Consider the following statement: The square of a prime number is not prime. (a) Write this as an if-then statement, using careful mathematical language and notation. (b) Prove or disprove the statement.

Answer:

The elements of given set are -1, 0 and 1.

Step-by-step explanation:

The given set is

[tex]\{x\in R:x^3-x=0\}[/tex]

We need to find all the elements of given set.

The given equation is

[tex]x^3-x=0[/tex]           .... (1)

Solve this equation o find the value of x.

Taking out common factors.

[tex]x(x^2-1)=0[/tex]

Using zero product property,

[tex]x=0[/tex]

[tex]x^2-1=0[/tex]

[tex]x^2=1[/tex]

[tex]x=\pm 1[/tex]

All rational and irrational numbers are real numbers.

On solving equation (1) we get x = -1, 0, 1. All these numbers are real number. So, the elements of given set are -1, 0 and 1. The set is defined as

{ -1, 0, 1}

Therefore, the elements of given set are -1, 0 and 1.

Answer:

Step-by-step explanation:

Suppose the dimensions of the playground are x and y.

The total amount of the fence used is given and it is 780 ft. In terms of x and y this would be 3x+2y=780 (we add 3x because we want it to be cut in the middle). Therefore,  y= 780/2-3/2x. Now, the total area (A )to be fenced is

A=x*y= x*(390-3/2x)=-3/2 x^2+390x

Calculating the derivative of A and setting it equals to 0 to find the maximum

A'= -3x+390=0

This yields x=130.

Therefore y=780/2-3/2*130=195

Thus, the maximum area is 130*195=25,350ft^2

The rectangle's dimensions that maximize the total enclosed area are 130 feet (length) and 260 feet (width), resulting in a maximum area of 33,800 square feet.

This problem is about optimization, specifically in the context of a rectangle’s dimensions and area. Here's how to solve it step-by-step:

Firstly, visualize the fenced area as a rectangle divided int two equal rectangles. The total fencing used makes up perimeter which consists of three lengths (L) and two widths (W), i.e. 3L + 2W = 780 feet.

To simplify, express one variable in terms of the other. From the equation above, we can express W as (780 - 3L)/2.

The area of a rectangle is given by L × W. Substituting W from the equation above, Area = L * (780 - 3L)/2.

To maximize this area, find its maximum point using differentiation: d(Area)/dL = 0. You will find that L = 130 feet.

Substitute L = 130 feet into the width equation to find W = 260 feet. So, the maximum enclosed area is L * W = 130 * 260 = 33,800 square feet.

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Answer:

  The product is approximately 5400.

Step-by-step explanation:

  9 1/16 rounded to the nearest whole number is 9.

  645 3/5 rounded to the nearest hundred is 600.

Multiplying these values gives a product of 9·600 = 5400.

The product is approximately 5400.

Answer:

5,400.

Step-by-step explanation:

9 1/6 = 9 to the nearest whole number.

645 3/5 = 600 to the nearest hundred.

The produce = 9 * 600 = 5400.

Answer:

The sum of the whole numbers from 1 to 720 is 259560.

Step-by-step explanation:

To find : The sum of the whole numbers from 1 to 720 ?

Solution :

The whole numbers from 1 to 720 form an arithmetic progression,

The first term is a=1

The last term is l=720

The number of terms n=720

The sum formula of A.P is

[tex]S_n=\frac{n}{2}[a+l][/tex]

Substitute the values in the formula,

[tex]S_{720}=\frac{720}{2}[1+720][/tex]

[tex]S_{720}=360\times 721[/tex]

[tex]S_{720}=259560[/tex]

Therefore, The sum of the whole numbers from 1 to 720 is 259560.

Answer: There are 2 strokes below par that the team has scored in the first round.

Step-by-step explanation:

Since we have given that

Scores of a team

69, 73, 70 and 74

on the first round on a part 72 course.

Now, we need to find the number of strokes above or below the par.

So, we will compare all the scores with 72.

So,

69-72 = -3(below par)

73-72 = 1 (above par)

70-72= - 2 (below par)

74-72 = 2 ( above par)

So, Number of strokes above or below par is given by

[tex]-3+1-2+2\\\\=-2[/tex]

Hence, there are 2 strokes below par that the team has scored in the first round.

Answer:

The answer is C, 3³.

Step-by-step explanation:

When you're dividing integers with exponents, you subtract the two exponent (and when multiplying them, you add them instead.)

In this case, you subtract 7 from 10 which gives you 3.

Answer:

[tex]\frac{3^{10}}{3^7}=3^3[/tex]

Step-by-step explanation:

The [tex]3^{10}[/tex] means we have ten copies of 3 on top; the [tex]3^{7}[/tex] means we have seven copies of three underneath.

[tex]\frac{3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3}{3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3}[/tex]

We have three extra 3's, and they are on top.

[tex]\frac{3\cdot 3\cdot 3}{1} =3^3[/tex]

Therefore,

[tex]\frac{3^{10}}{3^7}=3^3[/tex]

We can also use the The Quotient Rule for Exponents,

For any non-zero number x and any integers a and b [tex]\frac{x^a}{x^b}=x^{a-b}[/tex]

[tex]\frac{3^{10}}{3^7}=3^{10-7}=3^3[/tex]

Answer:

The area of square 2 is 130 units square

Step-by-step explanation:

see the attached figure to better understand the problem

we know that

The area of a square is

[tex]A=b^{2}[/tex]

where

b is the length side of the square

Let

b1 ----> the length side of square 1

b2 ----> the length side of square 2

b3 ----> the length side of square 3

Applying the Pythagoras Theorem

[tex]b1^{2}=b2^{2}+b3^{2}[/tex] -----> equation A

we have

[tex]A1=250\ units^2[/tex]

[tex]A3=120\ units^2[/tex]

Remember that

[tex]A=b^{2}[/tex]

so

[tex]A1=b1^2=250\ units^2[/tex]

[tex]A3=b3^2=120\ units^2[/tex]

substitute in the equation A and solve for b2^2

[tex]250=b2^{2}+120[/tex]

[tex]b2^{2}=250-120[/tex]

[tex]b2^{2}=130[/tex]

[tex]A2=b2^{2}[/tex]

therefore

The area of square 2 is 130 units square

Answer:

2x² + 4x + 3 = 0

Step-by-step explanation:

The function is said to be quadratic if it has highest degree = 2.

Further, The standard form of Quadratic Equation is:

ax² + bx + c = 0

where, a ≠ 0

a, b and c are constants

and x is unknown variable.

Thus, The Standard form of given Quadratic Equation is 2x² + 4x + 3 = 0

To express the quadratic function f(x) = 2x^2 + 4x + 3 in standard form, complete the square to get f(x) = 2(x + 1)^2 - 5, which reveals the vertex of the parabola at (-1, -5).

To express the quadratic function f(x) = 2x^2 + 4x + 3 in standard form, also known as vertex form, we need to complete the square. The standard form of a quadratic function is typically written as f(x) = a(x-h)^2 + k, where (h, k) is the vertex of the parabola formed by the graph of the quadratic equation.

Here's how we can rewrite the given quadratic function:

In this form, it is clear that the vertex of the parabola is at (-1, -5).

Answer:

Given dose (i.e 600 mg) lies outside the recommended range of  271.8-543.6 mg/day that too on the higher side

Hence, the prescription is not safe.

Step-by-step explanation:

Given:

Weight of medicine given per interval = 150 mg

time interval = 6 hours

thus, number of intervals per day = [tex]\frac{\textup{24}}{\textup{6}}[/tex] = 4

therefore,

the total dose of medicine provided per day = 4 × 150 = 600 mg

Now,

Recommended dosage =  20-40 mg/kg/day

weight of child = 30 lb

also,

1 lb = 0.453 kg

thus,

weight of child = 30 × 0.453 = 13.59 kg

Therefore, the recommended dose for the child

=  (  20-40 mg/kg/day ) × 13.59

= 271.8-543.6 mg/day

now,

the given dose (i.e 600 mg) lies outside the recommended range of  271.8-543.6 mg/day that too on the higher side

Hence, the prescription is not safe.

Answer:

0.9796

Step-by-step explanation:

Given that 88%of bolts and 83% of nails meet specifications.

This implies that for a randomly selected bowl the prob that it meets specifications = P(A) = 0.88

Similarly, for a randomly selected bolt, it meets specifications is P(B) = 0.83

We know that bolt and nail are independent of each other.

Hence [tex]P(A \bigcap B) = P(A)P(B)\\\\=0.88*0.83=0.7304[/tex]

Required probability = Probability that atleast one of them meets specifications)

[tex]= P(AUB)\\=P(A)+P(B)-P(A \bigcap B)\\=0.88+0.83-0.7304\\=1.71-0.7304\\=0.9796[/tex]

Final answer:

To find the probability that at least one of the chosen bolt or nail meets specifications, use the complement rule. Multiply the probabilities that each does not meet specifications and subtract from 1 to find the probability that at least one meets specifications.

Explanation:

To find the probability that at least one of the chosen bolt or nail meets specifications, we can use the complement rule. The complement of the event that at least one meets specifications is the event that none of them meet specifications. The probability that the bolt does not meet specifications is 1 - 0.88 = 0.12, and the probability that the nail does not meet specifications is 1 - 0.83 = 0.17. Since the events are chosen independently, we can multiply these probabilities together to get the probability that both do not meet specifications: 0.12 x 0.17 = 0.0204.

Using the complement rule, we subtract this probability from 1 to find the probability that at least one meets specifications: 1 - 0.0204 = 0.9796. Therefore, the probability that at least one of the bolt and nail meets specifications is approximately 0.9796.

Answer:

50,000 tablets may be prepared from 30g of colchicine

Step-by-step explanation:

This problem can be solved as a rule of three problem.

In a rule of three problem, the first step is identifying the measures and how they are related, if their relationship is direct of inverse.

When the relationship between the measures is direct, as the value of one measure increases, the value of the other measure is going to increase too.

When the relationship between the measures is inverse, as the value of one measure increases, the value of the other measure will decrease.

Unit conversion problems, like this one, is an example of a direct relationship between measures.

First Step:

The first step is knowing how many g are in a tablet.

Each gram has 1,000,000 mcg. So:

1g - 1,000,000 mcg

xg - 600 mcg

1,000,000x = 600

[tex]x = \frac{600}{1,000,000} = 0.0006[/tex]

Each tablet has 0.0006g

Final step:

How many tablets may be prepared from 30g of colchicine?

1 tablet - 0.0006g

x tablets - 30g

0.0006x = 30

[tex]x = \frac{30}{0.0006}[/tex]

x = 50,000

50,000 tablets may be prepared from 30g of colchicine

In part A, the next two numbers in the pattern are 70 and 700. In part B, the relationship between the terms in the pattern is that each term is 10 times as great as the term to its left.

Part A:

The pattern in part A involves multiplying each number by 10. Starting with 0.007, we multiply it by 10 to get 0.07. Then, we multiply 0.07 by 10 to get 0.7. Next, we multiply 0.7 by 10 to get 7. So, the next two numbers in the pattern are 70 and 700.

Part B:

The relationship between the terms in the pattern above is that each term is 10 times as great as the term to its left. For example, 0.07 is 10 times greater than 0.007, and 0.7 is 10 times greater than 0.07. This pattern continues, with 7 being 10 times greater than 0.7 and so on.

Learn more about pattern here:

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Answer:

Step-by-step explanation:

Given that,

Ken has 7 base ball

And each base ball have a weight of 0.3 pounds

If 1 ball = 0.3 pounds

Then, 7 baseball = 7 × 0.3 pounds

So, 7 baseball = 2.1 pounds

Then, the total weight of the 7base balls is 2.1 pounds

But, we want to to use the box to represent this data following the given information in the attachment

Note: since, the weight of the baseball is in decimal points

Let 0.1 pounds be 1 Square box.

Answer

1. To represent the weight of one base ball (0.3 pounds), ken should shade 3 Square box.

Since 1 pounds is 1 Square box

2. To represent the weight of all the seven base ball (2.1pounds), he should shade this amount seven times.

This means that he need to shade 3 square box seven times. This shows that the amount used in the second part of the question means by how much must the square be shaded compare to question the first part of the question

3. The shaded part of the model will represent the expression 0.3 × 7

4. The total weight of the base ball is 2.1 pounds.

Given that ;

Ken has no. of baseballs = 7

And each baseball have a weight is = 0.3 pounds

By unity method;

If 1 ball = 0.3 pounds

Then, 7 baseball = 7 × 0.3 pounds

So, 7 baseball = 2.1 pounds

Then, the total weight of the 7 baseballs is 2.1 pounds.

As per given in the question ;

We want to use the box to represent this data given in the following question .  

Since, the weight of the baseball is in decimal points

Let 0.1 pounds be 1 Square box.

Since 1 pounds is 1 Square box.

This means that he need to shade 3 square box seven times. This shows that the amount used in the second part of the question means by how much must the square be shaded compare to question .  

For the more information about the weight management follow the link given below .

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Answer:

Yes, The rational numbers are closed under multiplication.

Step-by-step explanation:

A rational number is a number which can be expressed in the form of a fraction [tex]\frac{x}{y}[/tex], where x and y are integers and y ≠ 0.

Now, closure property of multiplication states that if two rational numbers are multiplied then the product is also a rational number. Thus, if r and t are rational numbers, then

r×t = s, where s is the product of r and t

s is also a rational number.

Hence, the rational numbers are closed under multiplication.

This can be better explained with the help of an example [tex]\frac{3}{4} \times \frac{2}{5} = \frac{6}{20}[/tex],

It is clear that [tex]\frac{6}{20}[/tex] is a rational number.

Answer:

See step-by-step explanation below

Step-by-step explanation:

This problem is solved using the Euclidean algorithm; to prove that the integers 8n + 3 and 5n + 2 are relative prime we have to prove that:

gcd(8n + 3, 5n + 2) = 1

gcd (8n + 3, 5n + 2) = gcd (3n + 1, 5n + 2) = gcd (3n + 1, 2n + 1) = gcd(n, 2n + 1) = gcd(n,1) = 1

⇒gcd(8n + 3, 5n + 2) = 1

The integers 8n + 3 and 5n + 2 are relatively prime because their GCD is 1, as shown through the Euclidean algorithm. To compute the multipliers for the inverse of one number with respect to the other, one can backtrack through the Euclidean algorithm stages.

To prove that the integers 8n + 3 and 5n + 2 are relatively prime for any positive integer n, we need to show that their greatest common divisor (GCD) is 1. We can use the Euclidean algorithm to find the GCD of two numbers:

Let's say a = 8n + 3 and b = 5n + 2.

We compute a - b, which is [tex](8n + 3) - (5n + 2) = 3n + 1.[/tex]

Now we find the GCD of b and (a - b), which is GCD(5n + 2, 3n + 1).

Repeating the process, we have [tex](5n + 2) - (3n + 1) \times int((5n + 2) / (3n + 1)) = (5n + 2) - (3n + 1) \times 1 = 2n + 1.[/tex]

The GCD of (3n + 1) and (2n + 1) must now be found.

Continuing similarly, we eventually arrive at a difference of 1, demonstrating that the original two numbers are indeed relatively prime.

To find the multipliers for the inverse of one number with respect to the other, we can backtrack through the Euclidean algorithm steps, expressing each remainder as a linear combination of the two original numbers. This process will yield the required multipliers showing the inverse relationship.

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Answer:  The required answer is

[tex]f(x)=e^x+\cos x+\sin x.[/tex]

Step-by-step explanation:  We are given to find the inverse Laplace transform of the following function as a function of x :

[tex]F(s)=\dfrac{2s^2}{(s-1)(s^2+1)}.[/tex]

We will be using the following formulas of inverse Laplace transform :

[tex](i)~L^{-1}\{\dfrac{1}{s-a}\}=e^{ax},\\\\\\(ii)~L^{-1}\{\dfrac{s}{s^2+a^2}\}=\cos ax,\\\\\\(iii)~L^{-1}\{\dfrac{1}{s^2+a^2}\}=\dfrac{1}{a}\sin ax.[/tex]

By partial fractions, we have

[tex]\dfrac{s^2}{(s-1)(s^2+1)}=\dfrac{A}{s-1}+\dfrac{Bs+C}{s^2+1},[/tex]

where A, B and C are constants.

Multiplying both sides of the above equation by the denominator of the left hand side, we get

[tex]2s^2=A(s^2+1)+(Bs+C)(s-1).[/tex]

If s = 1, we get

[tex]2\times 1=A(1+1)\\\\\Rightarrow A=1.[/tex]

Also,

[tex]2s^2=A(s^2+1)+(Bs^2-Bs+Cs-C)\\\\\Rightarrow 2s^2=(A+B)s^2+(-B+C)s+(A-C).[/tex]

Comparing the coefficients of x² and 1, we get

[tex]A+B=2\\\\\Rightarrow B=2-1=1,\\\\\\A-C=0\\\\\Rightarrow C=A=1.[/tex]

So, we can write

[tex]\dfrac{2s^2}{(s-1)(s^2+1)}=\dfrac{1}{s-1}+\dfrac{s+1}{s^2+1}\\\\\\\Rightarrow \dfrac{2s^2}{(s-1)(s^2+1)}=\dfrac{1}{s-1}+\dfrac{s}{s^2+1}+\dfrac{1}{s^2+1}.[/tex]

Taking inverse Laplace transform on both sides of the above, we get

[tex]L^{-1}\{\dfrac{2s^2}{(s-1)(s^2+1)}\}=L^{-1}\{\dfrac{1}{s-1}\}+L^{-1}\{\dfrac{s}{s^2+1}+\dfrac{1}{s^2+1}\}\\\\\\\Rightarrow f(x)=e^{1\times x}+\cos (1\times x)+\dfrac{1}{1}\sin(1\times x)\\\\\\\Rightarrow f(x)=e^x+\cos x+\sin x.[/tex]

Thus, the required answer is

[tex]f(x)=e^x+\cos x+\sin x.[/tex]

Here's my step-by-step explanation:

Whenever you see the phrase "a number" in a problem like this, then they want you to use a variable. Let's use n for number and translate from English to Algebraic.

- The product of 2 more than a number and 10 is 36 more than 8 times the number.

- The product of 2 more than n and 10 is 36 more than 8 times n.

- The product of 2 + n and 10 is 36 + 8n.

- (2 + n)(10) is 36 + 8n.

- (2 + n)(10) = 36 + 8n

Let's solve.

(2 + n)(10) = 36 + 8n

20 + 10n = 36 + 8n

10n - 8n = 36 - 20

2n = 16

n = 8

Hope this helps, let me know if I made a mistake or if you have any questions!

Answer:

b+3(3-2b)=1-2(b+1)  

One solution was found :

                  b = 10/3 = 3.333

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

               b+3*(3-2*b)-(1-2*(b+1))=0  

Step by step solution :

Step  1  :

Equation at the end of step  1  :

 (b+(3•(3-2b)))-(1-2•(b+1))  = 0  

Step  2  :

Equation at the end of step  2  :

 (b +  3 • (3 - 2b)) -  (-2b - 1)  = 0  

Step  3  :

Equation at the end of step  3  :

 10 - 3b  = 0  

Step  4  :

Solving a Single Variable Equation :

4.1      Solve  :    -3b+10 = 0  

Subtract  10  from both sides of the equation :  

                     -3b = -10

Multiply both sides of the equation by (-1) :  3b = 10

Divide both sides of the equation by 3:

                    b = 10/3 = 3.333

One solution was found :

                  b = 10/3 = 3.333

Step-by-step explanation:

Answer:

P= -2x +46

Step-by-step explanation:

the relation between price and demand is

P= mx +b ........................1

when demand is 10 shirts price is $26

when demand is 20 shirts price is $6

firstly put P= 26 and x= 10 in 1

26= 10m + b.......................2

secondly put x= 20 and P= 6 in 1

6= 20m + b ............................3

solving  2 and 3 we get

m = -2

putting this value of m in either of 2 and 3 to get b

b= 46

so the final relation obtained by putting m= -2 and b= 46 in 1 we get

P= -2x +46

Answer:

2 mg/minute.

Step-by-step explanation:

We have been given that a drug is mixed 2 g in 500 ml.

First of all, we will convert 2 grams to milligrams. 1 gram equals 1000 milligrams.

2 grams = 2,000 mg.

Now, we will find amount of mg per ml as:

[tex]\text{The amount of mg per ml}=\frac{2000\text{ mg}}{\text{500 ml}}[/tex]

[tex]\text{The amount of mg per ml}=\frac{4\text{ mg}}{\text{ml}}[/tex]

We have been given that a lidocaine drip is infusing at 30 mL/hr on an infusion device, so amount of mg per hr would be:

[tex]\text{Amount of mg per hour}=\frac{30\text{ ml}}{\text{ hr}}\times \frac{4\text{ mg}}{\text{ml}}[/tex]

[tex]\text{Amount of mg per hour}=\frac{120\text{ mg}}{\text{ hr}}[/tex]

We know 1 hour equals 60 minutes.

[tex]\text{Amount of mg per hour}=\frac{120\text{ mg}}{\text{ hr}}\times \frac{\text{1 hour}}{\text{60 minutes}}[/tex]

[tex]\text{Amount of mg per hour}=\frac{120\text{ mg}}{\text{60 minutes}}[/tex]

[tex]\text{Amount of mg per hour}=\frac{2\text{ mg}}{\text{minute}}[/tex]

Therefore, the patient is receiving 2 mg/minute.

To calculate the mg/minute that the patient is receiving from the lidocaine drip, we need to convert the given flow rate from mL/hr to mL/minute, and then convert the drug concentration from grams to milligrams.

To calculate the mg/minute that the patient is receiving, we need to convert the given flow rate from mL/hr to mL/minute, and then convert the drug concentration from grams to milligrams. Here's how:

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Answer:

The probability that a transaction can be completed in less than 100 milliseconds if 30% of transactions are changes is 0.718

Step-by-step explanation:

Let A be the vent of new item

Let B be the event of transaction completed in less than 100 milliseconds

[tex]A^c = \text{change item}[/tex]

Since we are given that  30% of transactions are changes,

So, [tex]A^c =0.3[/tex]

We are given that The addition of an item can be completed less than 100 milliseconds 94% of the time

So, [tex]P(B|A)=0.94[/tex]

We are also given that only 20% of changes to a previous item can be completed in less than this time.

So,[tex]P(B|A^c)=0.2[/tex]

[tex]P(A)=1-P(A^c) = 1 - 0.3 = 0.7[/tex]

So, the probability that a transaction can be completed in less than 100 milliseconds :

= [tex]P(B|A)  \times P(A) +P(B|A^c) \times P(A^c)[/tex]

= [tex]0.94  \times 0.7 +0.2 \times 0.3[/tex]

= [tex]0.718[/tex]

Hence the probability that a transaction can be completed in less than 100 milliseconds if 30% of transactions are changes is 0.718

The overall probability that any transaction can be completed in less than 100 milliseconds is approximately 76%.

Given the probability that new additions are completed in less than 100 milliseconds is 94% and the changes in data are 20%. Also, we know that only 30% of transactions are changes. We are required to find the overall probability that a transaction can be completed in less than 100 milliseconds. This situation involves a mixed probability, where some transactions are additions (70% of them) and some are changes (30% of them). Therefore, we calculate as follows:

Probability (Transaction < 100 ms) = (0.7 * 0.94) + (0.3 * 0.2)

By calculating the expression above we find the overall probability of a transaction being completed in less than 100 milliseconds to be approximately 0.76 or 76%.

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Answer:

Step-by-step explanation:

Demontra has to divide negative 100 by negative 2, which means she has to do:

[tex]\Rightarrow \frac {-100}{-2}[/tex]

This can be written as:

[tex]\Rightarrow \frac {-1\times 100}{-1\times 2}[/tex]

-1 / -1 = 1

Also, 100 / 2 = 50

So,

[tex]\Rightarrow \frac {-1\times 100}{-1\times 2}=50\times 1=50[/tex]

She got -200 which is wrong as she muliplied the numbers which is also wrong as -100 × -2 = 200

She did not done anything right.

Answer:

The population in a statistical study is determined by all the individuals that could be part of the study, that is, all the individuals that have common characteristics that make them individuals of interest to the researcher.

In the study of the previous statement, the population is made up of all recruits from the US Army. UU. in Iraq at the time of the study.

Step-by-step explanation:

Final answer:

The population of interest in the study of stress levels among U.S. army recruits stationed in Iraq is all U.S. army recruits stationed in Iraq at the time of the study.

Explanation:

The question asks about the population of interest in a study of stress levels among U.S. army recruits stationed in Iraq. In this context, the population of interest refers to the entire group of individuals that the researchers aim to understand or make inferences about based on their study. Given the details of the study, the population of interest in this case includes all U.S. army recruits stationed in Iraq at the time of the study.

Answer:

a person should consume 1340 calories after eating ice cream

Step-by-step explanation:

Given :

The worst treat tested by the group was 1,910 total calories.

People need roughly 3,100 to 3,400 calories per day.

To Find : Using a daily average, how many additional calories should a person consume after eating ice cream?

Solution:

People need roughly 3,100 to 3,400 calories per day.

So, Average = [tex]\frac{3100+3400}{2}[/tex]

Average = [tex]3250[/tex]

So, one should take 3250 calories daily

The worst treat tested by the group was 1,910 total calories.

So, additional calories should a person consume after eating ice cream :

= 3250-1910

= 1340

Hence a person should consume 1340 calories after eating ice cream

Answer:  [tex]\sin(x+y)=\dfrac{1-8\sqrt{3}}{15}[/tex]

Step-by-step explanation:

Since we have given that

[tex]\sin x=\dfrac{1}{3}\\\\so,\\\\\cos x=\sqrt{1-\dfrac{1}{9}}=\sqrt{\dfrac{8}{9}}=\dfrac{2\sqrt{2}}{3}[/tex]

Since x ends in the 2 nd quadrant,

So, [tex]\cos x=\dfrac{-2\sqrt{2}}{3}[/tex]

Similarly,

[tex]\cos y=\dfrac{1}{5}\\\\So,\\\\\sin y=\sqrt{1-\dfrac{1}{25}}=\sqrt{\dfrac{24}{25}}=\dfrac{2\sqrt{6}}{5}[/tex]

So, sin(x+y) is given by

[tex]\sin x\cos y+\sin y\cos x\\\\\\=\dfrac{1}{3}\times \dfrac{1}{5}+\dfrac{2\sqrt{6}}{5}\times (-)\dfrac{2\sqrt{2}}{3}\\\\\\=\dfrac{1}{15}-\dfrac{8\sqrt{3}}{15}\\\\\\=\dfrac{1-8\sqrt{3}}{15}[/tex]

Hence, [tex]\sin(x+y)=\dfrac{1-8\sqrt{3}}{15}[/tex]

Answer:

Her weight is increase by 18 lbs over past five years and the slope is 3.6 lbs per year.

Step-by-step explanation:

Given information: Estelle weight is

At age 16 =  110 lbs

At age 21 = 128 ibs

Increase in her weight over the past 5 years is the difference of weight at age 21 and at age 16.

Increase in her weight over the past 5 years = 128 - 110 = 18

Her weight is increase by 18 lbs over past five years.

Let x=age and y=weight, then the weight function passes through the points (16,110) and (21,128).

If a line passes through two points [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex],  then the slope of the line is

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

Using the above formula we get

[tex]m=\frac{128-110}{21-16}[/tex]

[tex]m=\frac{18}{5}[/tex]

[tex]m=3.6[/tex]

Therefore the slope is 3.6 lbs per year.

Answer:

[tex] \frac{5\cdot 4\cdot 3}{10\cdot 9 \cdot 8}\approx 0.083[/tex]

Step-by-step explanation:

Getting all three marbles of green color only happens if every draw is a green marble. On the first marble draw, the urn has 10 marbles in it, out of which 5 are green. So the probability of drawing a green marble on this first draw is [tex]\frac{5}{10}[/tex]

Then, once this has happened, the second draw also needs to be a green marble. At this point in the urn there are only 9 marbles left, and only 4 of them are green. So the probability of drawing a green marble at this point is [tex] \frac{4}{9}[/tex]

Afterwards, on the last draw, a green marble also needs to be drawn. At this point there are only 8 marbles left on the urn, and only 3 of them are green. So the probability of drawing a green marble on this last draw is [tex] \frac{3}{8}[/tex]

Therefore the probability of drawing all three marbles of green color is

[tex] \frac{5}{10}\cdot\frac{4}{9}\cdot\frac{3}{8}\approx 0.083[/tex]

Answer:

The statement [tex]((p \lor q) \land (p \implies r) \land (\neg r)) \implies q[/tex] is a tautology.

Step-by-step explanation:

A truth table shows how the truth or falsity of a compound statement depends on the truth or falsity of the simple statements from which it's constructed.

We can see from the truth table that the last column contains only true values. Therefore, the statement is a tautology.

Logical equivalences are a type of relationship between two statements or sentences in propositional logic. To simplify an equivalency, start with one side of the equation and attempt to replace sections of it with equivalent expressions. Continue doing this until you have achieved the desired statement form.

[tex]((p \lor q) \land (p \implies r) \land (\neg r)) \implies q \\\equiv \neg[(p \lor q) \land (p \implies r) \land (\neg r)] \lor q[/tex] by implication law

[tex]\equiv \neg[(p \lor q) \land (\neg p \lor r) \land (\neg r))] \lor q[/tex] by implication law

[tex]\equiv \neg(p \lor q) \lor \neg (\neg p \lor r) \lor \neg(\neg r) \lor q[/tex] by de Morgan’s law

[tex]\equiv \neg(p \lor q) \lor \neg (\neg p \lor r) \lor r \lor q[/tex] by Double Negative

[tex]\equiv [(\neg p \land \neg q) \lor (p \land \neg r)] \lor r \lor q[/tex] by de Morgan’s law

[tex]\equiv [(\neg p \land \neg q) \lor q] \lor [(p \land \neg r) \lor r][/tex] by commutative and associative laws

[tex]\equiv [(\neg p \lor q) \land (\neg q \lor q)] \lor [(p \lor r) \land (\neg r \lor r)][/tex] by distributive laws

[tex]\equiv (\neg p \lor q) \lor (p \lor r)[/tex] by negation and identity laws

[tex]\equiv (\neg p \lor p) \lor (q \lor r)[/tex] by communicative and associative laws

[tex]\equiv T[/tex] by negation and domination laws

Therefore, the statement is a tautology.

The given logical statement is a tautology, as confirmed by a truth table and verified through logical equivalences, specifically equivalent to "q OR ~q," demonstrating its truth in all possible scenarios.

To determine whether the given logical statement "((p OR q) AND (p -> r) AND (~r)) -> q" is a contradiction, a tautology, or neither, we can create a truth table. The statement has three propositional variables: p, q, and r, so we need a truth table with 2^3 = 8 rows to cover all possible combinations of truth values for these variables.

p | q | r | (p OR q) | (p -> r) | (~r) | ((p OR q) AND (p -> r) AND (~r)) | (((p OR q) AND (p -> r) AND (~r)) -> q)

--|---|---|----------|----------|-----|---------------------------------|-----------------------------------------

T | T | T | T        | T        | F   | F                               | T

T | T | F | T        | F        | T   | F                               | T

T | F | T | T        | T        | F   | F                               | T

T | F | F | T        | F        | T   | F                               | T

F | T | T | T        | T        | F   | F                               | T

F | T | F | T        | T        | T   | T                               | T

F | F | T | F        | T        | F   | F                               | T

F | F | F | F        | T        | T   | F                               | T

In the last column, we evaluate the given logical statement "((p OR q) AND (p -> r) AND (~r)) -> q" for each row.

Now, let's analyze the results:

- The statement is True in all rows. Therefore, it is a tautology because it is always true, regardless of the truth values of p, q, and r.

We can also verify this using logical equivalences. The statement "((p OR q) AND (p -> r) AND (~r)) -> q" is logically equivalent to "q OR ~q," which is always true by the law of excluded middle. This confirms that the original statement is a tautology.

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Answer:

7 is both a factor and a multiple

7 is the factor of 14 and 7 has 14 as its multiple.

A factor is a number that completely divides another number. To put it another way, if adding two whole numbers results in a product, then the numbers we are adding are factors of the product because the product is divisible by them.

Given:

We have the Number 14.

So, the factors of 14 = 1, 2, 7, 14.

and, 7 can act as factor of 14.

also, 7 has multiple 14.

Hence, 7 is the required number.

Learn more about factors here:

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Answer:

This isn't true.

Step-by-step explanation:

Think of the case a=2, b=3 and c=6. We have that a|b, since 2|6.

We also have that ab=c, since 2*3=6. However, it is NOT true that a|b, as 2 does NOT divide 3. As this you can construct many other examples where a|c and ab=c BUT a does NOT divide b.

Other counterexamples:

a=2, b=5, c=10

a=2, b=7, c=14

a=2, b=11, c=22

a=2, b=13, c=26