the whole shipment would be accepted.

P(accepted) = P(either none or one defect) =  P(X=0)+P(X=1)

[tex]P(X=0)=(1-0.05)^{42}=(0.95)^{42}=0.115\\\\and\\\\P(X=1)=42\times (0.05)(0.95)^{41}=0.006\times 42=0.256[/tex]

So, P(Accepted) = 0.115+0.256=0.371

Hence, the probability that the whole shipment would be accepted is 0.371.

Many would be rejected.

Answer:

  $195.50

Step-by-step explanation:

  0.17% × $115,000 = $195.50

Jill's account will be charged $195.50 in expense fees.

Answer:

Jill will have to pay $195.5 in fees this year.

Step-by-step explanation:

This question may be solved by a simple rule of three.

This fund has an expense ratio of 0.17 percent. This means that for each investment in this fund, there is a fee of 17% percent of the value.

$115,000 is 100%, that is, decimal 1. How much is 0.17%, that is, 0.0017 of this value. So

1 - $115,000

0.0017 - $x

[tex]x = 115000*0.0017 = 195.5[/tex]

Jill will have to pay $195.5 in fees this year.

The correct answer is:

                   Option: E

   E.   None of the above

[tex]Ax=0[/tex] has infinite many solutions if det(A)=0

and is non-singular or invertible otherwise i.e. when det(A)≠0

The matrix that will be formed by the given set of vectors is:

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

Also, determinant i.e. det of matrix A is calculated by:

[tex]\begin{vmatrix}-1 &-2 &0 &-1 \\ -1&-1 &0 &-1 \\ 0 &0 &1 &0 \\ 0 &0 &0 &1 \end{vmatrix}=1(1(1+2))=3[/tex]

Hence, determinant is not equal to zero.

This means that the matrix is invertible and non-singular.

Answer:

b) An empirically testable statement that is an unproven supposition developed in order to explain phenomena.

Step-by-step explanation:

b) An empirically testable statement that is an unproven supposition developed in order to explain phenomena.

A hypothesis is an unproven supposition.This may be derived from previous research or theory and is developed prior to data collection.

Note:

Variance is the average squared deviations about the mean of a distribution of values.

Answer:

720

Step-by-step explanation:

Given : The word  ABSENT

To Find: How many different 6-letter arrangements can be formed using the letters in the word ABSENT, if each letter is used only once?

Solution:

Number of letters in ABSENT = 6

So, No. of arrangements can be formed using the letters in the word ABSENT, if each letter is used only once = 6!

                                                   = [tex]6 \times 5 \times 4\times 3 \times 2 \times 1[/tex]

                                                   = [tex]720[/tex]

So, Option C is true

Hence there are 720 different 6-letter arrangements can be formed using the letters in the word ABSENT.

Final answer:

There are 720 different 6-letter arrangements that can be formed from the word ABSENT, by applying the permutation formula 6! = 720. The correct option is c.

Explanation:

The question asks: How many different 6-letter arrangements can be formed using the letters in the word ABSENT, if each letter is used only once? To answer this, we need to calculate the number of permutations of 6 letters taken from 6. This is a simple permutation problem where we use the formula for permutations which is n!, where n is the total number of items to choose from and ! denotes a factorial, meaning the product of all positive integers up to n.

Given that the word ABSENT has 6 letters and we are arranging all 6, we have 6! = 6×5×4×3×2×1 = 720. Therefore, there are 720 different 6-letter arrangements that can be formed using the letters in ABSENT, with each letter used only once.

Step-by-step explanation:

given a slope and a point that the line passes through you have 2 options

Option 1: Solve for the equation of the line so you can just use that to graph the line. In this scenario it would be y=(-5/2)x - (20/13)

Option 2: plot the given point and, based on the slope, plot the next point that it crosses. In this case the next point would be (5, 7). Then you can just draw a line using these 2 points.

Answer:

12.85 cm

Step-by-step explanation:

Area ratio of 49:36 means a side length ration of 7:6.

7/6 = 15/x

7x = 90

x = 90/7

= 12.85

Answer: 12.85 cm

Step-by-step explanation:

Answer:

Step-by-step explanation:

The slope of the graph at x=0 is related to the value of b. It is also proportional to the value of a, which is the same for all but curve B. The red curve R has the largest slope at x=0, (much larger than 3/4 the slope of curve B), so curve R has the greatest value of b.

Similarly, the smallest value of b will correspond to the curve with the smallest (most negative) slope. That would be curve K. Curve K has the smallest value of b.

Step-by-step explanation:

Given:

Mean = 33700

Median = 34600

Mode = 35600

The mean is the average, the median is the middle number, and the mode is the most common number.

a)

First, we need to find the new mean (average) if one of the employees gets a 3500 raise.  The average is the total salary divided by number of employees:

(5 × 33700 + 3500) / 5 = 34400

b)

The mode is the most common number in a set.  Since there are only five employees, and the mode is different than the median, then the two highest earners must have the same salary.  The salaries from smallest to largest is therefore:

?, ?, 34600, 35600, 35600

When the median gets the 3500 raise, the set becomes:

?, ?, 35600, 35600, 38100

So the new median is 35600.

c)

The most common number is still 35600.  So the mode hasn't changed: 35600.

The Laplace Transform of a function f(t) is a tool used to solve differential equations by converting them into simpler algebraic equations. The transform itself is given by the integral ℒ{f(t)} = ∫∞₀ e−stf(t) dt, but without knowing the specific form of f(t), the exact transformation cannot be computed.

The Laplace Transform, as defined by Definition 7.1.1, is a mathematical tool often used to handle differential equations, especially in the fields of Physics and Engineering. The main idea behind the transform is to convert the differential equations, which are difficult to solve, into simple algebraic equations. These simple equations are relatively easy to solve. Once solved, the inverse Laplace Transform is employed to obtain the solution to the original differential equation.

The question asks to compute the Laplace Transform ℒ{f(t)} of a function. By definition, the Laplace transform of a function f(t), defined for t ≥ 0 is given by the integral ℒ{f(t)} = ∫∞₀ e−stf(t) dt. The question doesn't give an explicit form of the function f(t). In general, if f(t) = e^(-αt), where α is a constant, then its Laplace Transform is given by ℒ{e^(-αt)} = 1/(s + α) for s > α.

If you could kindly provide me with the specific function f(t), I can better determine the Laplace Transform ℒ{f(t)}

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Answer:

2400sq ft

Step-by-step explanation:

20×2 and 30×2

40×60

2400 sq ft

Answer:

Length = 40 feet

Width =30 feet

Step-by-step explanation:

It is given that a  rectangular patio measures 20 feet by 30 feet.

To find the area of first patio.

Area = length * breadth

 = 20 * 30 = 600 square feet

To find the value of x

It is given that new area will be the double the area of first patio.

Here area = 2 * 600 = 1200 square feet

Here length = 30 + x

width = 20 + x

Area = (30 +x)(20 + x) = 1200

x² + 50x + 600 =1200

x² + 50x - 600 = 0

By solving this quadratic equation we get,

x = 10 and x = -60

take positive value x = 10

To find the dimensions of new patio

New length = 30 + x = 30 + 10 = 40 feet

width= 20 + x = 20 + 10 = 30 feet

It would take 200 years (by definition of half-life). 50 is 1/2 of 100.

Answer: [tex]\dfrac{4}{165}[/tex]

Step-by-step explanation:

Given : The number of yellow balls in the box = 4

The total number of balls = [tex]4+2+5=11[/tex]

Since the given situation has dependent events.

Then, the probability of that all three balls are yellow is given by :-

[tex]\text{P(YYY)}==\dfrac{^4P_3}{^{11}P_3}=\dfrac{4\times3\times2}{11\times10\times9}\\\\\Rightarrow\ \text{P(YYY)}=\dfrac{4}{165}[/tex]

Hence, the probability of winning =[tex]\dfrac{4}{165}[/tex]

Answer:

4m²n

Step-by-step explanation:

The GCF is the factor by which all the terms among the given can be divided.

In the expression 8m²n³-24m²n²+4m³n,

The GCF between 8,24 and 4 is 4

The greatest common factor between m²,m² and m³ is m²

The greatest common factor between n³, n² and n is n

Thus multiplying the three we get:

4×m²×n

=4m²n

Answer:

[tex]\large\boxed{\text{if}\ m\geq0,\ \text{then}\ \sqrt{m^6}=m^3}\\\\\boxed{\text{if}\ m<0,\ \text{then}\ \sqrt{m^6}=-m^3}[/tex]

Step-by-step explanation:

[tex]\sqrt{m^6}=\sqrt{m^{3\cdot2}}\qquad\text{use}\ (a^n)^m=a^{nm}\\\\=\sqrt{(m^3)^2}\qquad\text{use}\ \sqrt{a^2}=|a|\\\\=|m^3|\\\\\text{if}\ m\geq0,\ \text{then}\ \sqrt{m^6}=m^3\\\\\text{if}\ m<0,\ \text{then}\ \sqrt{m^6}=-m^3[/tex]

Answer:M^3 is the square root of m6

Step-by-step explanation:

good luck!!!

Answer:   -0.36

Step-by-step explanation:

Given: Mean : [tex]\mu=10\text{ minutes}[/tex]

Standard deviation : [tex]\sigma=5\text{ minutes}[/tex]

We know that the formula to calculate the z-score is given by :-

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

For x=8.2 minutes, we have

[tex]z=\dfrac{8.2-10}{5}=0.36[/tex]

Hence, the z-score for this amount of advertising time = -0.36

Final answer:

The z-score for the observed amount of advertising time (8.2 minutes) compared to the radio station's average of 10 minutes with a standard deviation of 5 minutes is -0.36, when rounded to two decimal places.

Explanation:

To calculate the z-score for the amount of advertising time observed on the radio station (8.2 minutes), we use the formula: Z = (X - μ) / σ, where X is the value to calculate the z-score for, μ is the mean of the data, and σ is the standard deviation. Plugging in the given values: X = 8.2 minutes (amount of advertising time observed), μ = 10 minutes (average advertising time), σ = 5 minutes (standard deviation).

So, the z-score is calculated as follows:

Z = (8.2 - 10) / 5 = -1.8 / 5 = -0.36.

Thus, the z-score of the amount of advertising time (8.2 minutes) is -0.36, rounded to two decimal places.

I think to solve this question, subtract the rate of feet per second disappearing from the distance of feet between the 2 cars.

100-85=15fps

The car is traveling at 15 feet per second.

Hope this helps!

Answer:  George receive 44 % of his full dose .

Step-by-step explanation:

Given : The amount for full dose : 25  grams

The amount received by George = 11 grams

Now, the percent of his full dose received by George :-

[tex]\dfrac{\text{Dose taken by George}}{\text{Full dose}}\times100[/tex]

[tex]=\dfrac{11}{25}\times100=44\%[/tex]

Hence, George receive 44 % of his full dose .