URGENT PLZ HELP. DUE TOMORROW

Answer: the length of the second piece is 1 5/6 inches

Step-by-step explanation:

Mary had a link of yarn 4 1/3 inches. Converting 4 1/3 inches to improper fraction, it becomes 13/3 inches.

She cut it into two pieces for her art project. One of the pieces was 2 1/2 inches long. Converting 2 1/2 inches to improper fraction, it becomes 5/2 inches.

Therefore, the length of the second piece would be

13/3 - 5/2 = (26 - 15)/6 = 11/6 inches.

Converting to mixed fraction, it becomes

1 5/6 inches

Answer:

It will take about 35.49 hours for the water to leak out of the barrel.

Step-by-step explanation:

Let [tex]y(t)[/tex] be the depth of water in the barrel at time [tex]t[/tex],  where [tex]y[/tex] is measured in inches and [tex]t[/tex] in hours.

We know that water is leaking out of a large barrel at a rate proportional to the square root of the depth of the water at that time. We then have that

                                                 [tex]\frac{dy}{dt}=-k\sqrt{y}[/tex]

where [tex]k[/tex] is a constant of proportionality.

Separation of variables is a common method for solving differential equations. To solve the above differential equation you must:

Multiply by [tex]\frac{1}{\sqrt{y}}[/tex]

[tex]\frac{1}{\sqrt{y}}\frac{dy}{dt}=-k[/tex]

Multiply by [tex]dt[/tex]

[tex]\frac{1}{\sqrt{y}}\cdot dy=-k\cdot dt[/tex]

Take integral

[tex]\int \frac{1}{\sqrt{y}}\cdot dy=\int-k\cdot dt[/tex]

Integrate

[tex]2\sqrt{y}=-kt+C[/tex]

Isolate [tex]y[/tex]

[tex]y(t)=(\frac{C}{2} -\frac{k}{2}t)^2[/tex]

We know that the water level starts at 36, this means [tex]y(0)=36[/tex]. We use this information to find the value of [tex]C[/tex].

[tex]36=(\frac{C}{2} -\frac{k}{2}(0))^2\\C=12[/tex]

[tex]y(t)=(\frac{12}{2} -\frac{k}{2}t)^2\\\\y(t)=(6 -\frac{k}{2}t)^2[/tex]

At t = 1, y = 34

[tex]34=(6 -\frac{k}{2}(1))^2\\k=12-2\sqrt{34}[/tex]

So our formula for the depth of water in the barrel is

[tex]y(t)=(6 -\frac{12-2\sqrt{34}}{2}t)^2\\\\y(t)=\left(6-\left(6-\sqrt{34}\right)t\right)^2\\[/tex]

To find the time, [tex]t[/tex], at which all the water leaks out of the barrel, we solve the equation

[tex]\left(6-\left(6-\sqrt{34}\right)t\right)^2=0\\\\t=3\left(6+\sqrt{34}\right)\approx 35.49[/tex]

Thus, it will take about 35.49 hours for the water to leak out of the barrel.

The time it will take for all of the water to drain out of that barrel is 35.5 hours approx.

Let there are two variables p and q

Then, p and q are said to be directly proportional to each other if

[tex]p = kq[/tex]

where k is some constant number called constant of proportionality.

This directly proportional relationship between p and q is written as

[tex]p \propto q[/tex]  where that middle sign is the sign of proportionality.

In a directly proportional relationship, increasing one variable will increase another.

Now let m and n are two variables.

Then m and n are said to be inversely proportional to each other if

[tex]m = \dfrac{c}{n} \\\\ \text{or} \\\\ n = \dfrac{c}{m}[/tex]

(both are equal)

where c is a constant number called constant of proportionality.

This inversely proportional relationship is denoted by

[tex]m \propto \dfrac{1}{n} \\\\ \text{or} \\\\n \propto \dfrac{1}{m}[/tex]

As visible, increasing one variable will decrease the other variable if both are inversely proportional.

For the considered case, let we take three variables as:

Now, when  t = 0, x = 36 inches.

and at t = 1, x = 34 inches.

So depth of water is function of time passed. Let x = f(t)

Also, y is negative rate of change of x with respect to t(since depth is decreasing, and draining is measuring decrement rate, thus, negative of increment rate), or

[tex]y = -\dfrac{dx}{dt}[/tex]

We're given that: "Water is leaking out of a large barrel at a rate proportional to the square root of the depth of the water at that time"

That means, [tex]y \propto \sqrt{x}[/tex]

Let the constant of proportionality be k, then,

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

Since we've [tex]y = -\dfrac{dx}{dt}[/tex], therefore,

[tex]-\dfrac{dx}{dt} = k\sqrt{x}\\\\-\dfrac{dx}{\sqrt{x}} = kdt\\\\\text{Integrating both the sides, we get}\\\\\int -\dfrac{dx}{\sqrt{x}} = \int kdt\\\\-2\sqrt{x} = kt + C[/tex]

where C is integration constant.

Since at t = 0 hours passed, x = 36 inches, and at t = 1 hour passed, x = 34 inches, we get two equations as:

[tex]-2\sqrt{x} = kt + C\\-2\sqrt{36} = -12 = C\\-2\sqrt{34} = k + C[/tex]

Putting value of C from first equation in second, we get:

[tex]k = -2\sqrt{34} + C \\k = -2\sqrt{34} - 12[/tex]

Therefore, the relationship between depth of water and time we get is:

[tex]-2\sqrt{x} = (-2\sqrt{34} - 12)t -12\\\sqrt{x} = (\sqrt{34} - 6)t + 6[/tex]

When the whole barrel gets empty, the depth of water becomes 0. The time for it is calculated using above equation as:

[tex]\sqrt{x} = (\sqrt{34} - 6)t + 6\\0 = (\sqrt{34} - 6)t + 6\\t = \dfrac{6}{6 - \sqrt{34}} \approx 35.5[/tex](in hours)

Thus, the time it will take for all of the water to drain out of that barrel is 35.5 hours approx.

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The probability of having fewer than 4 successes (x < 4) in 8 independent trials with a success probability of 0.6 is 0.1758.

We have,

The binomial probability formula:

[tex]P(x) = (^nC_x) p^x (1-p)^{n-x}[/tex]

Where:

P(x) is the probability of x successes

n is the number of trials

p is the probability of success

nCx is the binomial coefficient, which represents the number of ways to choose x successes from n trials

For the given parameters:

n = 8

p = 0.6

x < 4

For x = 0:

[tex]P(0) = (^8C_0) (0.6^0) (1-0.6)^{8-0}\\= (1) (1) (0.4)^8[/tex]

= 0.0016

For x = 1:

[tex]P(1) = (^8C_1) (0.6^1) (1-0.6)^{8-1}\\= (8) (0.6) (0.4)^7[/tex]

≈ 0.0092

For x = 2:

[tex]P(2) = (^8C_2) (0.6^2) (1-0.6)^{8-2}\\= (28) (0.6^2) (0.4)^6[/tex]

≈ 0.0412

For x = 3:

[tex]P(3) = (^8C_3) (0.6^3) (1-0.6)^{8-3}\\= (56) (0.6^3) (0.4)^5\\= 0.1238[/tex]

Now, sum up these probabilities to find the probability of x < 4:

P(x < 4) = P(0) + P(1) + P(2) + P(3)

≈ 0.0016 + 0.0092 + 0.0412 + 0.1238

≈ 0.1758

Therefore,

The probability of having fewer than 4 successes (x < 4) in 8 independent trials with a success probability of 0.6 is 0.1758.

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Answer:

  A. 25% high

  B. 12.5% decrease

Step-by-step explanation:

A. The estimate relative to the actual turnout was ...

  7000/5600 = 1.25

The estimate was 25% high.

__

B. Relative to the previous election, the turnout was ...

  5600/6400 = 0.875 = 1 - 0.125

The percentage decrease from the previous election was 12.5%.

9 pairs of jeans and 12 shirts were sold on each day.

Let x be the number of jeans sold and y be the number of shirts sold. From the given information, we can form two equations:

25x + 18y = 441 (equation 1)

20x + 20y = 420 (equation 2)

Multiplying equation 2 by 5, we get 100x + 100y = 2100. Subtracting this equation from equation 1, we get -75x - 82y = -1659. Solving this equation, we find that x = 9 and y = 12.

Therefore, 9 pairs of jeans and 12 shirts were sold on each day.

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To get an average score of 90 in four subjects, Ami needs to score a combined total of 190 marks in Chemistry and Biology after having scored 75 in Maths and 95 in Physics.

Ami is aiming to achieve an average score of 90 across four subjects, with each subject having a maximum score of 100 marks. To determine the scores that she needs to achieve in Chemistry and Biology (the remaining two subjects), we begin by calculating the total marks she requires for the desired average.

Since the desired average is 90, for four subjects, the total marks needed would be: 90 marks/subject × 4 subjects = 360 marks.

Ami scored 75 in Maths and 95 in Physics, which sums up to: 75 + 95 = 170 marks.

To find out the remaining marks she needs, we subtract the marks she has already scored from the total marks needed for the average:

360 marks (total needed) - 170 marks (already scored) = 190 marks (needed for Chemistry and Biology).

Therefore, to ensure that she gets the desired average score of 90, she would need to score a combined total of 190 marks in Chemistry and Biology. This could be achieved by, for example, scoring 95 in both subjects or any other combination that adds up to 190.

To find out the minimum amount you need to save to buy the food truck (assuming a local business will donate three times your savings), divide the total cost of the truck by four. This is because the saved amount plus the donated amount (which is three times your savings) should be enough to cover the full cost of the truck.

The question is asking for the minimum amount that you need to save in order to buy a food truck, assuming a local business will donate 3 times the amount you have saved. To efficiently and accurately calculate this, it's best to start from the total cost of the truck and work backwards.

Firstly, let’s consider the cost to make the food truck fully functional as X (it's not specified whether it's the blue or green food truck). According to the information, the amount saved will be your contribution, and the local business will donate 3 times your saved amount. This means that the total money available to purchase the food truck will be 4 times the saved amount (the saved amount plus the three times donation).

To find out the minimum amount you need to save, we can use the formula:

minimum savings = total cost / 4

This equation shows that the minimum savings needed will be one-fourth of the total cost of the food truck.

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Answer: he had 68 cherries in the start

Step-by-step explanation:

In the beginning, Zak had 61 cherries and he gave away 18 cherries to Tim and 18 cherries to Janet.

The equation is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are equal.

Cherries are in a bag that Zak possesses. Tim and Janet each received 18 cherries from him. He currently has 25 cherries.

Let's assume the number of cherries Zak had at the start "x".

Zak gave away 18 cherries to Tim and 18 cherries to Janet, so he now has x - 18 - 18 = 25 cherries.

You can solve this equation for x by adding 18 + 18 to both sides, which gives you x = 25 + 18 + 18.

This simplifies to x = 61, so Zak had 61 cherries at the start.

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Answer:

EC = 1 ft

Step-by-step explanation:

DE // BC and AC and AD are transversal lines

∴ ∠E≅∠C  ⇒ corresponding angles are congruent

  ∠D≅∠B  ⇒ corresponding angles are congruent

 ∠A≅∠A   ⇒ Reflexive property

∴Δ ADE is similar to ΔABC  by AA postulate

So, The corresponding sides are in proportion.

[tex]\frac{AC}{AE} = \frac{AB}{AD}[/tex]

AE = 4 ft , AD = 8 ft , AB = 8 + 2 = 10 ft

AC = AE * AB/AD = 4*10/8 = 40/8 = 5 ft

EC = AC - AE = 5 - 4 = 1 ft

So, the Length of EC = 1 ft.

Answer:

3x+4x=35

Step-by-step explanation:

Answer:3x+4x=35

Step-by-step explanation:took the test and got it right

Answer:

13 square units

Step-by-step explanation:

You could calculate the length of each side using Pythagorean theorem, then use Heron's formula to find the area.  But there's an easier way: find the area of the rectangle that contains the triangle, and subtract the areas of the smaller triangles in the corners.

The area of the rectangle is 4 × 7 = 28.

The area of the upper left triangle is ½(2)(4) = 4.

The area of the upper right triangle is ½(3)(5) = 7.5.

The area of the lower right triangle is ½(1)(7) = 3.5.

So the area of the triangle is:

28 − 4 − 7.5 − 3.5 = 13

The sale price of a black coat is $ 2.4 less than sale price of a blue coat

Solution:

Given that,

Bluecoat usually cost $48 but it's on sale for 25% off

Cost price of blue coat = $ 48

Discount = 25 %

Therefore, discount price is given as:

Discount price = 25 % of 48

[tex]Discount\ price = 25 \% \times 48\\\\Discount\ price = \frac{25}{100} \times 48\\\\Discount\ price = 0.25 \times 48\\\\Discount\ price =12[/tex]

Thus sales price is given as:

Sales price = cost price - discount price

Sales price = 48 - 12

Sales price = 36

Thus sales price of blue coat is $ 36

Black coat is really cost $56 but it's on sale for 40%

Cost price of black coat = $ 56

Discount = 40 %

Therefore, discount price is given as:

Discount price = 40 % of 56

[tex]Discount\ price = 40 \% \times 56\\\\Discount\ price = \frac{40}{100} \times 56\\\\Discount\ price = 0.4 \times 56\\\\Discount\ price =22.4[/tex]

Thus sales price is given as:

Sales price = cost price - discount price

Sales price = 56 - 22.4

Sales price = 33.6

Thus sales price of black coat is $ 33.6

How much less is the sale price of a black coat in the sale price of a blue coat

Sales price of blue coat - sales price of black coat = 36 - 33.6 = 2.4

Thus sale price of a black coat is $ 2.4 less than sale price of a blue coat

Answer:

2.40

Step-by-step explanation:

Answer:

6cm

Step-by-step explanation:

Given

12J to stretch from 8cm to 10cm ---- Expression 1

Additional 48J from 10cm to 14cm ---- Expression 2

Let l represents the length of the spring

Let k represent string constant,

We can then write ( from expression 1)

12 = integral of kx dx

The lower bound being 8 - l

And the upper bound being 10 - l

Integrating kx dx

We have

½kx²

= ½k[x²]

= ½k[(10 - l)² - (8 - l)²].

So, 12 = ½k[(10 - l)² - (8 - l)²] ------ Equation 1

From expression 2, we can write

48 = integral of kx dx

The lower bound being 10 - l

And the upper bound being 14 - l

Integrating kx dx

We have

½kx²

= ½k[x²]

= ½k[(14 - l)² - (10 - l)²].

So, 48 = ½k[(14 - l)² - (10 - l)²] ------ Equation 2

Divide Equation 2 by Equation 1, so we get

48/12 = ½k[(14 - l)² - (10 - l)²] / ½k[(10 - l)² - (8 - l)²] ----- ½k cancel out ½k

So, w have

4 = [(14 - l)² - (10 - l)²] / [(10 - l)² - (8 - l)²]

Recall that a² - b² = (a + b)(a - b).

So,

4 = [(14 - l - 10 + l) (14 - l + 10 - l)] / [(10 - l - 8 + l) (10 - l + 8 - l)]

4 = [(4)(24 - 2l)]/[(2)(18-2l)]

4 = (96 - 8l)/36 - 4l) ---- Multiply both sides by 36 - 4l

4(36 - 4l) = 96 - 8l

144 - 16l = 96 - 8l ----- Collect Like Terms

144 - 96 = 16 - 8l

48 = 8l ----- Divide both sides by l

48/8 = l

6 = l --- Rearrange

l = 6

So, the natural length of the string is 6cm

Answer:

C. The chance to be selected into the sample was the same for all veterinarians

The statement that is NOT correct is D. The sample was a volunteer sample.

The statement that is NOT correct is D. The sample was a volunteer sample.

The given scenario describes a census survey, where questionnaires were sent to all veterinarians treating large animals. In a census survey, every member of the target population is included, so there is no sampling involved. Hence, there is no opportunity for the sample to be a volunteer sample. Therefore, option D is the incorrect statement.

A low response rate, as mentioned in option A, can lead to response bias because the respondents who choose to participate may have different characteristics than those who do not respond. The intended sample, as mentioned in option B, was the target population of veterinarians treating large animals. And option C is correct since all veterinarians on the list had an equal chance of being selected as part of the survey.

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Answer:

The population of interest is all the residents of the state.

Step-by-step explanation:

The term 'population of interest' is defined as the population under study from which the sample is drawn to make conclusions about the said population.

For instance, if a school principal wants to know the average SAT score for the students of his school, then his population of interest will be all the students of the school.

In this scenario a poll is conducted to determine which candidate, running for the governor's seat, has more support.

The sample of 500 registered voters are selected from the state for the poll.

This implies that the population under study consists of all the people of the state.

Thus, the population of interest in this case are all the residents of the state.

The correct option is: all the residents of the state.

The equation after the sum symbol is written in terms of ar^k-1

A = 12 and r = 0.7

To see if it converges use the formula a /1-r, if the answer is greater than 1 it converges, if it’s less than 1 it diverges

12 / 1 - 0.7 = 12/0.3 = 40

The answer C. Converges, 40

Answer:

Part a: The transformation matrix is the clockwise rotation matrix of π/4.

Part b: The hyperplane would move towards the mean of class 1.

Part c: The distance will remain in the Euclidean Space  due to the rotation transformation only.

Step-by-step explanation:

As the complete question is not available, the question is searched online and a reference question is obtained which has 3 parts as follows:

Part a:

The decision boundary  after transformation coincides with the line x1 = x2, the two class means must lie on a line that is normal to the decision boundary, i.e. on x1 = −x2. This implies that the transformation matrix Γ is a clockwise rotation transformation of π/4, given as

                                               [tex]\Gamma=\left[\begin{array}{cc}\frac{\sqrt{2}}{2}&\frac{\sqrt{2}}{2}\\\frac{-\sqrt{2}}{2}&\frac{\sqrt{2}}{2}\end{array}\right][/tex]

Part b:

If the prior probability of class 0 was increased after transformation, then the decision boundary of BDR would still have the same normal as before,  i.e.,[tex]w=(1/\sqrt{2},-/\sqrt{2})^T[/tex], but move toward the mean of class 1.

Part c:

Noting that

[tex]||\bold{T}_x-\bold{T}_y||^2=(x-y)^T \bold{T}^T\bold{T}(x-y)\\||\bold{T}_x-\bold{T}_y||^2=(x-y)^T(x-y)\\||\bold{T}_x-\bold{T}_y||^2=||x-y||^2[/tex]

This indicates that the distance is still the same and is in Euclidean space. This is due to the fact that rotation transformations does not affect the distances between the points.

Use a/ 1-r

Where a = 18 and r = 1.2

18/ 1 - 1.2 = 18/-.2 = -90

Because the answer is below 1. The sum does not exist therefore it diverges.

Answer: the 32nd term is - 3

Step-by-step explanation:

The formula for determining the nth term of an arithmetic sequence is expressed as

Tn = a + d(n - 1)

Where

a represents the first term of the sequence.

d represents the common difference.

n represents the number of terms in the sequence.

From the information given,

a = 15.6

d = 15 - 15.6 = 14.4 - 15 = - 0.6

n = 32

The explicit formula for the arithmetic sequence is

Tn = 15.6 - 0.6(n - 1)

We want to determine the value of the 32nd term, T32. Therefore,

T32= 15.6 - 0.6 (32 - 1)

T32 = 15.6 - 18.6

T32 = - 3

Yo sup??

we can solve this question by applying trigonometric ratios

let the angle S be x, then

sinx=2/5

x=23.6

Hope this helps.

Answer:

m∡S = 23.6 °

Step-by-step explanation:

We can see that this is a right angled triangle and therefore we can use trigonometric functions to determine an angle ∡S.

We know that:

                                              [tex]\sin \angle S = \frac{opposite}{hypotenuse}[/tex]

Therefore:

                                                   [tex]\sin \angle S = \frac{2}{5}[/tex]

It yields:

                                           [tex]\angle S = \sin^{-1} \frac{2}{5} =\sin^{-1}0.4[/tex]

Inserting that into the calculator we obtain m∡S = 23.6 degrees

Answer: I’m pretty sure $86

Step-by-step explanation:

If I’m correct it would just be $1,204 divided by 14 pairs of shoes.

Answer:

Claire traveled for 9 days.

Step-by-step explanation:

Given:

Total Distance traveled = 701 miles

Distance traveled each day = 80 miles

Distance traveled on last day = 61 miles

We need to find the number of days Claire traveled.

Solution:

Let the number of days Claire traveled be denoted by 'd'.

Now we can say that;

Total Distance traveled is equal to sum of Distance traveled each day multiplied by number of days and Distance traveled on last day.

framing in equation form we get;

[tex]80d+61=701[/tex]

Now Subtracting both side by 61 using Subtraction Property of Equality we get;

[tex]80d+61-61=701-61\\\\80d = 640[/tex]

Now Dividing both side by 80 we get;

[tex]\frac{80d}{80}=\frac{640}{80}\\\\d=8[/tex]

Hence Claire traveled 80 miles in 8 days and 61 miles on last day making of total 9 days of travel.

solution:

maximum acceleration produce by bag =60 g

time to come to stop = 36 ms

applying equation

[tex]V=u-at[/tex]

inserting values

[tex]0=u-(60)(36*10^-^3)[/tex]

[tex]u=(600*36)/1000=21.6 m/s\\[/tex]

now to find distance of penctration:

[tex]S=ut-1/2at^2[/tex]

inserting values

[tex]S= (21.6)(36*10^-^3)-1/2(600)(0.036)^2[/tex]

[tex]S=0.38m[/tex]

hence distance traveled by person before coming to rest is 0.38 m

Answer:

the truck will over load with 13 refrigerators and 8 pianos

Step-by-step explanation:

13 refrigerators is 2600 lbs

8 pianos is 4200 lbs

both added together is 6800 lbs

the truck can only hold 5900 so 13 refrigerators and 8 pianos is too heavy

Answer:

Step-by-step explanation:

The weight of r refrigerators is 200r. The weight of p pianos is 525p. The total weight must not exceed 5900 lb. So, the total weight of r refrigerators and p pianos must satisfy ...

  200r +525p ≤ 5900

In addition, we cannot have negative refrigerators or pianos, so we must also have ...

  r ≥ 0

  p ≥ 0

A graph is attached.

_____

The point (r, p) = (13, 8) is not in the solution space. 13 refrigerators and 8 pianos would overload the truck.

Answer:    [tex]\dfrac{12}{13}[/tex]

Step-by-step explanation:

Let A = he known the answer then A' = he guess the answer.

B = he answered it correctly

As per given , we have

[tex]P(A)=\dfrac{3}{4}\ \ ,\ \ P(A')=\dfrac{1}{4}[/tex]

[tex]P(B|A)=1[/tex]

[tex]P(B|A')=\dfrac{1}{4}[/tex]

By Bayes theorem , we have

[tex]P(A|B)=\dfrac{P(B|A)P(A)}{P(B|A)P(A)+P(B|A')P(A')}\\\\ P(A|B)=\dfrac{1\times\dfrac{3}{4}}{1\times\dfrac{3}{4}+\dfrac{1}{4}\times\dfrac{1}{4}}\\\\= \dfrac{12}{13}[/tex]

The probability that the student knows the answer given that he answered it correctly is  [tex]\dfrac{12}{13}[/tex] .

Answer: she has $37.2 left.

Step-by-step explanation:

Total cost of the peaches that Ms. Petrie bought is $4.95.

She some breakfast cereal for $7.85. The total amount that she spent in buying the peaches and cereals would be

4.95 + 7.85 = $12.8

Ms. Petrie had $50 before she went shopping. Therefore, the amount of money that Ms. Petrie have left after she buys the peaches and cereal would be

50 - 12.8 = $37.2

To calculate the total cost of the patio, multiply the area of the patio (96 square yards) by the cost per square yard ($0.95), resulting in a total cost of $91.20.

To find the total cost of the concrete patio that Evan wants to build, we need to calculate the total area of the patio first and then multiply this area by the cost per square yard.

First, the area of the patio is calculated by multiplying the length by the width:

Then, we determine the total cost by multiplying the area by the cost per square yard:

Therefore, the total cost to build the patio will be $91.20.

Answer:

B) ∫₂⁵ ∜x dx

Step-by-step explanation:

The factor is 3/n, so b − a = 3

The expression under the radical is 2 + 3k/n, so a = 2.  Therefore, b = 5.

The function is f(x) = ∜x.

Plugging into a definite integral:

∫₂⁵ ∜x dx

Answer:

a. a × b > 0 ∀ a,b ∈ R : a,b < 0

b. a - a = 0 ∀ a ∈ R

Step-by-step explanation:

a. Let a and b be the numbers. Since it says product of two numbers is greater than zero, we write a × b > 0. Since a and b are real numbers, we write a,b ∈ R where ∈ denotes element of a set and R is the set of real numbers. We then use the connective ∀ which denotes "for all" to join a × b > 0 with a,b ∈ R. So, we write a × b > 0 ∀ a,b ∈ R. Since a and b are negative, we write a,b < 0. We now use the connective : which denotes "such that" to combine a × b > 0 ∀ a,b ∈ R with a,b < 0 to give a × b > 0 ∀ a,b ∈ R : a,b < 0. So, the expression is

a × b > 0 ∀ a,b ∈ R : a,b < 0

b. Let a be the number. Since we are looking for a difference, we write a - a. Since it is equal to zero, we write a - a = 0. Since a is an element of real numbers,R, we write a ∈ R, where ∈ denotes "element of". So, a ∈ R denotes a is an element of real numbers R. We combine these two expressions with the connective ∀ which denotes "for all" to give a - a = 0 ∀ a ∈ R. So, the expression is

a - a = 0 ∀ a ∈ R

The mathematical statements given are translated into logical expressions with predicates, quantifiers, logical connectives, and mathematical operators. The first statement is written as ∀x∀y ((x < 0 ∧ y < 0) → P(x, y)), and the second as ∀x (D(x, x)).

The mathematical statements can be expressed using predicates, quantifiers, logical connectives, and mathematical operators as follows:

For the statement 'The product of two negative real numbers is positive':

Let the predicate P(x, y) represent 'the product of x and y is positive', where x and y are real numbers. Then, the statement can be written as:

∀x∀y ((x < 0 ∧ y < 0) → P(x, y))

This translates to: 'For all real numbers x and y, if x and y are both negative, then the product of x and y is positive.'.

For the statement 'The difference of a real number and itself is zero':

Let D(x, y) be a predicate that states 'the difference of x and y is zero'. Then, the statement can be formulated as:

∀x (D(x, x))

Which reads as: 'For all real numbers x, the difference of x and itself is zero.'.

Therefore,  The first statement is written as ∀x∀y ((x < 0 ∧ y < 0) → P(x, y)), and the second as ∀x (D(x, x)).