What value of x will make parallelogram ABCD a rhombus?

Answer:

0 < t < [tex]\frac{5}{3}[/tex]

After 1.67 days the stocks would be sold out.

Step-by-step explanation:

The price of a certain computer stock after t days is modeled by

p(t) = 100 + 20t - 6t²

Now we will take the derivative of the given function and equate it to zero to find the critical points,

p'(t) = 20 - 12t = 0

t = [tex]\frac{20}{12}[/tex]

t = [tex]\frac{5}{3}[/tex] days

Therefore, there are two intervals in which the given function is defined

(0, [tex]\frac{5}{3}[/tex]) and ([tex]\frac{5}{3}[/tex], ∞)

For the interval (0, [tex]\frac{5}{3}[/tex]),

p'(1) = 20 - 12(1) = 20

For the interval ([tex]\frac{5}{3}[/tex], ∞),

p'(2) = 20 - 12(2) = -4

Positive value of p'(t) in the interval (0, [tex]\frac{5}{3}[/tex]) indicates that the function is increasing.

0 < t < [tex]\frac{5}{3}[/tex]

Since at the point t = 1.67 days curve is showing the maximum, so the stocks should be sold after 1.67 days.

The price of the computer stock rises for the first 5/3 days after it is issued. Ideally, if you owned the stock, you should sell it after 5/3 days to maximize your profit.

The price of the stock becomes maximum when the rate of change of the price is 0. To find when that is, we need to take the derivative of the price function, which is p'(t) = 20 - 12t, and set it equal to 0. Solving for t, we get t = 20/12 = 5/3. So, the price of the stock rises for t in (0, 5/3) days. Therefore, if you owned the stock, you should sell it after 5/3 days to maximize your profit.

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Answer:

20/7 days (just less than 3 days)

Step-by-step explanation:

Recall that (1 job) = (rate)(time), so time = (1 job) / (rate).

Set up and solve the following equation:

  1 job

------------------------------- = time required for 2 pumps working together

1 job         1 job

---------- + -------------

4 days      10 days

This comes out to:

              1 job

------------------------------------------- = time required

10 job-days      4 job-days

------------------ + -----------------

   40 days           40 days

or:

    1 job

-------------------- = (40/14) days, or 20/7 days (just less than 3 days)

14 job·days

-----------------

   40 days  

The number of days required to complete the work together by the pump is 20 / 7 days.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that one pump can empty a pool in 4 days, whereas a second pump can empty the pool in 10 days.

The number of days will be calculated as,
1 / N  = ( 1 / 10 ) + ( 1 / 4 )

1 / N = (10 + 4 ) / 40

1 / N = 14 / 40

1 / N = 7 / 20

N = 20 / 7 days

Therefore, the number of days required to complete the work together by the pump is 20 / 7 days.

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The total cost to join the website for one year is $34.00

Step-by-step explanation:

Onetime membership fee = $10

Per month newsletter charges = $2.00

One year = 12 months

Newsletter charges for 1 year = [tex]2.00*12=\$24[/tex]

Total cost = One time fee + newsletter charges for 1 year

[tex]Total\ cost=10+24.00=\$34.00[/tex]

The total cost to join the website for one year is $34.00

Keywords: addition, multiplication

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Final answer:

To demonstrate that E[1/X] for a geometric random variable X with success probability p equals −p log(p)/(1−p), we use the pmf of a geometric distribution and turn the sum into an integral, ultimately showing the expected result through integration.

Explanation:

To show that for a geometric random variable X with parameter p, the expected value E[1/X] is −p log(p)/(1−p), we begin by recognizing that the probability mass function (pmf) of a geometric distribution with success probability p is given by P(X = x) = p(1-p)^(x-1) for x = 1, 2, 3, ... . To find E[1/X], we sum the values of 1/x multiplied by the probability of each x, which is ∑ (1/x)*p(1-p)^(x-1).

To evaluate this, we express each term α_i/i, where α_i = p(1-p)^(i-1), as an integral from 0 to 1-p of x^(i-1) dx. We then change the order of summation and integration, which is permissible under the conditions of the monotone convergence theorem:
∑ (α_i / i) = ∑ ∫_{0}^{1-p} x^(i-1) dx

= ∫_{0}^{1-p} (∑ x^(i-1)) dx

= ∫_{0}^{1-p} (p / (1 - x)) dx

= p ∫_{0}^{1-p} 1/(1 - x) dx

= -p log(1 - (1-p))

= -p log(p) / (1-p).

Answer:

8. [tex]\displaystyle \frac{9[x + 5]}{x - 14}[/tex]

7. [tex]\displaystyle -\frac{2x - 1}{2[3x - 5]}[/tex]

6. [tex]\displaystyle \frac{2[x - 4]}{5[x + 3]}[/tex]

5. [tex]\displaystyle \frac{2x + 7}{x + 3}[/tex]

4. [tex]\displaystyle 3x^{-1}[/tex]

Step-by-step explanation:

All work is shown above from 8 − 4.

I am joyous to assist you anytime.

Answer:

  24

Step-by-step explanation:

Counting Li Ana, there are 6 friends distributing fliers, so each one is distributing an amount calculated as ...

  (144 fliers)/(6 persons) = 24 fliers/person

Each person will distribute 24 fliers.

Answer:

Part (A): There are 9 integers between 5 and 31 which are divisible by 3.

Part (B): There are 6 integers between 5 and 31 which are divisible by 4.

Part (C): There are 2 integers between 5 and 31 which are divisible by 3 and by 4.

Step-by-step explanation:

Consider the provided information.

Part (A) we need to find how many integers between 5 and 31 are divisible by 3.

Between 5 and 31 there are 25 integers.

According to quotient rule: [tex]\frac{25}{3} \approx8.33[/tex]

That means either 8 or 9 integers are divisible by 3 as 8.33 lies between 8 and 9.

The integers are: 6, 9, 12, 15, 18, 21, 24, 27, 30

Hence, there are 9 integers between 5 and 31 which are divisible by 3.

Part (B) we need to find how many integers between 5 and 31  are divisible by 4.

Between 5 and 31 there are 25 integers.

According to quotient rule: [tex]\frac{25}{4} \approx6.25[/tex]

That means either 6 or 7 integers are divisible by 4, as 6.25 lies between 6 and 7.

The integers are: 8, 12, 16, 20, 24, 28

Hence, there are 6 integers between 5 and 31 which are divisible by 4.

Part (C) we need to find how many integers between 5 and 31 are divisible by 3 and by 4

Between 5 and 31 there are 25 integers.

Integers should be divisible by 3 and by 4, that means integers should be divisible by 3×4=12.

According to quotient rule: [tex]\frac{25}{12} \approx2.08[/tex]

That means either 2 or 3 integers are divisible by 3 and by 4 or 12, as 2.08 lies between 2 and 3.

The integers are: 12, 24,

Hence, there are 2 integers between 5 and 31 which are divisible by 3 and by 4.

To find positive integers that are divisible by 3, 4, or both between 5 and 31, we can determine the multiples of each number. The multiples of 3 are: 6, 9, 12, 15, 18, 21, 24, 27, and 30. The multiples of 4 are: 8, 12, 16, 20, 24, and 28. The multiples of both 3 and 4 (or their least common multiple, 12) are: 12 and 24.

a) To find the positive integers between 5 and 31 that are divisible by 3, we need to look for numbers that are multiples of 3. Starting with 6, the first multiple of 3 in this range, we continue adding 3 to each number until we reach the highest multiple less than or equal to 31. So the multiples of 3 between 5 and 31 are: 6, 9, 12, 15, 18, 21, 24, 27, and 30.

b) To find the positive integers between 5 and 31 that are divisible by 4, we need to look for numbers that are multiples of 4. Starting with 8, the first multiple of 4 in this range, we continue adding 4 to each number until we reach the highest multiple less than or equal to 31. So the multiples of 4 between 5 and 31 are: 8, 12, 16, 20, 24, and 28.

c) To find the positive integers between 5 and 31 that are divisible by both 3 and 4, we need to find the common multiples of 3 and 4. This can be done by finding the multiples of the least common multiple (LCM) of 3 and 4, which is 12. So the multiples of 12 between 5 and 31 are: 12 and 24.

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Answer:

k=545/43948.8

Step-by-step explanation:

widgets per second

The constant of proportionality for the scenerio given will be 11.53

The general equation of a straight line is -

[y] = [m]x + [c]

where -

[m] is slope of line which tells the unit rate of change of [y] with respect to [x].

[c] is the y - intercept i.e. the point where the graph cuts the [y] axis.

The equation of a straight line can be also written as -

Ax + By + C = 0

By = - Ax - C

y = (- A/B)x - (C/A)

We have 545 workers who worked 7 hours and  produced 43948.8 widgets and the number of widgets that a manufacturing plant can produce varies jointly as the number of workers and the time that they have worked.

The equation can be represnted using the equation of a straight line. We can write -

y = kx

Now, x = 545 x 7 = 3815

So -

43948.8 = (3815)k

k = 43948.8/3815

k = 11.52

Therefore, the constant of proportionality for the scenerio given will be 11.53

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Answer:

D. X= ± square root of 8 - 2

Step-by-step explanation:

Given quadratic equation is \[x^{2}+4x=4\]

Rearranging the terms: \[x^{2}+4x-4=0\]

This is the standard format of quadratic equation of the form \[ax^{2}+bx+c=0\]

Here, a=1 , b=4 and c=-4.

Roots of the quadratic equation are given by \[\frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\]

Substituting the values and calculating the roots:

\[\frac{-4 \pm \sqrt{(-4)^{2}-4*1*(-4))}}{2*1}\]

= \[\frac{-4 \pm \sqrt{32}}{2}\]

= \[\frac{-2*2 \pm 2*\sqrt{8}}{2}\]

= \[-2 \pm \sqrt{8}\]

Hence option D is the correct option.

The solution of x² + 4x = 4 is x =  ± √8 - 2,

Hence, option D is correct.

The given quadratic equation is,

x² + 4x = 4

Here we have to solve it by completing square method

Now proceed the expression,

⇒ x² + 4x = 4

Adding 4 both sides,

⇒ x² + 4x + 4 = 4 + 4

⇒ x² + 4x + 4 = 8

Since we know that 2² is equal to 4, then

⇒ x² + 4x + 2² = 8

Since we know that, Formula of complete square,

(a+b)² = a² + 2ab + b²

Therefore,

⇒ (x+2)² = 8

Taking square root both sides we get,

⇒ (x+2) = ±√8

Hence,

x =  ± √8 - 2

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The subject of this question is mathematics, specifically quadratic functions. The student needs to find the interval of the domain that represents the airplane while it was in the air.

The subject of this question is Mathematics, specifically quadratic functions. The question asks about a quadratic function that models the height of a toy airplane relative to the ground. The student needs to find the interval of the domain that represents the airplane while it was in the air.

To find the interval of the domain, we need to determine the maximum horizontal distance the airplane traveled before landing. The horizontal distance is represented by the variable x in the quadratic function. Since the horizontal distance was 50 feet, the interval of the domain should be [0, 50]. This means the airplane was in the air for a horizontal distance between 0 and 50 feet.

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Answer:

Converse: If the circumference of the two circle is same then they have same diameter.

Inverse: If two circle do not have same diameter then they do not have same circumference.

Contrapositive: If two circle do not have same circumference then they do not have same diameter.

Step-by-step explanation:

Let the statement's be,

p : Two circles have same diameter.

q : Two circles have same circumference.

The given statement is a conditional statement logically is given as

p → q

It means p implies q, that is if p then q

where p is called antecedent or hypothesis and

           q is called consequent or conclusion.

The converse of this statement logically is given as

q → p

It means q implies p, that is If q then p.

Converse: If the circumference of the two circle is same then they have same diameter.

The Inverse of this statement logically is given as

~p → ~q

It means negation p implies negation q, that is If not p then not q.

Inverse: If two circle do not have same diameter then they do not have same circumference.

The Contrapositive of this statement logically is given as

~q → ~p

It means negation q implies negation p, that is If not q then not p.

Contrapositive: If two circle do not have same circumference then they do not have same diameter.

Final answer:

The converse, inverse, and contrapositive statements of the original mathematical statement concerning circles and their diameters and circumferences are explained in relation to the formula C = πd.

Explanation:

The original statement given is: If two circles have the same diameter, then they have the same circumference.

The converse of this statement would be: If two circles have the same circumference, then they have the same diameter.

The inverse of the original statement is: If two circles do not have the same diameter, then they do not have the same circumference.

The contrapositive is: If two circles do not have the same circumference, then they do not have the same diameter.

In mathematics, particularly in geometry, a circle's circumference (C) is related to its diameter (d) by the formula C = πd, where π is a constant approximately equal to 3.14159. Hence, two circles with the same diameter will indeed have the same circumference.

Answer:

Step-by-step explanation:

The equation of a straight line is usually represented in the slope-intercept form, y = mx + c

Where c = y intercept

m = slope

We want to determine the slope of the line parallel to 2y=3x+6

Rearranging 2y=3x+6 in the slope intercept form, it becomes

2y/2 = 3x/2 + 6/2

y = 3x/2 + 3

The slope = 3/2

If two lines are parallel to each other, the their slopes are equal.

So the slope of the line parallel to 2y=3x+6 is 3/2

To determine slope of the line perpendicular to y=8x+24

Comparing y=8x+24 with the slope intercept form, y = m+ c,

Slope, m = 8

If two limes are perpendicular, then the product of the slopes is -1

Let the slope of the perpendicular line to the one given by the above equation be m1. Therefore,

8 × m1 = -1

8 m1 = -1

m1 = -1/8

,

Answer:

The linear inequalities representing the given situation is -

x≥0 , y≥0

13x + 9y ≥ 368

x + y ≤ 40

Step-by-step explanation:

Let Rolando work 'x' hours a week at starbucks and 'y' hours a week at the youth soccer league.

∵ No. of hours cannot be negative -

x≥0 ,  y≥0

Given that Rolando earns $13 an hour at starucks -

Total earning at starbucks per week = 13x

Also Rolando earns $9 an hour at the youth soccer league -

Total earning at the youth soccer league per week = 9y

Total earnings for Rolando per week = 13x+9y

Since his total earnings needs to be atleast $368 -

13x+9y ≥ 368

Total number of hours for which Rolando can work per week is -

x+y

Since he cannot work for more than 40 hours a week -

x+y ≤ 40.

∴ x≥0 , y≥0

  13x + 9y ≥ 368

  x + y ≤ 40

Final answer:

Rolando's work situation can be modeled by the linear inequalities 13x + 9y ≥ 368 and x + y ≤ 40, where x is the number of hours at Starbucks and y is the number at the youth soccer league.

Explanation:

To represent Rolando’s work situation with linear inequalities, let us define two variables: let x be the number of hours Rolando works at Starbucks, and let y be the number of hours he works at the youth soccer league. Given that Rolando earns $13 per hour at Starbucks and $9 per hour at the youth soccer league, the inequality to ensure he makes at least $368 per week is 13x + 9y ≥ 368.

Furthermore, because Rolando can work a maximum of 40 hours per week, we have another inequality: x + y ≤ 40. Thus, the system of linear inequalities that models Rolando’s situation is:

13x + 9y ≥ 368

x + y ≤ 40

These inequalities define the range of hours Rolando can work at both jobs to meet his weekly earnings goal without exceeding the maximum number of allowed hours.

Answer:

Answer is reflection across line. When transformation maps X"Y"Z" .Then first transformation for this composition is , and the second transformation is 90° rotation about point X'.

Step-by-step explanation:

When you transform a point across a line y=x then x-coordinate and y-coordinate changes.

Similarly when you reflect a point across the line y=-x then signs are changed and both coordinate change places.

It is the first transformation across the line.

Answer:

d, a reflection across line m

Step-by-step explanation:

Just did it on edge 2020

(its right)

Answer: (0.076, 0.140)

Step-by-step explanation:

Confidence interval for population proportion (p) is given by :-

[tex]\hat{p}\pm z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]

, where [tex]\hat{p}[/tex] = sample proportion.

n= sample size.

[tex]\alpha[/tex] = significance level .

[tex]z_{\alpha/2}[/tex] = critical z-value (Two tailed)

As per given , we have

sample size : n= 500

The number of Independents.:  x= 54

Sample proportion of Independents[tex]\hat{p}=\dfrac{x}{n}=\dfrac{54}{500}=0.108[/tex]

Significance level 98% confidence level : [tex]\alpha=1-0.98=0.02[/tex]

By using z-table , Critical value : [tex]z_{\alpha/2}=z_{0.01}=2.33[/tex]

The 98% confidence interval for the true percentage of Independents among Haywards 50,000 registered voters will be :-

[tex]0.108\pm (2.33)\sqrt{\dfrac{0.108(1-0.108)}{500}}\\\\=0.108\pm2.33\times0.013880634\\\\=0.108\pm0.03234187722\\\\\approx0.108\pm0.032=(0.108-0.032,\ 0.108+0.032)=(0.076,\ 0.140)[/tex]

Hence, the 98% confidence interval for the true percentage of Independents among Haywards 50,000 registered voters.= (0.076, 0.140)

Final answer:

To calculate a 98% confidence interval for the percentage of Independents among registered voters in Hayward, we use the sample proportion and z-score to find that the true percentage is likely between 8% and 13.6%.

Explanation:

To calculate a 98% confidence interval for the true percentage of Independents among Hayward's 50,000 registered voters, based on a sample of 54 Independents out of 500 voters, we use the formula for the confidence interval for a proportion, which is p ± z*[tex]\sqrt{ ((p(1-p))/n)}[/tex], where p is the sample proportion, z is the z-score corresponding to the confidence level, and n is the sample size.

First, we find the sample proportion (p) of independents: p=54/500 = 0.108 or 10.8%.

Next, we look up the z-score for a 98% confidence level, which is approximately 2.33.

We then plug the values into the formula: 0.108 ± 2.33*[tex]\sqrt{((0.108(1-0.108))}[/tex]/500), and calculate the confidence interval.

Calculating the margin of error: 2.33*[tex]\sqrt{((0.108(1-0.108))}[/tex])/500) ≈ 0.028 or 2.8%.

Therefore, the 98% confidence interval is 10.8% ± 2.8%, which means it ranges from 8% to 13.6%.

The true percentage of Independents among Hayward's registered voters is likely between 8% and 13.6% with 98% confidence.

Answer:

40 hours will it take for the pool to fill.

Step-by-step explanation:

A pump can fill a swimming pool in 8 hours.

Work done by pump to fill in 1 hour is  [tex]\frac{1}{8}[/tex]

The pool also has a drain that can empty the pool in 10 hours.

Work done by pump to drain in 1 hour is  [tex]\frac{1}{10}[/tex]

If someone turns on the pump to fill the pool, but forgets to shut the drain.

Work done by both pipe in 1 hour is

[tex]W=\frac{1}{8}-\frac{1}{10}[/tex]

[tex]W=\frac{10-8}{80}[/tex]

[tex]W=\frac{2}{80}[/tex]

[tex]W=\frac{1}{40}[/tex]

Both pipe filled [tex]\frac{1}{40}[/tex] part of pool in hours = 1

Both pipe filled complete pool in hours = [tex]\frac{1}{\frac{1}{40}}=40[/tex]

Therefore, 40 hours will it take for the pool to fill.

The possible values of X and their associated probabilities are:

4: with probability 6/91

1: with probability 64/91

4: with probability 28/91

To determine the probabilities associated with each possible value of X (winnings), we need to consider all the possible combinations of selecting two balls from the urn:

Possible Cases:

Two Black Balls:

Probability = (4C2 / 14C2) * (4C2 / 14C2) = 6/91, where 4C2 represents choosing 2 balls from 4 black balls and 14C2 represents choosing 2 balls from a total of 14 balls.

Winnings (X) = $2 * 2 = $4.

One Black Ball, One White Ball:

There are two scenarios: Black-White and White-Black.

Probability (Black-White) = (4C1 * 8C1) / 14C2 = 32/91.

Probability (White-Black) = (8C1 * 4C1) / 14C2 = 32/91.

Combining both scenarios, total probability = 64/91.

Winnings (X) = $2 * 1 - $1 * 1 = $1.

Two White Balls:

Probability = (8C2 / 14C2) = 28/91.

Winnings (X) = $1 * 0 - $2 * 2 = -$4.

Therefore, the possible values of X and their associated probabilities are:

4: with probability 6/91

1: with probability 64/91

4: with probability 28/91

Final answer:

To determine the probabilities for X, the winnings from selecting two balls out of an urn with 8 white, 4 black, and 2 orange balls, one must calculate the probability of each possible combination of draws and the associated winnings.

Explanation:

To complete the explanation and provide accurate calculations for the probabilities associated with each possible value for X (the winnings from selecting two balls), we need to adjust and clarify the possible outcomes and their associated winnings based on the given scenario:

1. Two Black Balls:

  - Winning: $4 (since each black ball is worth $2)

  - Probability [tex]: \( \frac{4}{14} \times \frac{3}{13} \)[/tex] because there are 4 black balls out of 14 total, and then 3 out of 13 after the first is drawn.

2. One Black and One White Ball:

  - Winning: $1 (since a black ball is +$2 and a white ball is -$1)

  - Probability:[tex]\( 2 \times \frac{4}{14} \times \frac{8}{13} \)[/tex] because there are two ways this can happen (black then white or white then black).

3. Two White Balls:

  - Winning: -$2 (since each white ball is -$1)

  - Probability:[tex]\( \frac{8}{14} \times \frac{7}{13} \)[/tex] because there are 8 white balls initially, then 7 out of the remaining 13.

4. One Black Ball and One Orange Ball:

  - Winning: $2 (since a black ball is +$2 and an orange ball has no change)

  - Probability:[tex]\( 2 \times \frac{4}{14} \times \frac{2}{13} \)[/tex] because there are two ways this can occur (black then orange or orange then black).

5. One White Ball and One Orange Ball:

  - Winning: -$1 (since a white ball is -$1 and an orange ball has no change)

  - Probability: [tex]\( 2 \times \frac{8}{14} \times \frac{2}{13} \)[/tex] because there are two sequences for this outcome (white then orange or orange then white).

6. Two Orange Balls

  - Winning: $0 (since orange balls have no change)

  - Probability:[tex]\( 2 \times \frac{8}{14} \times \frac{2}{13} \)[/tex]  because there are only 2 orange balls.

Answer:

5

Step-by-step explanation:

To answer this, we must figure out the minimum amount we can add 32 to itself to reach 150. This means that we must multiply 32 by our desired amount, x, to reach the amount of hours closest to 150 (but above/equal to it). To find x, we can divide 150 by 32, and if it gives us a number with no decimals or remainders, that is x, as it is the smallest amount of people to get above/equal to 150. If it gives us a remainder/decimal, we must round up, as we cannot have a part of a person and we must get the work done, so we must get at least 150 hours. 150/32 is 4.6 something, so we must round up. 5 is our answer

The number of people required to do the work of 150 hours in a week is 5.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that the work in an office takes 150 hours to complete every week. Each person in the office works 32 hours a week.

The number of people will be calculated as,

N = ( 150 / 32 )

N = 4.7 or 5 people

Therefore, the number of people required to do the work of 150 hours in a week is 5.

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Answer:

  It takes the second worker 11 hours to do the job alone  

Step-by-step explanation:

The problem tells you how to work it.

  [tex]\dfrac{1}{11}+\dfrac{1}{x}=\dfrac{2}{11}\\\\x+11=2x \qquad\text{multiply by 11x}\\\\11=x \qquad\text{subtract x}[/tex]

x = 11 tells you the second worker takes 11 hours to do the job alone.

_____

You can also work directly with the fractions. Subtract 1/11 and you have ...

  1/x = 1/11

It should not be a real stretch to see that x=11. If you need, you can multiply both sides by 11x to get 11=x.

Answer:

1/4

Step-by-step explanation:

Take my word for it.

(Yes I know this answer is late)

To solve the equation 3(‒4x + 5) = 12, apply the distributive property, then use subtraction to isolate the variable, and lastly, divide to find the value of x, which is 0.25.

To solve the linear equation 3(−4x + 5) = 12, we start by applying the distributive property to eliminate the parentheses:

Next, we use the subtraction property of equality to isolate the variable:

Finally, we apply the division property of equality to solve for x:

Answer:

Step-by-step explanation:

1) Twelve more than the product of 5 and a number x. This is expressed as

y = 5x + 12

when x = 2,

y = 5×2 +12 = 22

and when y = 10

10 = 5x + 12

5x = - 2

x = -2/5

2) 15 decreased by the product of a number x and 4. This is expressed as

y = 15 - 4x

When x = 2

y = 15 - 4×2 = 7

When y = 10

10 = 15 - 4x

4x = 15 - 10 = 5

x = 5/4

3) You eat 5 slices of bread. Your friend eats 2 slices fewer than you eat.

Let y represent the number of slices that your friend ate. The expression will be

y = 5 - 3 = 2

y = 2

4) Your uncle is 2 years older than 3 times your age. A. You are x years old. Let your uncle be y years old. Therefore,

y = 3x + 2

If you are 12 years old, your uncle well be

3×12 + 2 = 38 years old

Answer:

an = 4/3n - 13/3.

Step-by-step explanation:

The first term is a1,

a24 = a1 + 23d

83/3 = a1 + 4/3* 23

a1 =    83/3 - 92/3

a1 = -9/3 = -3.

So the nth term an =  -3 + 4/3(n - 1)

an = -3 + 4/3 n - 4/3

an = 4/3n - 13/3

To find the explicit formula for the general nth term of an arithmetic sequence, use the formula a_n = a_1 + (n - 1)d, where a_n represents the nth term, a_1 is the first term, and d is the common difference. In this case, the explicit formula is a_n = -3 + (n - 1)(4/3), based on given information a_24 = 83/3 and d = 4/3.

To find the explicit formula for the general nth term of an arithmetic sequence, we use the formula: a_n = a_1 + (n - 1)d, where a_n represents the nth term, a_1 is the first term, and d is the common difference. In this case, we are given that a_24 = 83/3 and d = 4/3. We can substitute these values into the formula and solve for a_1. From there, we can simplify the formula and express it in its lowest terms.

Given: a_24 = 83/3 and d = 4/3

We can rearrange the formula and solve for a_1 as follows:

Now that we have found a_1 = -3, we can simplify the formula and express it as:

a_n = -3 + (n - 1)(4/3)

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Answer:

Option B

Step-by-step explanation:

multiplying 2/5 with 5/2 will give 1 on the left hand side of the equation

multiplying 10 with 5/2 will give 25 on the right hand side of the equation, ultimately resulting in  x=25

The number that we must use to multiply each side of the equation 2/5x = 10 to produce the equivalent equation x = 25 is; B: 5/2

We are given the equation;

(2/5)x = 10

Multiplying both sides by the same number is same as the original equation.

Thus, to make the left hand side only x, let us multiply both sides by the inverse of 2/5 which is 5/2 to get;

x = 10 × 5/2

x = 25

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(a) The formula for linear approximation: Δg ≈ g × (Δr / r). (b) The acceleration will be negative. (c) The percentage change in acceleration due to gravity when moving from sea level to the top of Mount Elbert is approximately 0.06703%.

(a) To find the linear approximation to the change in acceleration due to gravity, Δg, when increasing the distance from the centre of the Earth by Δr = x, use the formula for linear approximation:

Δg ≈ g × (Δr / r)

Here, g is the original acceleration due to gravity, Δr is the change in distance (x), and r is the original distance from the centre of the Earth.

(b) The change in acceleration, Δg, will be negative. This is because as you move farther away from the centre of the Earth, the gravitational acceleration decreases, which means Δg is a negative value.

(c) Given the height of Mount Elbert, Δr = 4.29 km, and assuming the radius of the Earth is r = 6400 km, we can use the linear approximation formula from part (a) to find the percentage change in acceleration due to gravity:

Δg ≈ g × (Δr / r)

Δg ≈ g × (4.29 / 6400)

Take the absolute value to find the percentage change:

Percentage Change = |(Δg / g)| × 100

Percentage Change = (Δg / g) × 100

Percentage Change = [(g × 4.29 / 6400) / g] × 100

Percentage Change = (4.29 / 6400) × 100

Calculate the percentage change:

Percentage Change ≈ (0.0006703125) × 100

Percentage Change ≈ 0.06703%

So,  the formula for linear approximation: Δg ≈ g × (Δr / r), the acceleration will be negative, and the percentage change in acceleration due to gravity when moving from sea level to the top of Mount Elbert is approximately 0.06703%.

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The gravitational acceleration decreases as you move away from the Earth, with the rate of change proportional to the amount you increase your distance away from the Earth. The decrease is negative and is very minuscule even when moving from sea level to the top of Mount Elbert in Colorado.

The acceleration due to gravity, g, is inversely proportional to the square of the distance from the center of the Earth, r. Therefore, as we move away from the Earth, the value of g decreases.

For part (a), the change in acceleration due to an increase in distance (Δg) would be the derivative of g concerning r, times Δr=x. Therefore, Δg is approximately equal to -GM/r3x, where G is the gravitational constant and M is the mass of the Earth. When divided by g (which is GM/r2), this gives Δg/g = -x/r.

For part (b), this means that as you increase your distance away from the Earth, the gravitational acceleration decreases, therefore Δg is negative.

For part (c), the percentage change in g is given by the change in distance (4.29 km in this case), divided by the original radius of the Earth (6400 km), times 100%. Therefore, the percentage change is approximately 0.067% or an extremely small change.

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Answer: Option (C)

Explanation:

Linear programming is referred to as the mathematical method designed in order to assist individuals to plan and thus make decisions which are necessary in order to allocate the resource. Under linear programming, first an individual defines the objective, thereby also looking at the limits i.e. constraints that are being put forth. Also defining the alternatives and the linearity.

A linear programming problem requires an objective, constraints, alternatives, and linearity. The objective is what you're trying to achieve, constraints are limitations or restrictions, alternatives are options available to reach your objective, while linearity necessitates that all functions in the problem must be linear.

The correct answer is C: An objective, constraints, alternatives, and linearity. These are the four requirements of a linear programming problem. The objective refers to what the problem is attempting to achieve, such as minimizing cost or maximizing profit. Constraints are the limitations or restrictions of the problem. Alternatives refer to the different options or decisions available to reach the objective. Finally, the requirement of linearity is that all functions in the problem, whether they are objective or constraints, must be linear.

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Answer: the other zeros are -3/2 and 5/3

Step-by-step explanation:

A) The function F(x) becomes F(x) = 500(sec x)²

B) The average force exerted by the press over the interval [0, pi/3] is 825.7 N.

Given that,

The force F (in newtons) of the hydraulic cylinder in a press is proportional to the square of sec x.

And, The domain of F is [0, pi/3], and F(0) = 500

A) For F as a function of x,

Since F is proportional to the square of sec x.

We are also given that F(0) = 500.

Let's denote the constant of proportionality as k.

Hence write the equation as:

F(x) = k(sec x)²

To find the value of k, substitute x = 0:

500 = k(sec 0)²

Since sec 0 = 1, we get:

k = 500

So, the function F(x) becomes:

F(x) = 500(sec x)²

B) For the average force exerted by the press over the interval [0, pi/3],  evaluate the average value of F(x) over this interval.

The average value of a function f(x) over an interval [a, b] is given by:

Average value = (1 / (b - a)) × ∫[a to b] f(x) dx

In this case, a = 0 and b = pi/3.

Average value = (1 / (pi/3 - 0)) × ∫[0 to pi/3] 500(sec x)² dx

Simplifying, we get:

Average value = (3/pi) × ∫[0 to pi/3] 500(sec x)² dx

Integrating (sec x)², we have:

Average value = (3/pi) [500 tan x] [from 0 to pi/3]

Evaluating this expression, we get:

Average value ≈ 825.7 N

Therefore, the average force exerted by the press over the interval [0, pi/3] is 825.7 N.

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The force F of the hydraulic cylinder is defined by the equation F(x)=500(secx)^2. The average force over the interval [0, pi/3] can be found by integrating this function over the interval and dividing by the interval length.

The force F of the hydraulic cylinder is directly proportional to the square of sec x, where sec x = 1/cos x. Given that F(0)=500, it implies that when x=0, F=500, so the proportionality constant k can be determined by substituting these values into the equation, as F=k(secx)^2. Where cos 0 = 1, therefore sec 0 = 1. So we get F=k*(1)^2 => k=500. Therefore, the equation that defines F as a function of x is F(x)=500(secx)^2.

To find the average force exerted by the press over the interval [0,pi/3], we need to integrate this function over the given interval and divide by the length of the interval. Therefore: F_avg = (1/(pi/3 - 0))∫ from 0 to pi/3 (500(secx)^2) dx. Solving this definite integral equation will yield the average force.

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Answer:

p*(3/10)+14

Step-by-step explanation:

There are 10 girls for every 3 boys. Therefore, if we have the number of girls, p, the number of boys is equal to p*(3/10). Then, adding the 14 boys that left, our answer is p*(3/10)+14

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