17,210,000,000 written in scientific notation?

Answer:

Is needed 16 fluid oz of water to fill in 2 cups of water

Step-by-step explanation:

1 cup have 8 fluid oz

2 cups = 2 * 8 fluid oz = 16 fluid oz

Final answer:

The recipe requires 2 cups of water, which is equivalent to 16 fluid ounces since there are 8 fluid ounces in one cup.

Explanation:

To calculate how many fluid ounces of water is needed if a recipe requires 2 cups of water, you need to understand the unit equivalence between cups and fluid ounces. One cup is equivalent to 8 fluid ounces. Since the recipe requires 2 cups, you will multiply the number of cups by the unit equivalence to find the total number of fluid ounces.

Here is the calculation:

2 cups × 8 fluid ounces/cup = 16 fluid ounces

Therefore, the recipe requires 16 fluid ounces of water.

Answer:  [tex]224.9\ m^2[/tex]

Step-by-step explanation:

An isosceles right triangle is a right triangle having two legs (other than hypotenuse ) of same length  .

Given : The length of the hypotenuse of an isosceles right triangle  is 30 meters.

Let x be the side length of the other two legs, then by using the Pythagoras theorem for right triangle , we have

[tex](30)^2=x^2+x^2\\\\\Rightarrow\ 900=2x^2\\\\\Rightarrow\ x^2=\dfrac{900}{2}\\\\\Rightarrow\ x^2=450\\\\\Rightarrow\ x=\sqrt{450}=\sqrt{9\times25\times2}=\sqrt{3^2\times5^2\times2}\\\Rightarrow\ x=3\times5\sqrt{2}=15(1.414)=21.21[/tex]

Thus, the other two legs have side length of 21.21 m each.

Now, the area of a right triangle is given by :-

[tex]A=\dfrac{1}{2}\times base\times height\\\\\Rightarrow\ A=\dfrac{1}{2}(21.21)\times(21.21)=224.93205\approx224.9\ m^2[/tex]

Hence, the area of the given isosceles right triangle= [tex]224.9\ m^2[/tex]

Answer:

The deep body temperature of a healthy person is 310.15 K

Step-by-step explanation:

The formula we are going to use is [tex]T_{K}=T_{C}+273.15[/tex]

We know that the deep body temperature is 37°C, putting this value into the formula we have that [tex]T_{K}=37+273.15 = 310.15 K[/tex]

Answer:

Step-by-step explanation:

If we assume that [tex][(x \vee y) \wedge (x \rightarrow z) \wedge (\neg z)][/tex] is true, then:

[tex](x \vee y)[/tex] is true

[tex](x \rightarrow z)[/tex] is true

[tex](\neg z)[/tex] is true

If [tex](\neg z)[/tex] is true, then [tex]z[/tex] is false.

[tex](x \rightarrow z) \equiv (\neg x \vee z)[/tex], since [tex](x \rightarrow z)[/tex] is true, then [tex](\neg x \vee z)[/tex] is true

If [tex]z[/tex] is false and [tex](\neg x \vee z)[/tex] is true, then [tex]\neg x[/tex] is true.

If [tex]\neg x[/tex] is true, then [tex]x[/tex] is false, as [tex](x \vee y)[/tex] is true and [tex]x[/tex] is false, then [tex]y[/tex] is true.

Conclusion [tex]y[/tex] it's true.

Answer:

a) [tex]\neg \forall x :x>5 \equiv \forall x:x\leq 5[/tex]

b) [tex]\neg [x^2+2x+1=0]\equiv x^2+2x+1\neq0[/tex] or the set [tex]\{x:x\neq-1\}[/tex]

Step-by-step explanation:

First, notice that in both cases we have to sets:

a) is the set of all real numbers which are higher than 5 and in

b) the set is the solution of the equation [tex]x^2+2x+1=0[/tex] which is the set [tex]x=-1[/tex]

De Morgan's Law for set states:

[tex]\overline{\rm{A\cup B}} = \overline{\rm{A}} \cap \overline{\rm{B}}\\[/tex], being [tex]\overline{\rm{A}}[/tex] and [tex]\overline{\rm{B}}[/tex] are the complements of the sets [tex]A[/tex] and [tex]B[/tex]. [tex]\cup[/tex] is the union operation and [tex]\cap[/tex] the intersection.

Thus for:

a) [tex]\neg \forall x :x>5 \equiv \forall x:x\leq 5[/tex]. Notice that [tex]\forall x:x\leq 5[/tex] is the complement of the given set.

b) [tex]\neg [x^2+2x+1=0]\equiv x^2+2x+1\neq0[/tex] which is the set [tex]B = \{x:x\neq-1\}[/tex]

Answer:

If he uses 1 hour to practice harmonica, one our on homework, and another hour doing chores. He will be spending twice as much time practicing harmonica and doing homework than he would spend on chores.

Answer:

2

Step-by-step explanation:

UP

Answer:

Johnny is wrong.

[tex]A \cup B = \left\{a,b,c,d,e,f,g,h\right\}[/tex]

Step-by-step explanation:

Johnny is wrong.

A better definition would be: [tex]A \cup B[/tex] is the set of all elements that belong to at least one A or B.

So, the elements that belong to both A and B, like c and d in this exercise, also belong to [tex]A \cup B[/tex].

So:

[tex]A \cup B=\left\{a,b,c,d,e,f,g,h\right\}[/tex]

Answer:

1) The tournament can be played in 10 different ways

2) The probability of 5 games being played is 0.40

3) The probability of a team winning two games in a row is 0.80  

Step-by-step explanation:

a) From the tree diagram below we can observe that the tournament can be played in 10 different ways.

b)The probability of 5 games being played is

P= (number of possibilities where 5 games are being played) / (Total games)

P = 4 / 10

P= 0.40

c) The probability of a team winning two games in a row is

P = (number of possibilities where a team wins two games in a row) / Total games

P = 8 / 10

P = 0.80

Answer:

ohhh you have the right time and I am so glad to see it in a better time and if it was not to do you have no doubt

Step-by-step explanation:

your body has to have your body to have the best results you need for kids in a group that can help with a lot to learn about it as you do so much of the time it works out to you to get the job you need for the next few years or 8īijjhguu is not just the best person but the meaning and skills of a person is the meaning of life and the meaning is the only 3AM in a school that you need a lot of a job 3to to do with you need to get your mind off to get your job done with your mind and then you have to do it right now and.

Answer:

A. Representative

Step-by-step explanation:

The sample is a subset of the population. We take samples because it is easy to analyze samples as compared to population and it is less time taken. Further, Sample is said to be good if the sample is the representative of the population.

Simple Random Sampling is the sampling where samples are chosen randomly, where each unit has an equal chance of being selected in a sample.

Since, observer is taken the weight of men over age 18. Thus, it is good sample which represent whole population.

Answer:

20 mg/ml.

Step-by-step explanation:

We have been given that a liquid oral concentrate of morphine sulfate contains 2.4 g of morphine sulfate in a 120-mL bottle.

[tex]\text{Concentration of morphine sulfate on g/mL basis}=\frac{\text{2.4 g}}{\text{120 ml}}[/tex]

To convert the concentration of morphine sulfate on a mg/mL basis, we need to convert 2.4 grams to milligrams.

1 gram equals 1000 milligrams.

[tex]\text{Concentration of morphine sulfate on mg/mL basis}=\frac{\text{2.4 g}}{\text{120 ml}}\times\frac{\text{1,000 mg}}{\text{1 g}}[/tex]

[tex]\text{Concentration of morphine sulfate on mg/mL basis}=\frac{2.4\times\text{1,000 mg}}{\text{120 ml}}[/tex]

[tex]\text{Concentration of morphine sulfate on mg/mL basis}=\frac{2400\text{ mg}}{\text{120 ml}}[/tex]

[tex]\text{Concentration of morphine sulfate on mg/mL basis}=\frac{20\text{ mg}}{\text{ml}}[/tex]

Therefore, the concentration of morphine sulfate would be 20 mg per ml.

Answer:

Step-by-step explanation:

Answer: 0.1498 square units.

Step-by-step explanation:

Let x be any random variable that follows normal distribution.

Given : For a normal distribution with mean equal to - 30 and standard deviation equal to 9.

i.e.  [tex]\mu=-30[/tex] and [tex]\sigma=9[/tex]

Use formula [tex]z=\dfrac{x-\mu}{\sigma}[/tex] to find the z-value corresponds to -34.5 will be

[tex]z=\dfrac{-34.5-(-30)}{9}=\dfrac{-34.5+30}{9}=\dfrac{-4.5}{9}=-0.5[/tex]

Similarly,  the z-value corresponds to -38 will be

[tex]z=\dfrac{-39-(-30)}{9}=\dfrac{-39+30}{9}=\dfrac{-9}{9}=-1[/tex]

By using the standard normal table for z-values , we have

The  area under the curve that is between - 34.5 and - 39. will be :-

[tex]P(-1<z<-0.5)=P(z<-0.5)-P(z<-1)\\\\=(1-P(z<0.5))-(1-P(z<1))\\\\=1-P(z<0.5)-1+P(z<1)\\\\=P(z<1)-P(z<0.5)\\\\=0.8413-0.6915=0.1498[/tex]

Hence, the area under the curve that is between - 34.5 and - 39 = 0.1498 square units.

Answer:

Use the formula [tex]Area(S)=\iint_{S} 1 dS= \iint_{D} \lVert r_{u}\times r_{v} \rVert dudv[/tex]

Step-by-step explanation:

Let [tex]r(x,y)=(x,y,5-3x^2-2y^2)[/tex] be the explicit parametrization of the paraboid. The intersection of this paraboid with the xy plane is the ellipse given by

[tex]\dfrac{x^2}{\frac{5}{3}}+\dfrac{y^{2}}{\frac{5}{2}}=1[/tex]

The partial derivatives of the parametrization are:

[tex]\begin{array}{c}r_{x}=(1,0,-6x)\\r_{y}=(0,1,-4y)\end{array}[/tex]

and computing the cross product we have

[tex]r_{x}\times r_{y}=(6x,4y,1)[/tex]. Then

[tex]\lVert r_{x}\times r_{y}\rVert =\sqrt{1+36x^{2}+16y^{2}}[/tex]

Then,  if [tex]R[/tex] is the interior region of the ellipse the superficial area located above of the xy  is given by the double integral

[tex]\iint_{R}\sqrt{1+36x^2+16y^2}dxdy=\int_{-\sqrt{5/3}}^{\sqrt{5/3}}\int_{-\sqrt{5/2}\sqrt{1-\frac{x^2}{5/3}}}^{\sqrt{5/2}\sqrt{1-\frac{x^2}{5/3}}}\sqrt{1+36x^2+16y^2}dy dx=30.985[/tex]

The last integral is not easy to calculate because it is an elliptic integral, but with any software of mathematics you can obtain this value.

Answer:

The cost for 17 ft length with diameter 5 in is $408.

Step-by-step explanation:

Consider the provided information.

It is given that the cost of stainless steel tubing varies jointly as the length and the diameter of the tubing. If a 5 foot length with a diameter 2 inches costs $48.00.

let [tex]C\propto L\cdot D[/tex]

[tex]C=k\cdot L\cdot D[/tex]

[tex]48=k\cdot 5\cdot 2[/tex]

[tex]\frac{48}{10}=k[/tex]

[tex]k=4.8[/tex]

For 17 ft length with diameter 5in the cost will be:

[tex]C=k\cdot L\cdot D[/tex]

Substitute the respective values in the above formula.

[tex]C=4.8\cdot 17\cdot 5[/tex]

[tex]C=408[/tex]

Cost = $408

Hence, the cost for 17 ft length with diameter 5in is $408.

Answer:

[tex]x=36.772[/tex]  

Step-by-step explanation:

Given : Expression [tex]x=25.36+0.45(25.36)[/tex]

To find : Solve the expression ?

Solution :

Step 1 - Write the expression,

[tex]x=1(25.36)+0.45(25.36)[/tex]

Step 2 - Apply distributive, [tex]ab+ac=a(b+c)[/tex]

Here, a=25.36, b=1, c=0.45

[tex]x=25.36(1+0.45)[/tex]

Step 3 - Solve the expression,

[tex]x=25.36\times 1.45[/tex]

[tex]x=36.772[/tex]

Therefore, [tex]x=36.772[/tex]

Answer:

1) Distance in feet equals 1640.4 feet

2) Distance in miles equals 0.3107 miles.

Step-by-step explanation:

From the basic conversion factors we know that

1 meter = 3.2808 feet

Thus by proportion we conclude that

[tex]500meters=500\times 3.2808feet\\\\\therefore 500meters=1640.4feet[/tex]

Similarly using the basic conversion factors we know that

1 mile = 1609.34 meters

thus we conclude that 1 meter =[tex]\frac{1}{1609.34}[/tex] miles

Hence by proportion 500 meters = [tex]500\times \frac{1}{1609.34}=0.3107[/tex] miles

Answer:

76075, 76076, 76077

Step-by-step explanation:

There are 3 teams; Team A, Team B and Team C

Team A has most wins

Team C has least wins

Team B is in between

All these will be consecutive numbers.

Team B: x

Team A: x + 1 (most wins)

Team C: x - 1 (least wins)

Team A + Team B + Team C = Total number of wins

x + x + 1 + x - 1 = 228228

3x = 228228

x = 76076

Wins of Team B : x = 76076

Wins of Team A : x + 1 = 76076 + 1 = 76077

Wins of Team C : x - 1 = 76076 - 1 = 76075

Therefore, in the 2010 season, Team A had 76077 wins, Team B had 76076 wins and Team C had 76075 wins.

!!

Answer:

Your account is going to be worst $405,142.92

Step-by-step explanation:

This is a compound interest problem.

The compound interest formula is given by:

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

Where A is the amount of money, P is the principal(the initial sum of money), r is the interest rate(as a decimal value), n is the number of times that interest is compounded per unit t and t is the time the money is invested or borrowed for.

So, for this problem:

We want to find A

[tex]P = 55,000[/tex]

[tex]n = 1[/tex]

[tex]r = 0.105[/tex]

[tex]t = 20[/tex]

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

[tex]A = 55000(1 + \frac{0.105}{1})^{20}[/tex]

[tex]A = 405,142.92[/tex]

Your account is going to be worst $405,142.92

Answer:

238.25 mg

Step-by-step explanation:

Given:

Molar mass of MgCl₂ = 95.3

atomic weight of Mg₂ = 243

Atomic weight of Cl = 35.5

Volume of solution required = 5.0 mEq of magnesium

Now,

mEq = [tex]\frac{\textup{Weight in mg\timesValency}}{\textup{Atomic mass}}[/tex]

on substituting the values, we get

5 = [tex]\frac{\textup{Weight in mg\times2}}{\textup{95.3}}[/tex]

or

weight of magnesium chloride = 238.25 mg

Therefore,

the required mass of MgCl₂ is 238.25 mg

Answer: 29

Step-by-step explanation:

In statistics, the degrees of freedom gives the number of values that have the freedom to vary.

If the sample size is 'n' then the degree of freedom is given by:-

df= n-1

Given: The number of participants for the sample =30

Then the degree of freedom for this sample=30-1=29

Hence, The total degree of freedom= 29

In the context of the experimental design, the total degrees of freedom would be calculated based on the between-group and within-group variations. With three groups and three repeats, the total degrees of freedom would be 62.

When calculating the total degrees of freedom in an experiment where 30 participants each completed a memory task three times under different levels of caffeine, we can determine the degrees of freedom by considering the number of groups and the number of repeats within each group. Since there are three groups (no, moderate, and high levels of caffeine), and each participant completes the task three times, this setup hints at an Analysis of Variance (ANOVA) scenario with repeated measures. However, without the detailed design for the within-subject factors, we can only calculate the between-group degrees of freedom. For three groups, the between-group degrees of freedom would be the number of groups minus one, which would give us 2 degrees of freedom (dfbetween = number of groups - 1).

For within-group or repeated measures, you typically have dfwithin = (number of repeats - 1) × number of subjects. Since each participant repeats the task three times, and there are 30 participants, the calculation would be dfwithin = (3 - 1) × 30 = 2 × 30 = 60 degrees of freedom. Hence, the total degrees of freedom would be the sum of between-group and within-group degrees of freedom, which is 2 + 60 = 62.

Answer:

Step-by-step explanation:

Let A, B and C be square matrices, let [tex]D = ABC[/tex]. Suppose also that D is an invertible square matrix. Since D is an invertible matrix, then [tex]det (D) \neq 0[/tex]. Now, [tex]det (D) = det (ABC) = det (A) det (B) det (C) \neq 0[/tex]. Therefore,

[tex]det (A) \neq 0[/tex]

[tex]det (B) \neq 0 [/tex]

[tex]det (C) \neq 0[/tex]

which proves that A, B and C are invertible square matrices.

Answer:

To see the steps to the diagonal form see the step-by-step explanation. The solution to the system is [tex]x =  -\frac{1}{9}[/tex], [tex]y= -\frac{1}{9}[/tex], [tex]z= \frac{4}{9}[/tex] and [tex]w = \frac{7}{9}[/tex]

Step-by-step explanation:

Gauss elimination method consists in reducing the matrix to a upper triangular one by using three different types of row operations (this is why the method is also called row reduction method). The three elementary row operations are:

To solve the system using the Gauss elimination method we need to write the augmented matrix of the system. For the given system, this matrix is:

[tex]\left[\begin{array}{cccc|c}1 & 1 & 1 & 1 & 1 \\1 & 1 & 0 & -1 & -1 \\-1 & 1 & 1 & 2 & 2 \\1 & 2 & -1 & 1 & 0\end{array}\right][/tex]

For this matrix we need to perform the following row operations:

After this step, the system has an upper triangular form

The triangular matrix looks like:

[tex]\left[\begin{array}{cccc|c}1 & 0 & 0 & -0.5 & -0.5  \\0 & 1 & 0 & -0.5 & -0.5\\0 & 0 & 1 & 2 &  2 \\0 & 0 & 0 & 1 &  \frac{7}{9}\end{array}\right][/tex]

If you later perform the following operations you can find the solution to the system.

After this operations, the matrix should look like:

[tex]\left[\begin{array}{cccc|c}1 & 0 & 0 & 0 & -\frac{1}{9}  \\0 & 1 & 0 & 0 &   -\frac{1}{9}\\0 & 0 & 1 & 0 &  \frac{4}{9} \\0 & 0 & 0 & 1 &  \frac{7}{9}\end{array}\right][/tex]

Thus, the solution is:

[tex]x =  -\frac{1}{9}[/tex], [tex]y= -\frac{1}{9}[/tex], [tex]z= \frac{4}{9}[/tex] and [tex]w = \frac{7}{9}[/tex]

Answer:

R = {(10,2), (10,4), (10,6), (10,8), (2,10), (2,2), (2,4), (2,6), (2,4), (4,10), (4,2), (4,6), (6,10), (6,2), (6,4), (6,8)}

Step-by-step explanation:

A = (10, 1, 2, 3, 4, 5, 6)

B = (10, 1, 2, 3, 4, 5, 6, 7, 8)

R is a relation and defined as

R: A→B

R: The greatest common divisor of a and b is 2

To find the elements of R we need to find pair from A and B respectively.

R = {(10,2), (10,4), (10,6), (10,8), (2,10), (2,2), (2,4), (2,6), (2,4), (4,10), (4,2), (4,6), (6,10), (6,2), (6,4), (6,8)}

Now, these pairs were found by picking one element from A and checking with elements of B if they have a gcd of 2. If yes, they fall under the defined relation.

Answer:

[tex](-\frac{85}{28},2)[/tex]

Step-by-step explanation:

We are asked to find the focus of the parabola that has a vertex [tex](-3,2)[/tex] opens horizontally and passes through the point [tex](-10,1)[/tex].

We know that equation of a horizontal parabola is [tex](y-k)^2=4p(x-h)[/tex], where p is not equal to zero.

[tex](h,k)[/tex] = Vertex of parabola.

Focus: [tex](h+p,k)[/tex]

Upon substituting the coordinates of vertex and given point in equation of parabola, we will get:

[tex](1-2)^2=4p(-10-(-3))[/tex]

[tex](-1)^2=4p(-10+3)[/tex]

[tex]1=4p(-7)[/tex]

[tex]1=-28p[/tex]

[tex]\frac{1}{-28}=\frac{-28p}{-28}[/tex]

[tex]-\frac{1}{28}=p[/tex]

Focus: [tex](h+p,k)[/tex]

[tex](-3+(-\frac{1}{28}),2)[/tex]

[tex](-3-\frac{1}{28},2)[/tex]

[tex](\frac{-3*28}{28}-\frac{1}{28},2)[/tex]

[tex](\frac{-84}{28}-\frac{1}{28},2)[/tex]

[tex](\frac{-84-1}{28},2)[/tex]

[tex](\frac{-85}{28},2)[/tex]

[tex](-\frac{85}{28},2)[/tex]

Therefore, the focus of the parabola is at point [tex](-\frac{85}{28},2)[/tex].

Answer:

$37,800.

Step-by-step explanation:

We have been given that the cost, in dollars, of making x items is given by the function [tex]C(x)=25x+300[/tex]. We are asked to find [tex]C(1500)[/tex].

To find [tex]C(1500)[/tex], we will substitute [tex]x=1500[/tex] in our given function as:

[tex]C(1500)=25(1500)+300[/tex]

[tex]C(1500)=37,500+300[/tex]

[tex]C(1500)=37,800[/tex]

Therefore, the cost of making 1500 items would be $37,800 and option D is the correct choice.

Final answer:

To find the total kilograms of ibuprofen, multiply 36 tablets by 200 mg each to get 7200 mg, and then divide by 1,000,000 to convert to kilograms, resulting in 0.0072 kg.

Explanation:

To calculate the total amount of ibuprofen in kilograms, you start by finding the total amount in milligrams. Multiply the number of tablets (36) by the amount of ibuprofen per tablet (200 mg).

36 tablets × 200 mg/tablet = 7200 mg

Now, since there are 1,000,000 milligrams in a kilogram, you'll need to divide the total milligrams by 1,000,000 to convert it to kilograms.

7200 mg ÷ 1,000,000 mg/kg = 0.0072 kg

Therefore, the package contains 0.0072 kilograms of ibuprofen.

Answer:

>>syms t

>>f=sin(t)*cos(t);

>>int(f)

Step-by-step explanation:

It's actually pretty easy, just use symbolic variables.

First, create the symbolic variable t using this command:

syms t

Now define the function

f=sin(t)*cos(t)

Finally use the next command in order to calculate the indefinite integral:

int(f)

I attached a picture in which you can see the procedure and the result.

To evaluate the integral of sintcost tdt in MATLAB, use the symbolic math toolbox and the 'int' function.

To evaluate the integral sintcost tdt in MATLAB, you can use the symbolic math toolbox. First, define the extended variable t using 'syms t'. Then, determine the integrand using the symbolic expression 'f = sin(t) * cos(t)'. Finally, use the 'int' function to find the integral, 'int(f, t)'.

Here's the MATLAB code:

The resulting 'integral_value' will be the evaluated integral.

https://brainly.com/question/34616100

#SPJ12

Answer:

602 tourists visited only the LEGOLAND.

Step-by-step explanation:

To solve this problem, we must build the Venn's Diagram of this set.

I am going to say that:

-The set A represents the tourists that visited LEGOLAND

-The set B represents the tourists that visited Universal Studios

-The set C represents the tourists that visited Magic Kingdown.

-The value d is the number of tourists that did not visit any of these parks, so: [tex]d = 58[/tex]

We have that:

[tex]A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)[/tex]

In which a is the number of tourists that only visited LEGOLAND, [tex]A \cap B[/tex] is the number of tourists that visited both LEGOLAND and Universal Studies, [tex]A \cap C[/tex] is the number of tourists that visited both LEGOLAND and the Magic Kingdom. and [tex]A \cap B \cap C[/tex] is the number of students that visited all these parks.

By the same logic, we have:

[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]

[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]

This diagram has the following subsets:

[tex]a,b,c,d,(A \cap B), (A \cap C), (B \cap C), (A \cap B \cap C)[/tex]

There were 1,107 tourists suveyed. This means that:

[tex]a + b + c + d + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 1,107[/tex]

We start finding the values from the intersection of three sets.

The problem states that:

36 tourists had visited all three theme parks. So:

[tex](A \cap B \cap C) = 36[/tex]

72 tourists had visited both LEGOLAND and Universal Studios. So:

[tex](A \cap B) + (A \cap B \cap C) = 72[/tex]

[tex](A \cap B) = 72 - 36[/tex]

[tex](A \cap B) = 36[/tex]

79 tourists had visited both the Magic Kingdom and Universal Studios

[tex](B \cap C) + (A \cap B \cap C) = 79[/tex]

[tex](B \cap C) = 79 - 36[/tex]

[tex](B \cap C) = 43[/tex]

68 tourists had visited both the Magic Kingdom and LEGOLAND

[tex](A \cap C) + (A \cap B \cap C) = 68[/tex]

[tex](A \cap C) = 68 - 36[/tex]

[tex](A \cap C) = 32[/tex]

258 tourists had visited Universal Studios:

[tex]B = 258[/tex]

[tex]B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)[/tex]

[tex]258 = b + 43 + 36 + 36[/tex]

[tex]b = 143[/tex]

268 tourists had visited the Magic Kingdom:

[tex]C = 268[/tex]

[tex]C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)[/tex]

[tex]268 = c + 32 + 43 + 36[/tex]

[tex]c = 157[/tex]

How many tourists only visited the LEGOLAND (of these three)?

We have to find the value of a, and we can do this by the following equation:

[tex]a + b + c + d + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 1,107[/tex]

[tex]a + 143 + 157 + 58 + 36 + 32 + 43 + 36 = 1,107[/tex]

[tex]a = 602[/tex]

602 tourists visited only the LEGOLAND.

Answer:

1) [tex]L(y)=\frac{2}{s^{3}}[/tex]

2) [tex]L(y)=\frac{6}{s^{4}}[/tex]

Step-by-step explanation:

To find : Calculate the Laplace transforms of the following from the definition ?

Solution :

We know that,

Laplace transforms of [tex]t^n[/tex] is given by,

[tex]L(t^n)=\frac{n!}{s^{n+1}}[/tex]

1) [tex]y=t^2[/tex]

Laplace of y,

[tex]L(y)=L(t^2)[/tex] here n=2

[tex]L(y)=\frac{2!}{s^{2+1}}[/tex]

[tex]L(y)=\frac{2}{s^{3}}[/tex]

2) [tex]y=t^3[/tex]

Laplace of y,

[tex]L(y)=L(t^3)[/tex] here n=3

[tex]L(y)=\frac{3!}{s^{3+1}}[/tex]

[tex]L(y)=\frac{3\times 2}{s^{4}}[/tex]

[tex]L(y)=\frac{6}{s^{4}}[/tex]