Find a counter example to show that the following statement is false:

For all nonzero real numbers a, b, c, d, a/b + c/d = a+c/b+d

A. a=3, b=5,c=-3,d=5
B. a=0, b=4,c=0,d=9
C. a=-2, b=1,c=2,d=1
D.a=1, b=2,c=1,d=2

Answer:

42 in

Step-by-step explanation:

108 in = 2l + 2w

l = 12 in

108 = 24 + 2w

84 = 2w

w = 42

The width is 21 inches, leading to a length of 33 inches, which corresponds to option c.

To find the length of the rectangular poster, we will use the formula for the perimeter of a rectangle, which is P = 2l + 2w, where P represents the perimeter, l represents the length, and w represents the width. According to the question, we have a perimeter P of 108 inches and the length is 12 inches greater than the width. If we let w represent the width, then the length can be represented as w + 12. Plugging into the perimeter formula we get: 108 = 2(w + 12) + 2w.

Now, let's solve for w:

108 = 2w + 24 + 2w

108 = 4w + 24

108 - 24 = 4w

84 = 4w

w = 84 / 4

w = 21 inches

Now that we have the width, we can find the length by adding 12 inches:

Length = w + 12

Length = 21 + 12

Length = 33 inches

Therefore, the length of the poster is 33 inches, which corresponds to option c.

Answer:

see below

Step-by-step explanation:

a

y = 2x -3

Standard form is Ax + By =C  where A is a positive integer and B and C are integers

Subtract 2x from each side

y-2x = 2x-2x -3

-2x+y = -3

We want A to be positive

Multiply each side by -1

2x -y = 3

This is in standard form

b

y = 2/3 x -7

Subtract 2/3x from each side

y-2/3x = 2/3x-2/3x -7

-2/3x+y = -7

We want A to be positive integer

Multiply each side by -3

-3*(-2/3x+y) = -7*-3

2x -3y = 21

c

y = -3x +1/2

Add 3x to each side

3x +y = -3x+3x +1/2

3x+y = 1/2

Multiply each side by 2

2(3x+y) = 1/2*2

6x+2y = 1

Answer:

f(x) = -logx + 4.

Step-by-step explanation:

y = -log x  approaches  the point (0, 1)  but this approaches (0, 5)

Looks like the graph of - log x + 4.

Answer:

[tex]y = 5-\sqrt{x}[/tex]

Step-by-step explanation:

Note that the graph shown is a transformation of the parent function

[tex]y = \sqrt{x}[/tex]

The main function cuts the x-axis at x = 0 and has no negative values of y. In addition, the main function is always growing.

Note that the function shown is decreasing and intercepts the y-axis at y = 5.

Therefore, the graph of the main function has been reflected on the x-axis and displaced 5 units upwards.

If we do y = f(x) then to reflect the main function on the  x-axis and move it upwards 5 units we make the following transformation

[tex]y = -f(x) + 5[/tex].

Then the graphed function is:

[tex]y = 5-\sqrt{x}[/tex]

Answer:

total return is -34.4828 %  and The annual return is - 8.1094 %

Step-by-step explanation:

given data

principal amount of share = $2900

selling amount pay = $1900

time period (t) = 5 years

to find out

total return and The annual return

solution

we know interest formula

amount = principal [tex](1 + r/100)^{t}[/tex]

put all these value amount principal, time n we get rate

1900 = 2900 [tex](1 + r/100)^{5}[/tex]

[tex](1 + r/100)^{5}[/tex]  = 1900/2900

1 + r/100 = 0.9189

r = −8.1094

The annual return is - 8.1094 %

and total return is ((selling amount pay - principal amount ) /  principal  amount )  × 100

put these value and we get total return

total return  = ((1900 - 2900) / 2900 )  × 100

total return  =  -0.344828  × 100

total return  =  -34.4828 %

Answer: a) [tex]t=\frac{2.303}{k}\log\frac{a}{a-x}[/tex]

b) 3.15 weeks.

c) 0.11 grams

Step-by-step explanation:

a) radioactive decay follows first order kinetics and thus Expression for rate law for first order kinetics is given by:

[tex]k=\frac{2.303}{t}\log\frac{a}{a-x}[/tex]

where,

k = rate constant  = ?

t = time taken for decomposition = 1 week

a = initial amount of the reactant  = 1 g

a - x = amount left after decay process  = 0.8 g

Now put all the given values in above equation, we get

[tex]k=\frac{2.303}{1}\log\frac{1}{0.8}[/tex]

[tex]k=0.22weeks^{-1}[/tex]

2) To calculate the half life, we use the formula :

[tex]t_{\frac{1}{2}=\frac{0.693}{k}[/tex]

[tex]t_{\frac{1}{2}=\frac{0.693}{0.22}=3.15weeks[/tex]

Thus half life of Thorium 234 is 3.15 weeks.

3) amount of Thorium 234 present after 10 weeks:

[tex]10=\frac{2.303}{0.22}\log\frac{1}{a-x}[/tex]

[tex](a-x)=0.11g[/tex]

Thus amount of Thorium 234 present after 10 weeks is 0.11 grams

Answer:

The probability that mike will guess​ correctly is 0.0027397 or [tex]\frac{1}{365}[/tex].

Step-by-step explanation:

Consider the provided information.

The number of days in a year is 365 (Ignore leap​ years).

[tex]Probability = \frac{favorable\ outcomes}{possible\ outcomes}[/tex]

Here, favorable outcomes is 1 and total number of outcomes are 365.

Substitute these value in above formula.

[tex]Probability = \frac{1}{365}[/tex]

[tex]Probability = 0.0027397[/tex]

Thus, the probability that mike will guess​ correctly is 0.0027397 or [tex]\frac{1}{365}[/tex].

Probability of an event represents the chances of occurrence of that event.

The probability that Mike will guess Kelly's birth date correctly (ignoring leap years) is [tex]\dfrac{1}{365} \approx 0.00027[/tex]

Suppose that there are finite elementary events in the sample space of the considered experiment, and all are equally likely.

Then, suppose we want to find the probability of an event E.

Then, its probability is given as

[tex]P(E) = \dfrac{\text{Number of favorable cases}}{\text{Number of total cases}}[/tex]

Where favorable cases are those elementary events who belong to E, and total cases are the size of the sample space.

Since we know that in a year (non leap year), there are 365 days, and Kelly's birthday can be on any one of those date, so the total number of days to chose from is 365 and since birthday of Kelly is going to be on single day of whole year, so favorable case is only single.

Thus,

if E = Selecting Kelly's birth date correctly,

Then

[tex]P(E) = \dfrac{1}{365} \approx 0.00027[/tex]

Thus,

The probability that Mike will guess Kelly's birth date correctly (ignoring leap years) is [tex]\dfrac{1}{365} \approx 0.00027[/tex]

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Step-by-step explanation:

Probability, P = [tex]\frac{no. of favourable outcomes}{Total no. of possible outcomes}[/tex]

Out of 8846 heart pacemaker malfunctions cases, caused by firmware cases are 375

then, no. of cases not caused by firmware are:

8846 - 375 = 8471

Probability for three different pacemakers respectively is given by:

P(1) = [tex]\frac{8471}{8846}[/tex]

we select the next malfunctioned pacemaker from the remaining i.e., out of 8471, excluding the chosen malfunctioned pacemaker

P(2|1) =  [tex]\frac{8470}{8845}[/tex]

Therefore, the events are not independent of each other

Now, if the selection is without replacement, then

P(3|1 & 2) =  [tex]\frac{8469}{8844}[/tex]

Now, by general multiplication rule(as the events are not independent):

P(none of the 3 are caused by malfunction) =

[tex]\frac{8471}{8846}\times\frac{8470}{8845}\times\frac{8469}{8844}[/tex]

= 0.9019

P(none of the 3 are caused by malfunction)  = 90.19%

The probability is high, therefore the whole batch will be accepted.

Probability of an event represents the chances of that event to occur.

Suppose there are n mutually independent events.

The probability of their simultaneously occurrence is given as

[tex]P(A_1 \cap A_2 \cap ... \cap A_n) = P(A_1) \times P(A_2) \times ... \times P(A_n)[/tex]

(This is true only if all those events are mutually independent).

The probability of getting a failure(malfunction) in a randomly selected pacemaker for the given case is

[tex]\dfrac{375}{8846} \approx 0.042[/tex]

The probability of a pacemaker working fine = 1- its failure probability  = 1 - 0.042 = 0.958

Since each of the 3 selected pacemakers are independent for their failure or success of each other, thus, if we have:

[tex]A_1[/tex] = Event of properly working of first pacemaker

[tex]A_2[/tex] = Event of properly working of second pacemaker

[tex]A_3[/tex] = Event of properly working of third pacemaker

Then, as all three are independent for their working from each other, thus,

[tex]P(A_1 \cap A_2 \cap A_3) = P(A_1) \times P(A_2)\times P(A_3) = (0.958)^3 \approx 0.879\\\\\begin{aligned}{P(\text{batch getting selected}) &= P(\text{All three pacemaker working})\\& = P(A_1 \cap A_2 \cap A_3) \\&\approx 0.879\\\end{aligned}[/tex]

Thus, as this probability is high, thus, this procedure is likely to result in the entire batch being accepted.

Thus,

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The probability that exactly two telephones will end up being replaced under warranty is found to be 0.147.

The binomial distribution is based on the binomial theorem which can be written as (a + b)ⁿ = ⁿC₀aⁿb⁰ + ⁿC₁aⁿ⁻¹b¹ + ⁿC₂aⁿ⁻²b²+ ....+ ⁿCₙa₀bⁿ.

In order to use in the probability, there should be events independent of each other and the sum of there probabilities is 1.

Suppose the total number of telephones be x.

Then, the telephones submitted under warranty is 20% × x = 0.2x.

The number of telephones to be repaired is 60% × 0.2x = 0.12x

And, those to be replaced  are given as 40% × 0.2x = 0.08x.

Now, the probability of exactly two units being replaced out of 10 is given by binomial distribution as,

P(Exactly two units being replaced) = ¹⁰C₂(0.08)²(0.92)⁸

⇒ P(Exactly two units being replaced) = 0.147

Hence, the probability for the given case is obtained as 0.147.

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To solve this problem, we can use the concept of conditional probability. Given that 20% of telephones are submitted for service while under warranty and of these, 60% can be repaired and 40% must be replaced with new units, we can calculate the probability that exactly two telephones will be replaced under warranty.

To solve this problem, we can use the concept of conditional probability. Let's break down the information given:

Now, let's calculate the probability that exactly two telephones will be replaced:

Therefore, the probability that exactly two telephones will end up being replaced under warranty is approximately 0.0572, or 5.72%.

Answer:

Step-by-step explanation:

In order to solve the inequality, follow the simple steps:

–3(5y – 4) ≥ 17 .

Dividing both sides with -3:

5y - 4 ≤ -17/3 (the sign of the inequality becomes opposite whenever a negative number is either multiplied or divided on both the sides of the inequality).

Adding 4 on both sides:

5y ≤ -5/3

Dividing 5 on both sides:

y ≤ -1/3.

This shows that all the values of y less than and equal to -1/3 satisfy the inequality.

The number line has been attached. Since it involves ≤ sign, therefore, a filled circle will be used to plot the inequality. Less than means that all the values on the left hand side of the number line will be included. This is denoted with an arrow (see the diagram)!!!

Answer with explanation:

 It is given that, abc=1

[tex]\rightarrow \frac{1}{1+a+b^{-1}}+\frac{1}{1+b+c^{-1}}+\frac{1}{1+c+a^{-1}}\\\\\rightarrow \frac{b}{b+ab+1}+\frac{c}{c+bc+1}+\frac{a}{a+ac+1}\\\\abc=1\\\\\rightarrow \frac{b}{b+ab+abc}+\frac{c}{c+bc+abc}+\frac{a}{a+ac+abc}\\\\\rightarrow \frac{1}{1+a+ac}+\frac{1}{1+b+ab}+\frac{1}{1+c+bc}\\\\\rightarrow \frac{1}{abc+a+ac}+\frac{1}{1+b+ab}+\frac{1}{1+c+bc}\\\\\rightarrow \frac{1+a}{a(bc+1+c)}+\frac{c}{c+bc+1}\\\\\rightarrow\frac{1+a+ac}{a(bc+1+c)}\\\\\rightarrow\frac{1+a+ac}{abc+a+ac)}\\\\\rightarrow\frac{1+a+ac}{1+a+ac)}\\\\=1[/tex]

Hence proved.

Answer:

ABC=1

Step-by-step explanation:

Answer:$2.7 dollars per piece

Step-by-step explanation: Divide 113.40 by 42

$2.7 per piece is the answer.

$113.40 / 42 piece set of stainless steel = $2.7

Problem-solving is the act of defining a problem; figuring out the purpose of the trouble; identifying, prioritizing, and selecting alternatives for an answer; and imposing an answer.

Problem-solving starts with identifying the issue. As an example, a trainer may need to parent out a way to improve a scholar's overall performance on a writing talent test. To do that, the instructor will overview the writing exams looking for regions for improvement.

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Answer:

=SUM(C2:C4, C7)

Step-by-step explanation:

In excel 'C' represent the cell ( intersection of row and column ),

Also, '=sum( )' represents the function of addition in the excel,

Also, when we operate all consecutive cells from [tex]Cn[/tex] to [tex]Cm[/tex]

Then under the function formula we write [tex]Cn:Cm[/tex]

And, when we operate a cell [tex]Cn[/tex] with non consecutive cell [tex]Cl[/tex]

Then under the function formula we write [tex]Cn, Cl[/tex]

Hence, C2+C3+C4+C7 can be written as,

=SUM(C2:C4, C7)

Answer:

We need to put $801.88 amount in the bank.

Step-by-step explanation:

given that

t=3 yr

Need amount $850 after 3 yr so P= $850

Interest rate=2%

We know that for simple interest

[tex]P=A\left(1+\dfrac{rt}{100}\right)[/tex]

Where r is the Interest rate,t is the time,A is the present amount and P is principle amount after t time.

Here given that  P= $850

So now putting the values

[tex]850=A\left(1+\dfrac{2\times 3}{100}\right)[/tex]

So A=$801.88

We need to put $801.88 amount in the bank.

Final answer:

Madison can choose from 4368 different combinations of 9 movies, including all 4 dramas and 5 out of the 16 non-drama movies.

Explanation:

Madison is picking out movies to rent and wants all 4 dramas out of her 9 selections. With 7 foreign films, 3 horror films, 6 action movies, and 4 dramas, the total number of non-drama movies is 16 (7+3+6). Since she must select all 4 dramas, she needs to choose the remaining 5 movies from the 16 non-drama movies available.

To find how many different combinations of 9 movies she can rent, we use the combination formula C(n, r) = n! / [r!(n-r)!], where n is the total number of items to choose from, r is the number of items to choose, and ! denotes factorial, representing the product of all positive integers up to that number.

The question asks us to calculate C(16, 5) because Madison is choosing 5 movies out of the 16 non-drama movies. This calculation yields:

C(16, 5) = 16! / [5!(16-5)!] = 4368

Therefore, there are 4368 different combinations of 9 movies Madison can rent if she wants all 4 dramas.

Answer with explanation:

Given : The probability of grocery bags are classified as plastic = 0.97

(a) f two bags are chosen at random.

Then , the probability that both grocery bags are plastic​ is given by :-

[tex]^2C_2(0.97)^2(1-0.97)^0=0.9409[/tex]

(b) If five grocery bags are chosen at random.

Then , the probability that all five grocery bags are plastic​ is given by :-

[tex]^5C_5(0.97)^5(1-0.97)^0\approx0.8587[/tex]

(c) The probability of getting paper = 1-0.97=0.03

The probability that at least one of five randomly selected grocery bags is paper :-

[tex]P(x\geq1)=1-P(0)\\\\1-^5C_0(0.03)^0(0.97)^5\\\\=1-(0.03)^0(0.97)^5=0.1412659\approx0.14>0.05[/tex]

Thus , it would not be unusual that at least one of five randomly selected grocery bags is paper.

Probability is used to determine the chances of an event

[tex]p = 97\%[/tex] -- chances that a bag is plastic

(a) Both plastics selected are plastic

This is calculated as:

[tex]P(2) = p^2[/tex]

So, we have:

[tex]P(2) = (97\%)^2[/tex]

[tex]P(2) = 0.9409[/tex]

Hence, the probability that both grocery bags are plastic is 0.9409

(b) All five are plastic

This is calculated as:

[tex]P(5) = p^5[/tex]

So, we have:

[tex]P(5) = (97\%)^5[/tex]

[tex]P(5) = 0.8587[/tex]

Hence, the probability that all five grocery bags are plastic is 0.8587

(c) At least one of the five is paper

The probability that none of the five is paper is the same as the probability that all five is plastic.

So:

[tex]P(None) = 0.8587[/tex]

Using complement rule,

[tex]P(At\ least\ 1) = 1 - P(None)[/tex]

So, we have

[tex]P(At\ least\ 1) = 1 - 0.8587[/tex]

[tex]P(At\ least\ 1) = 0.1413[/tex]

Hence, the probability that at least one grocery bags is paper is 0.1413

The above probability is greater than 0.05

Hence, it would not be unusual that at least one of five randomly selected grocery bags is paper.

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Answer:

Option A

Step-by-step explanation:

Given that in a study of the relationship between geographical mobility (number of times a person has changed residences) and number of friends, Pearson's r^2 is reported as .40.

r square, being the coefficient of determination explains the variablity of one variable due to the variability of the other.  Here Mobility explains 40% variation in the number of friends is right answer.

Option B is wrong because r = ±6324, so cannot say positive or negative.

Similarly option c is wrong because we are unsure whether negative correlation.  Option d is wrong since 16% is not right .

Answer:

The average daily sales rate for candy is 37.4.

Step-by-step explanation:

We know that,

[tex]\text{Inventory turnover }= \frac{\text{Total sale}}{\text{Average inventories}}[/tex]

[tex]\implies \text{Total sale}=\text{Inventory turnover }\times \text{Average inventories}[/tex]

Given,

Inventory turnover of candies = 3.3,

Average inventories = 340

So, sale of candies = 3.3 × 340 = 1122

Now,

[tex]\text{Average daily sales rate}=\frac{\text{total sale}}{\text{Number of days}}[/tex]

Since, 1 month = 30 days ( approx ),

Hence, the  average daily sales rate for candy = [tex]\frac{1122}{30}[/tex]=37.4

First,

We are dealing with parabola since the equation has a form of,

[tex]y=ax^2+bx+c[/tex]

Here the vertex of an up - down facing parabola has a form of,

[tex]x_v=-\dfrac{b}{2a}[/tex]

The parameters we have are,

[tex]a=-5,b=-10, c=6[/tex]

Plug them in vertex formula,

[tex]x_v=-\dfrac{-10}{2(-5)}=-1[/tex]

Plug in the [tex]x_v[/tex] into the equation,

[tex]y_v=-5(-1)^2-10(-1)+6=11[/tex]

We now got a point parabola vertex with coordinates,

[tex](x_v, y_v)\Longrightarrow(-1,11)[/tex]

From here we emerge two rules:

So our vertex is minimum value since,

[tex]a=-5\Longleftrightarrow a<0[/tex]

Hope this helps.

r3t40

Answer: 0.0427

Step-by-step explanation:

Given : The area under the standard normal curve to the left of z = -1.72 is​ 0.0427

We know that the normal curve is a bell shaped curve that is symmetric such that half of the data falls to the left of the mean (Mean lies at the middle of the curve) and half of data falls to the right.

Now, z=-1.72 lies on the left side and z=1.72 lies right side.

Since, normal curve is symmetric and the magnitude of the values if same , then the area under the standard normal curve to the left of z = -1.72 is​  equals to the area to the right of z = 1.72 is 0.0427

Therefore, the area under the standard normal curve to the right of z = 1.72  is 0.0427

The area under the standard normal curve to the right of z=1.72 is 0.9573 or 95.73%, which can be calculated by subtracting the area to the left of z from 1 (1 - 0.0427 = 0.9573).

The area under the standard normal curve to the left of z=-1.72 is 0.0427. This essentially means that 4.27% of all observations fall under this score. Conversely, the area to the right of the z-score represents the proportion of observations that are greater than z. In a normal distribution, the sum of the areas to the left and to the right of a z-score must always equal to 1 (0.0427 + x = 1), because all possible outcomes are accounted for by a normal distribution. Hence, we can find out the area to the right by simply subtracting the area to the left from 1. So, the area under the standard normal curve to the right of z=1.72 is 1 - 0.0427 = 0.9573 or 95.73%.

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Answer:

Option D.

Step-by-step explanation:

Arianna kicks a soccer ball into the air with an initial velocity = 42 feet per second

If the starting height of the ball is 0 feet then we have to find the approximate maximum height the soccer ball attached.

Since we know vertical motion is represented by

v² = u² - 2gh

Where v = final velocity of the object

u = initial velocity of the object

g = gravitational force

h = h = maximum height achieved by the object

Here v = 0, u = 42 feet per second and g = 32 feet per second²

Now we plug in these values in the formula

0 = (42)² - 2(32)(h)

1764 = 64h

[tex]h=\frac{1764}{64}[/tex]

h = 27.56 feet

  ≈ 27.60 feet

Therefore, maximum height achieved by the soccer ball is 27.6 feet.

Option D is the answer.

Answer:

Option D is 2.76 ft

Step-by-step explanation:

Its correct

Answer:

g(-4) = -4

g(-2) = 1

g(1) = -4

Step-by-step explanation:

The value of the given function is -4 for all values of x other than -2 and 1 if x=-2

So,

For x=-4 the value of function will be -4.

g(-4) = -4

For x=-2

The value of function is -2.

g(-2) = 1

And for x=1, the value will be -4.

g(1) = -4 ..

Answer with explanation:

Suppose, A={(a,b),(b,a), (c,d),(d,c),(p,q),(q,p),(a,a),(b,b)}

A Relation M is Symmetric , if (p,q)∈M , then (q,p)∈M.

⇒It is given that, R and S are symmetric Relation  on a Set A.

⇒If R is symmetric, then if (a,b)∈R, means,(b,a)∈R.So, R={(a,b),(b,a)}.

⇒If S is Symmetric, then if (c,d)∈S, means,(d,c)∈S.So, S={(c,d),(d,c)}.

⇒R ∩ S ={(a,b),(b,a),(c,d),(d,a)}

⇒If you will look at the elements of Set , R∩S, there is (a,b)∈ R∩S,so as (b,a)∈ R∩S.Also, (c,d)∈ R∩S,so as (d,a)∈ R∩S.

Which shows Relation in the set ,  R∩S is symmetric.

Answer:

Faster worker takes 4 hours and slower worker takes 12 hours.

Step-by-step explanation:

Let x be the time ( in hours ) taken by faster worker,

So, according to the question,

Time taken by slower worker = (x+8) hours,

Thus, the one day work of faster worker = [tex]\frac{1}{x}[/tex]

Also, the one day work of slower worker = [tex]\frac{1}{x+8}[/tex]

So, the total one day work when they work together = [tex]\frac{1}{x}+\frac{1}{x+8}[/tex]

Given,

They take 3 hours in working together,

So, their combined one day work = [tex]\frac{1}{3}[/tex]

[tex]\implies \frac{1}{x}+\frac{1}{x+8}=\frac{1}{3}[/tex]

[tex]\frac{x+8+x}{x^2+8x}=\frac{1}{3}[/tex]  ( Adding fractions )

[tex]3(2x+8)=x^2+8x[/tex]    ( Cross multiplication )

[tex]6x+24=x^2+8x[/tex]       ( Distributive property )

[tex]x^2+2x-24=0[/tex]          ( Subtraction property of equality )

By quadratic formula,

[tex]x=\frac{-2\pm \sqrt{100}}{2}[/tex]

[tex]x=\frac{-2\pm 10}{2}[/tex]

[tex]\implies x=4\text{ or }x=-6[/tex]

Since, hours can not negative,

Hence, time taken by faster worker = x hours = 4 hours,

And, the time taken by slower worker = x + 8 = 12 hours.

To solve this work rate problem, we set up an equation with combined work rates and find that the faster worker takes 3 hours alone, while the slower worker takes 11 hours alone.

The question states that two secretaries can stuff envelopes together in 3 hours. The slower worker takes 8 hours more than the faster worker to complete the job alone. To find how long it takes each secretary to complete the job alone, we can set up an equation using the reciprocal of their work rates.

Let x be the time it takes for the faster worker to stuff the envelopes alone. Then, the slower worker will take x + 8 hours. The work rate of the faster worker is 1/x and the slower worker's rate is 1/(x + 8). Working together, their combined work rate is 1/3 per hour (since they complete the task in 3 hours).

The combined work rate equation will be:

1/x + 1/(x + 8) = 1/3

To solve this equation:

As a result, the faster worker takes 3 hours to complete the job alone, and the slower worker takes 11 hours to complete the job alone.

Answer:   The required general solution is

[tex]y(x)=Ae^{-x}+e^x(B\cos x+C\sin x).[/tex]

Step-by-step explanation:  We are given to find the general solution of the following differential equation :

[tex]y^{\prime\prime\prime}-y^{\prime\prime}+2y=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]

Let y = y(x) and [tex]y=e^{mx}[/tex] be an auxiliary solution of equation (i).

Then, we have

[tex]y^\prime=me^{mx},~~~y^{\prime\prime}=m^2e^{mx},~~~y^{\prime\prime\prime}=m^3e^{mx}.[/tex]

Substituting these values in equation (i), we have

[tex]m^3e^{mx}-m^2e^{mx}+2e^{mx}=0\\\\\Rightarrow (m^3-m^2+2)e^{mx}=0\\\\\Rightarrow m^3-m^2+2=0~~~~~~~~~~~~~~~~~~~~~~~~~~[\textup{since }e^{mx}\neq0]\\\\\Rightarrow m^2(m+1)-2m(m+1)+2(m+1)=0\\\\\Rightarrow (m+1)(m^2-2m+2)=0\\\\\Rightarrow m+1=0~~~~~\Rightarrow m=-1[/tex]

and

[tex]m^2-2m+2=0\\\\\Rightarrow (m^2-2m+1)+1=0\\\\\Rightarrow (m-1)^2=-1\\\\\Rightarrow m-1=\pm\sqrt{-1}\\\\\Rightarrow m=1\pm i~~~~~~~~~~~~~~~~~~~~~~~~~~~[\textup{where }i^2=-1][/tex]

So, we get

[tex]m_1=-1,~~m_2=1+i,~~m_3=1-i.[/tex]

Therefore, the general solution of the given equation is given by

[tex]y(x)=Ae^{m_1x}+e^{1\times x}(B\cos 1x+C\sin 1x)}\\\\\Rightarrow y(x)=Ae^{-x}+e^x(B\cos x+C\sin x).[/tex]

Thus, the required general solution is

[tex]y(x)=Ae^{-x}+e^x(B\cos x+C\sin x).[/tex]

Answer with explanation:

⇒(2 x y )d x+(x-6 y) d y=0

P= 2 x y

Q=x-6 y

[tex]P_{y}=2 x\\\\Q_{x}=1[/tex]

So this Differential Equation is exact.

To solve this, we will first evaluate,[tex]\varphi (x,y)[/tex].

[tex]\varphi_{x}=P\\\\\varphi_{y}=Q\\\\\varphi=\int P d x\\\\= \int 2 x y  dx\\\\\varphi=x^2 y\\\\\varphi(x,y)=x^2y+k(y)------(1)[/tex]

Differentiating with respect to , y

[tex]\varphi'(x,y)=x^2+k'(y)=Q=x-6 y\\\\\rightarrow x-6 y-x^2=k'(y)\\\\ k(y)=\int (x-6 y -x^2) dy\\\\k(y)=x y-3 y^2-x^2 y+f\\\\\varphi(x,y)=x^2y+x y-3 y^2-x^2 y+f\\\\\varphi(x,y)=x y-3 y^2+f[/tex]

Substituting the value of , k(y) in equation 1.

This is required Solution of exact differential equation.

Answer:

125 mL/h

Step-by-step explanation:

Intravenous pumps are used when there is a need of constant pumping of fluids into the body. Injecting every hour with the required volume causes damage.

Volume of 0.45 saline = 1000 mL

Time to be infused = 8 hours

The Intravenous pump should be programmed in the following way

[tex]\frac{Volume}{time}=\frac{1000}{8}\\\Rightarrow flow\ rate=125\ mL/hr[/tex]

Hence, the IV pump should be programmed for 125 mL/hr.

Answer:

Step-by-step explanation:

Divisibility rule when a number is divisible by 11:

Take the alternating sum of digits in a number read from left to right .If that is divisible by 11 then the the given number is divisible by 11.

It is a divisibility rule of 11.

Let 1342 is a number .The number is divisible by 11 or not

Alternating sum of digits  =1-3+4-2=0

The alternating sum of digits is divisible by 11 .

Therefore, the number 1342 is divisible by 11.

Let a number 2728

Alternating sum of digits = 2-7+2-8=-11.

The alternating sum of digits is divisible by 11 .

Therefore, the number 2728 is divisible by 11.

The rule for determining divisibility by 11 involves alternately adding and subtracting a number's digits; if the result is 0 or divisible by 11, the original number is too. Examples illustrate the application of the divisibility test.

Rule for Divisibility by 11

The rule for determining when a number is divisible by 11 is quite straightforward. To apply the rule, you alternate between adding and subtracting the digits of the number in question from left to right. If the resulting sum (which may be negative) is 0 or divisible by 11 itself, then the original number is divisible by 11. For example, consider the number 2728. Apply the rule like this: 2 - 7 + 2 - 8 = -11. Since -11 is divisible by 11, 2728 is also divisible by 11.

Examples

Another example: Take the number 5831. Using the rule: 5 - 8 + 3 - 1 = -1, which is not divisible by 11, hence 5831 is not divisible by 11.

This divisibility test is a fantastic example of how certain mathematical rules are used to quickly determine properties of numbers without needing to perform long divisions.

Answer:

He sold 200 lower floor tickets.

Step-by-step explanation:  

Consider the provided information.

Pierre sold 340 tickets for a concert.

Let Balcony tickets represents by x and lower floor ticket represents by y.

As we know he sold 340 tickets in total, i.e.

x + y = 340 ......(1)

Balcony tickets cost $5 while tickets for the lower floor cost $10.

If he sold x balcony tickets, so the money he got after selling x balcony tickets is 5x. Similarly, If he sold y lower floor tickets, so the money he got after selling y lower floor tickets is 10y.

Pierre sold $2,700 worth of tickets, i.e.

5x + 10y = 2700 ......(2)

Now use elimination method to solve the system of equations:

Multiply the equation (1) with 5 and subtract them as shown below:

5x + 5y = 1700

5x + 10y = 2700

_____________

-5y = -1000

  y = 200

Thus, he sold 200 lower floor tickets.

The area is given by the integral,

[tex]\displaystyle\int_{-1/2}^{1/2}(1-2x^2-|x|)\,\mathrm dx[/tex]

The integrand is even, so we can simplify the integral somewhat as

[tex]\displaystyle2\int_0^{1/2}(1-2x^2-|x|)\,\mathrm dx[/tex]

When [tex]x\ge0[/tex], we have [tex]|x|=x[/tex], so this is also the same as

[tex]\displaystyle2\int_0^{1/2}(1-2x^2-x)\,\mathrm dx[/tex]

which has a value of

[tex]2\left(x-\dfrac23x^3-\dfrac12x^2\right)\bigg|_0^{1/2}=2\left(\dfrac12-\dfrac1{12}-\dfrac18\right)=\boxed{\dfrac7{12}}[/tex]

so that A = 7.

Answer: Slope would be,

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

Step-by-step explanation:

Here, the given curve,

[tex]x^3 + y^3=C^3[/tex]

[tex]\implies x^3 + y^3 - C^3=0[/tex]

In Barrow's method,

Steps are as follows,

Step 1 : put, x = x - e, y = y - a

[tex](x-e)^3 + (y-a)^3 - C^3=0[/tex]

[tex]x^3-3x^2e+3xe^2-e^3+y^3-3y^2a+3ya^2-a^3+C^3=0[/tex]

Step 2 : Reject terms which do not contain a or e,

[tex]-3x^2e+3xe^2-e^3-3y^2a+3ya^2-a^3=0[/tex]

Step 3 : Reject all terms in which a or e have exponent greater than 1,

[tex]-3x^2e-3y^2a=0[/tex]

Step 4 : Find the ratio of a : e,

[tex]-3y^2a=3x^2e[/tex]

[tex]\implies \frac{a}{e}=-\frac{x^2}{y^2}[/tex]

Hence, the slope of the given curve is [tex]-\frac{x^2}{y^2}[/tex]