if 4 divides a^2-3b^2, then at least one of the integers a and b is even.

Answer:

Not always we can use a calculator to determine if a number is rational or irrational.

Step-by-step explanation:

Consider the provided information.

Can you ever use a calculator to determine if a number is rational or irrational.

Irrational   number: A   number is irrational if it cannot   be   expressed by dividing two     integers. The decimal expansion of     Irrational numbers are neither terminate nor     periodic.

The calculators gives the approximate answer, whether the number is irrational or rational.

So, it would be difficult to identify whether a large number produced by the calculator is irrational or not. As we know that many rational numbers can be incredibly large.

So, we can say that not always we can use calculator to determine if a number is rational or irrational.

Thus, Not always we can use a calculator to determine if a number is rational or irrational.

Answer:

In order to have a consistent linear system represented by the augmented matrix:

[tex]\left[\begin{array}{ccc}2&-3&h\\-6&9&5\end{array}\right][/tex]

the value of h must be:

[tex]h=-\frac{5}{3}[/tex]

Step-by-step explanation:

A system is consistent if it has a solution, this solution can be unique or a set of infinite solutions.  

First, you take the augmented matrix and find the equivalent row echelon form using Gaussian-Jordan elimination:

To do this, you have to multiply the 1st row by 3 and add it to the 2nd row, the resulting matrix is:

[tex]\left[\begin{array}{ccc}2&-3&h\\0&0&5+3h\end{array}\right][/tex]

Now, write the system of equations:

[tex]2x_1-3x_2=h\\0x_1+0x_2=5+3h[/tex]

The only way this system has a solution is if 5+3h=0, then, to satisfy this, the value of h must be:

[tex]h=-\frac{5}{3}[/tex]

Answer:

The set of solutions is [tex]\{\left[\begin{array}{c}a\\b\\x\\y\\z\end{array}\right] = \left[\begin{array}{c}-26+503y+543z\\-37+655y+724z\\-4+80y+90z\\y\\z\end{array}\right] : \text{y, z are real numbers}\}[/tex]

Step-by-step explanation:

The matrix associated to the problem is [tex]A=\left[\begin{array}{ccccc}1&-1&2&-8&1\\2&-1&-4&1&-2\\-4&1&4&-3&-1\end{array}\right][/tex] and the vector of independent terms is (3,1,-1)^t. Then the matrix equation form of the system is Ax=b.

The vector equation form is [tex]a\left[\begin{array}{c}1\\2\\-4\end{array}\right]+b\left[\begin{array}{c}-1\\-1\\1\end{array}\right] + x\left[\begin{array}{c}2\\-4\\4\end{array}\right]+y\left[\begin{array}{c}-8\\1\\-3\end{array}\right] + z\left[\begin{array}{c}1\\-2\\-1\end{array}\right]=\left[\begin{array}{c}3\\1\\-1\end{array}\right][/tex].

Now we solve the system.

The aumented matrix of the system is [tex]\left[\begin{array}{cccccc}1&-1&2&-8&1&3\\2&-1&-4&1&-2&1\\-4&1&4&-3&-1&-1\end{array}\right][/tex].

Applying rows operations we obtain a echelon form of the matrix, that is [tex]\left[\begin{array}{cccccc}1&-1&2&-8&1&3\\0&1&-8&-15&-4&-5\\0&0&1&-80&-9&-4\end{array}\right][/tex]

Now we solve for the unknown variables:

Since the system has two free variables then has infinite solutions.

The set of solutions is [tex]\{\left[\begin{array}{c}a\\b\\x\\y\\z\end{array}\right] = \left[\begin{array}{c}-26+503y+543z\\-37+655y+724z\\-4+80y+90z\\y\\z\end{array}\right] : \text{y, z are real numbers}\}[/tex]

Answer:

20 cm

Step-by-step explanation:

We are given a trapezoid, where the length of shorter base or on of the parllel line is 16 cm and the length of other parallel side is 24 cm.

Let the two parallel sides be x and y that is x = 16 cm and y = 24 cm.

A median of a trapezoid is a line segment that divides the non parallel sides of a trapezoid equally or a line segment that passes through the mid points of non-parallel sides of a trapezoid.

The length of median of a trapezoid = [tex]\frac{\text{Sum of parallel sides}}{2}[/tex] = [tex]\frac{16+24}{2}[/tex] = 20 cm.

Thus, the length of median of trapezoid is 20 cm.

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

Final answer:

To solve the system of inequalities, first, solve each inequality separately. Then, combine the solutions to find the common range of values for x that satisfy all the inequalities.

Explanation:

To solve the system of inequalities:

2x - 1 < x + 3

5x - 1 > 6 - 2x

x - 5 < 0

So, the solution to the system of inequalities is: x < 4, x > 1, x < 5

Answer:    a) 45    b) 60

Step-by-step explanation:

Given : A small restaurant has a menu with 2 appetizers, 5 main courses, and 3 desserts.

a) Number of meals includes an main course , a dessert and a appetizer, :-

[tex]2\times5\times3=30[/tex]

Number of meals includes an main course and a dessert and but not appetizer , then total possible meals:-

[tex]5\times3=15[/tex]

Then, the number of meals are possible if each includes an main course and a dessert, but may or may not include an appetizer= 30+15=45

b) Number of meals includes an main course and appetizer but not dessert:

[tex]5\times2=10[/tex]

Number of meals includes only main course =5

Now, the number of meals if dessert is also not required= 45+5+10=60

Answer:

1360.78 g

Step-by-step explanation:

1 lb = 453.592 g

3 lbs = 3 * 453.592 g = 1360.78 g

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.

To learn more about tautology

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Answer:

(a) The probability is 9.49%

(b) The probability is 38.28%

Step-by-step explanation:

The probability that a failure is due to induced substance is calculated as a multiplication as:

(13%) * (73%) = 9.49%

Where 13% is the percentage of heart failures that are due outside factors and 73% is the percentage of outside factors that are due induced substances.

On the other hand, the probability that a failure is due to disease or infection is the sum of the probability that a failure is due to disease and the probability that a failure is due to infection.

Then, the probability that a failure is due to disease is calculated as:

(87%) * (27%) =  23.49%

Where 87% is the percentage of heart failures that are due natural factors and 27% is the percentage of natural factors that are due disease.

At the same way, the probability that a failure is due to infection is calculated as:

(87%) * (17%) =  14.79%

So, the probability that a failure is due to disease or infection is:

23.49% + 14.79% = 38.28%

Answer:

The correct answer is B. : 10 sq mm = 0.1 sq cm

Step-by-step explanation:

It is just a matter of changing the units. The equivalence we need to know is 1cm = 10 mm. Also, we need to have in mind that we can write 10 sq mm as 10 mm*mm, because : 10 sq mm = 10 mm² = 10 mm*mm

Now we multiply two times by the fraction (1cm / 10 mm), which does not alter our measurement because the fraction is the same as multiplying by 1.

10 sq mm = 10 mm* mm = (10 mm*mm)*(1 cm / 10 mm)*(1 cm / 10 mm) = (10 mm*mm*cm*cm/ 10*10 mm*mm) =10/100 cm*cm = 0.1 cm² = 0.1 sq cm

Therefore, we have the equivalency : 10 sq mm = 0.1 sq cm

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:

$ 4933.2 ( approx )

Step-by-step explanation:

∵ Future value formula is,

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

Where,

P = principal amount,

r = annual rate,

n = number of periods,

t = number of years,

Given,

P = $ 4,000, r = 3.5 % = 0.035, t = 6 years n = 12 ( number of months in 1 year = 12 ),

Hence, the future value would be,

[tex]A=4000(1+\frac{0.035}{12})^{72}=4933.20414683\approx \$ 4933.2[/tex]

By using the principles of linear interpolation, the yield of strawberries with 75 cubic feet of fertilizer can be calculated as approximately 616.25 pounds.

The yield of strawberries based on the amount of fertilizer fed to the plants can be estimated using linear interpolation. We can establish two points based on the given information: (70, 770) and (100, 1100), where the first number represents the amount of fertilizer and the second one, the yield. The interpolation line equation can be formulated as y = mx + c where m = (y2 - y1) / (x2 - x1); as such, m = (1100 - 770) / (100 - 70) = 8.25.

To find the value of c (y-intercept), we use the equation with one of the known points and solve c = y1 - m * x1 = 770 - 8.25 * 70 = -5.

The yield, y at 75 cubic feet of fertilizer can be calculated as y = 8.25 * 75 - 5 = 616.25. Therefore, the estimated yield of strawberries when 75 cubic feet of fertilizer is applied is approximately 616.25 pounds.

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Answer:  a.    320 books

Step-by-step explanation:

Given : The total number of books  were placed on the discount shelf for 70% off the regular price = 960

The fraction of books sold = [tex]\dfrac{2}{3}[/tex]

Then, the number of books sold = [tex]\dfrac{2}{3}\times960=640[/tex]

Now, the number of books remain on the discount shelf = [tex]960-640=320[/tex]

Hence, the number of books remain on the discount shelf =320

Answer: A programmer is the person who is responsible for making  computer programs.He/she makes sure that the program is created according to the requirement and  accurate performing operations .The goals of the programmer are as follows:-

Programmer is indulged in these goals because there are always upcoming new technologies in the field of programming so, to keep theirselves updates and maintain their skill they improve theirselves time to time. Also it can affect the job of the programmer if they are not aware about programming skills quite well or might end up losing the job.

Answer:

a) Since the sphere intersects the xy-plane then the set of points of the sphere nearest to the xy-plane is the set of points in the circumference [tex](x+5)^2+y^2=18[/tex].

b)(-14.9, 0, 9 )

Step-by-step explanation:

a) The centre of the sphere is (-5,0,-9) and the radio of the sphere is [tex]\sqrt{99} \sim 9.9[/tex]. Since |-9|=9 < 9.9,  then the sphere intersect the xy-plane and the intersection is a circumference.

Let's find the equation of the circumference.

The equation of the xy-plane is z=0. Replacing this in the equation of the sphere we have:

[tex](x+5)^2+y^2+9^2=99[/tex], then [tex](x+5)^2+y^2=18[/tex].

b) Observe that the point (-9,0,9) has the same y and z coordinates as the centre and the x coordinate of the point is smaller than that of the x coordinate of  the centre. Then the point of the sphere nearest to the given point will be at a distance of one radius from the centre, in the negative x direction.

(-5-[tex]\sqrt{99}[/tex], 0, 9)= (-14.9, 0, 9 )

The 99% confidence interval estimate for the genuine proportion of families who own at least one DVD player is:

Lower bound: 0.287.

Upper bound: 0.563.

To find the 99% confidence interval estimate of the true proportion of families who own at least one DVD player, follow these steps:

Step 1. Determine the sample proportion [tex](\( \hat{p} \))[/tex]:

Number of families surveyed ( n ) = 85

Number of families owning at least one DVD player ( x ) = 36

Sample proportion [tex](\( \hat{p} \)) = \( \frac{x}{n} = \frac{36}{85} = 0.4247 \)[/tex]

Step 2. Find the standard error (SE) of the sample proportion:

Standard error formula: [tex]\( SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \)[/tex]

Plug in the values: [tex]\( SE = \sqrt{\frac{0.4247 \times (1 - 0.4247)}{85}} = \sqrt{\frac{0.4247 \times 0.5753}{85}} = \sqrt{\frac{0.2443}{85}} = \sqrt{0.002875} = 0.0536 \)[/tex]

Step 3. Determine the z-value for a 99% confidence interval:

The z-value for a 99% confidence interval is approximately 2.576.

Step 4. Calculate the margin of error (ME):

Margin of error formula: [tex]\( ME = z \times SE \)[/tex]

Plug in the values: [tex]\( ME = 2.576 \times 0.0536 = 0.1381 \)[/tex]

Step 5. Determine the confidence interval

Lower limit: [tex]\( \hat{p} - ME = 0.4247 - 0.1381 = 0.2866 \)[/tex]

Upper limit: [tex]\( \hat{p} + ME = 0.4247 + 0.1381 = 0.5628 \)[/tex]

Therefore, the 99% confidence interval estimate of the true proportion of families who own at least one DVD player is:

Lower limit: 0.287

Upper limit: 0.563

Complete Question:

A survey of 85 families showed that 36 owned at least one DVD player. Find the 99 \% confidence interval estimate of the true proportion of families who own at least on DVD player. Place your limits, rounded to 3 decimal places, in the blanks. Do not use any labels or symbols other than the decimal point. Simply provide the numerical values. For example, 0.123 would be a legitimate entry.

Lower limit (first blank) [tex]$=$ $\qquad$[/tex] ______ , Upper limit (second blank) = _______

Answer: Hi!

First, if you think that a compass has degrees as units, then N50E would be

50 degrees from north in the direction of the east, so if you put our 0 in east and count counterclockwise this will be an angle of 40 degrees.

If you think north has te Y axis positive direction, and east as the X axis positive direction. then the first boat has an angle of 40° counterclockwise from the +x

so the velocity in y is Vy=30mph*sin(40°) and in x is Vx= 30mph*cos(40°)

then the total displacement will be 22.98m to east and 19.28 north

the second one goes to s 70 e, so using the same notation as before, you can write this has -20° degrees count counterclockwise.

so decomposing the velocity will give us

Vy = 26*sin(-20°) and the displacement in Y is -8.89m

Vx = 26*cos(-20°) and the displacement in X is 24.43m

so the distance between the boats in y will be 19.28m - (-8.99)m = 28.27m

and in x: 24.43m - 22.98m = 1.45m

and the total distance is [tex]D^{2} = 1.45^{2} + 28.27^{2}[/tex]

so D = 28.30 m

Hi!

You know that if f(p) = 0, then (x-p) is a factor of the polynomial f(x)

Then, f(3)=0 is the case p=0, son the factor is (x-3)

Answer: Since we are told that f(3) = 0, we know that (x-3) is a factor.

Answer:

1. g(x)=2x+1-3 --> g(x)=2x-2, which is also y=2x-2, so you can graph it.

Step-by-step explanation:

Question 1: If f(x) = 2x+1, then you can see that all you have to do is substitute the equation for f(x) into the g(x) equation because g(x)= f(x)-3. So, if you substitute it, the equation will be g(x) = (2x+1) -3, then you just solve the rest of the equation. Put it into slope intercept form, y=mx+b, and then graph the equation.

Sorry, I don't really understand number 2 myself, so hopefully I could help with he first one.

Answer:

Step-by-step explanation:

All the right triangles are similar, so all will have the same angles.

The missing angle (B) in ΔABC is the complement of the given one:

  ∠DBC = 90° - 65° = 25°

The missing angles in the smaller triangles are the complements of the known acute angles in those triangles.

A diagram can help you see this.

Answer:

3.055 m

Step-by-step explanation:

In this solution we will use next notation:

[tex]t_1[/tex]=  time elapsed since oco serves the ball until it reaches its opponent.

[tex]t_2[/tex]=  time elapsed since the opponent returns the ball until it reaches oco.

d= Total distance traveled by Oco since serving the ball until  meeting the return.

We know that oco serves at vs = 50 m/s and her opponent is x=25 m away. Then, t_1 is given by

[tex]t_1=\frac{25m}{50m/s}=0.5s[/tex]

To compute t_2 observe that the return speed is 12.5 m/s and the distance that the ball will travel is [tex]25-(10t_1+10t_2)[/tex]. Then,

[tex]t_2=\frac{25-10t_1-10t_2}{12.5}=\frac{20-10t_2}{12.5}\implies t_2=\frac{20}{22.5}=\frac{8}{9}s[/tex].

Therefore,

[tex]d=10(t_1+t_2)=10(0.5+\frac{8}{9})=10(\frac{17}{18})=\frac{85}{9}m[/tex]

Finally, as Oco started 12.5m away from the net, when she meets the return she will be

[tex]12.5-\frac{85}{9}=\frac{55}{18}=3.055m[/tex]

away from the net.

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

Answer:

Hello my friend! The answer is 1.75X10^-15

Step-by-step explanation:

If you multiply 2.5 x7 = 17.5

When we do the product of exponential terms with the same base, we can sum de  exponents. In this case (-10) + (-6) = -16.  

However, to scientific notation, we have to use 1.75

So, the final result wich were 17.5x10-16, will be "1.75x10^-15"

Answer:

  see below

Step-by-step explanation:

The "roster method" means you simply list them all:

  {John Boozman, Doug Jones, Martha McSally, Lisa Murkowski, Tom Cotton, Richard C. Shelby, Kyrsten Sinema, Dan Sullivan}

_____

There are no senators from these states with the same name, so repeated elements is not an issue here.

The set S includes the Senators John Boozman, Tom Cotton from Arkansas, Richard C. Shelby, Doug Jones from Alabama, and Lisa Murkowski, Dan Sullivan from Alaska. Each state has two unique senators, thus there are no repeated elements in the set.

The set S of current U.S. Senators from states that begin with 'A' using the roster method can be written as follows:

John Boozman (Arkansas)

Tom Cotton (Arkansas)

Richard C. Shelby (Alabama)

Doug Jones (Alabama)

Lisa Murkowski (Alaska)

Dan Sullivan (Alaska)

Each state beginning with 'A' (Alabama, Alaska, and Arkansas) contributes two senators to the set. S, as defined, would not have repeated elements since senators are unique to each state they represent, and no senator represents more than one state.

Answer:

C

Step-by-step explanation:

A can be from 2- 9 ( 8 digits)

B can be 0 to 9 (10 digits)

C can be 0 to 9 (10 digits)

Each of the X's can be 0 to 9 (10 digits)

To get the number of possibilities, we multiply them to get:

8 * 10 * 10 * 10 * 10 * 10 * 10 = 8,000,000

But now, 1 number (867-5309) is restricted, so the number of possibilities decrease by 1:

8,000,000 - 1= 7, 999, 999

Correct answer is C

Answer:

30 micro grams

Step-by-step explanation:

1 ml contains 0.05 mg (milligram) of digoxin

So, 0.6 ml contains digoxin = [tex]0.6 \times 0.05[/tex]

                                          = [tex]0.03 mg[/tex]

Now 1 mg contains 1000 μg (micro grams)

So, 0.03 mg contains micro grams= [tex]0.03 \times 1000[/tex]

                                                      = [tex]30[/tex]

Hence 30 micro grams of digoxin would be delivered in each dose of 0.6 ml .

Answer:

D

Step-by-step explanation:

Because ordering in ratios is important, so it must stay constant like 6,15.

Answer:

The answer is: D) 15/6

Step-by-step explanation:

The ratio of two given numbers such as X and Y is expressed by the symbol ':' Therefore, the ratio of X and Y or X:Y can be referred to as X is to Y and can also be expressed as a fraction X/Y or X÷Y.

Therefore, the ratio can be expressed in a number of ways, 6:15 = 6 to 15 = 6/15

Whereas, 15/6 = 15:6 ≠ 6:15