Multiples of 3 and 5 Problem 1 If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23. Find the sum of all the multiples of 3 or 5 below 1000.

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

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

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:

  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%.

Answer:

(a) and (b) are correct but (c) is not correct

Step-by-step explanation: (c) P(nA)*P(nB) + P(nA/nB)

0.20*0.30 + 0.10 = 0.060 + 0.10 = 0.160

Answer: d. 25

Step-by-step explanation:

Let x be the total number of students in the class.

As per given :

Number of students want ice cream party = 20% of x= 0.20x

[we divide percent by 100 to convert it into decimal.]

Actual number of students want ice cream party =5

⇒  0.20x=5

Divide both sides by 0.20 , we get

⇒ x=  [tex]\dfrac{5}{0.20}=\dfrac{500}{20}=25[/tex]

Hence, the total number of students in the class.= 25

Thus , the correct answer is d. 25

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:

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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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:

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.

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.

The temperature at 10 pm was -1 degree.

To find the temperature at 10 pm, we need to start with the temperature at 8am. Since the temperature was 3 degrees below zero at 8 am, we can add the rise of 14 degrees by 1 am to get the temperature at 1 am. Then, we need to subtract the 12 degree drop by 10 pm to find the temperature at that time.

Starting with 8 am: -3 degrees

Add 14 degrees: -3 + 14 = 11 degrees at 1 am

Subtract 12 degrees: 11 - 12 = -1 degree at 10 pm

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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)).

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:

a) Mean = 1

b) Variance = 0.9      

Step-by-step explanation:

We are given the following in the question:

P(Success of exploration) = 0.1

Then the number of adults follows a binomial distribution, where

[tex]P(X=x) = \binom{n}{x}.p^x.(1-p)^{n-x}[/tex]

All the explorations are independent.

where n is the total number of observations, x is the number of success, p is the probability of success.

Here n = 10, p = 0.1

a) Mean number of successful explorations

[tex]\mu = np = 10(0.1) = 1[/tex]

b) Variance number of successful explorations

[tex]\sigma^2 = np(1-p) = (10)(0.1)(1-0.1) = 0.9[/tex]

Answer:

Scully will catch up with Mulder by 7:30 PM.

Step-by-step explanation:

Consider the provided information.

Mulder leaves the office at 9​:30a.m. averaging 57mph.

Scully leaves at 10​:00​a.m., following the same path and averaging 60 mph.

Distance covered by Mulder between 9:30 AM and 10:00 AM:

[tex]\frac{57}{2}= 28.5[/tex] miles

Let t is the time taken by Scully to catch up with​ Mulder.

After leaving office Scully needs to cover extra distance of 28.5 miles because Mulder leaves the office earlier.

Therefore total distance cover by them is:

[tex]28.5+57t=60t[/tex]

[tex]28.5=60t-57t[/tex]

[tex]28.5=3t\\t=9.5[/tex]

Hence, it would take 9 hours 30 minutes to catch up Mulder.

10 am + 9 hours 30 minutes = 7:30 pm

Therefore, Scully will catch up with Mulder by 7:30 PM.

Final answer:

Scully will catch up with Mulder at 7:30 p.m. by closing the initial 28.5 miles gap at a rate of 3 mph faster than Mulder.

Explanation:

To determine when Scully will catch up with Mulder, we need to calculate the relative speeds at which the two are traveling and how long it will take for Scully to close the gap that Mulder has created by leaving earlier. Since Mulder left at 9:30 a.m. and Scully left at 10:00 a.m., Mulder has a half-hour head start. In that half-hour, traveling at 57 mph, Mulder will have covered 28.5 miles (because 0.5 hours × 57 mph = 28.5 miles).

Scully is traveling at 60 mph, which is 3 mph faster than Mulder. So, Scully closes the gap by 3 miles every hour. To catch up 28.5 miles at a rate of 3 miles per hour, Scully will need 9.5 hours (because 28.5 miles ÷ 3 mph = 9.5 hours).

We need to add this time to Scully's departure time to find out when she will catch up. Therefore, adding 9.5 hours to 10:00 a.m. yields 7:30 p.m. So, Scully will catch up with Mulder at 7:30 p.m.

Answer: Total amount invested in both accounts is $1875

Step-by-step explanation:

Let x represent the amount invested at 11%.

Let y represent the amount invested at 5%.

She invests four times as much in an account paying 11% as she does in an account paying 5%. This means that

x = 4y

The formula for determining simple interest is expressed as

I = PRT/100

Where

P represents the principal

R represents the rate of investment

T represents the time in years.

Considering the amount invested at 11%,

I = (x × 11 × 1)/100 = 0.11x

Considering the amount invested at 5%,

I = (y × 5 × 1)/100 = 0.05y

If she earns $183.75 in interest in one year from both accounts combined, it means that

0.11x + 0.05y = 183.75 - - - - - - - - - -1

Substituting x = 4y into equation 1, it becomes

0.11 × 4y + 0.05y = 183.75

0.44y + 0.05y = 183.75

0.49y = 183.75

y = 183.75/0.49

y = 375

x = 4y = 375 × 4

x = 1500

Total amount invested in both accounts is

1500 + 375 = $1875

Amy invested a total of $1875 in both simple interest accounts. She placed $375 into the account with a 5% interest rate and $1500 into the account with an 11% interest rate to achieve the total interest income of $183.75 in one year.

Amy invests money in two simple interest accounts. One account pays 11% interest, while the other pays 5%. If she puts four times as much into the 11% account as the 5% account, we can set up the following equations to find out how much she invested altogether:

We can use the formula for simple interest which is Interest = Principal × Rate × Time, where the principal is the initial amount of money invested, the rate is the interest rate, and the time is the period of time over which the money is invested.

So, the total interest earned from both accounts is:

Interest from 5% account + Interest from 11% account = $183.75

(x × 0.05 × 1) + (4x × 0.11 × 1) = $183.75

0.05x + 0.44x = $183.75

0.49x = $183.75

Now, solve for x:

0.49x = $183.75

x = $183.75 / 0.49

x = $375

Since x represents the amount invested at 5%, Amy invested $375 in the 5% account. To find the total investment:

Total investment = x + 4x

Total investment = $375 + 4(×$375)

Total investment = $375 + $1500

Total investment = $1875

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.

Learn more about binomial probability here:

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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

Based on the simple interest formula, the annual interest rate paid when borrowing $960 and repaying $1170 a month later is approximately 262.5%.

This problem can be solved by using the formula for calculating simple interest, which is Interest = Principal (P) * Rate (R) * Time (T). In this case, the difference between what you paid and what you borrowed, $1170 - $960 = $210, is the interest you paid. Therefore, we can use the given information to set up the equation 210 = 960 * R * (1/12), because the time period T is 1 month, or 1/12 of a year. By solving this equation, we find that the annual interest rate is approximately 262.5%.

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Answer: It is not possible that two triangles that are similar and not congruent in spherical geometry.

Step-by-step explanation:

For instance, taking a circle on the sphere whose diameter is equal to the diameter of the sphere and inside is an equilateral triangle, because the sphere is perfect, if we draw a circle (longitudinal or latitudinal lines) to form a circle encompassing an equally shaped triangle at different points of the sphere will definately yield equal size.

in other words, triangles formed in a sphere must be congruent and also similar meaning having the same shape and must definately have the same size.

Therefore, it is not possible for two triangles in a sphere that are similar but not congruent.

Two triangles in sphere that are similar must be congruent.

Answer:

3x+4x=35

Step-by-step explanation:

Answer:3x+4x=35

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

Answer:

Reflection about either the x- axis or y- axis.

Step-by-step explanation:

It is likely that the triangle shows a mirror image when it has been rotated at point C so that it matches ΔADC.

The triangles are congruent if the sides and angles are equal. The only way to effect the change is through a rotation.

In fact, two objects are said to be congruent if they can be transformed into another shape by translation, rotation, and reflection.

The resulting image is a distinct figure that can fit and match the other completely.

In this case, the triangle is rotated through 180⁰ so the resulting image is a reflection or mirror image.

Answer:

If two pairs of angles and the included side are congruent, the triangles are congruent.

Step-by-step explanation:

itsLearning

Answer: 15

Step-by-step explanation:Do 27-12 and you get 15

Answer:

15 students don't have a home computer

Step-by-step explanation:

Take the total amount of students, which is 27 and take the amount that do have home computers, which is 12, and subtract.

So, 27 - 12 = 15

The answer is 1/4. Set up a proportion to solve for x. Another way is to use unit analysis.

The answer is 1/4.

The scale factor is 1:4.

First, set up a proportion.
1 gallon/4 quarts=3 gallons/x quarts
Next, cross multiply to solve for x.
1/4-3/x
1x=3x4

Another way to solve the same problem is to use unit analysis.
First, write the unit conversion as a fraction.

One more framing of fractions and their relationship to multiplication and division: dividing by 8 is the same as multiplying by. Multiplying by is the same as dividing by 2. Multiplication and division are thus essentially the same, only having to flip the number or fraction upside-down into its reciprocal.

Use the quadratic equation to find x
x= -.0024, .00139
.001-.0024 is negative, which can't happen in real life so we know x actually equals .00139

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.

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.

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.

Learn more about differential equations here:

https://brainly.com/question/14658115