if the GCF of two numbers is 1, they are called relatively prime. Find three sets of relatively prime numbers

The amount of cobalt-60 will decrease with each half-life, which is 5.26 years. After 5 half-lives (26.3 years), the 200g sample will reduce to 6.25g.

The subject of the question is the half-life of a radioactive substance, in this case, cobalt-60. Half-life is the time required for half the atoms in a sample to decay. Given that the half-life of cobalt-60 is 5.26 years, and we are interested in a period of 26.3 years, we first classify this period into 'half-life' units, which equals 26.3 / 5.26  = 5 half-lives.

With each half-life, the quantity of cobalt-60 is cut in half. We start with a 200g sample and after 5 half-lives, the amount of cobalt-60 would decrease as follows:

Therefore, after 26.3 years, only 6.25g of the original 200g sample of cobalt-60 will remain.

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The probability of having at most three defects in a four-hour period is approximately 0.8572.

The Poisson probability distribution is used to model the number of events that occur in a fixed interval of time or space, given the average rate at which the events occur. In this problem, the rate of errors is given as one error every two hours. We need to find the probability of having at most three defects in a four-hour period.

To solve this problem, we can use the Poisson probability formula, which is:

P(x; λ) = (e^(-λ) * λ^x) / x!

where P(x; λ) is the probability of having exactly x events occur in the given interval, λ is the average rate of events, e is the base of the natural logarithm, and x! is the factorial of x.

First, we need to calculate the average rate of events in a four-hour period. Since the rate is given as one error every two hours, we can calculate the rate for a four-hour period as follows:

Average rate = (4 hours) / (2 hours/error) = 2 errors

Now, we can plug this average rate (λ = 2) into the Poisson probability formula for x = 0, 1, 2, and 3 to find the probabilities of having at most three defects:

P(0; 2) = [tex](e^{(-2)} * 2^0)[/tex] / 0! ≈ 0.1353

P(1; 2) = ([tex]e^{(-2)} * 2^1[/tex]) / 1! ≈ 0.2707

P(2; 2) = [tex](e^{(-2)} * 2^2)[/tex] / 2! ≈ 0.2707

P(3; 2) = [tex](e^{(-2)} * 2^3)[/tex] / 3! ≈ 0.1805

To find the probability of at most three defects, we sum up these individual probabilities:

P(at most 3 defects) = P(0; 2) + P(1; 2) + P(2; 2) + P(3; 2) ≈ 0.8572

So, the probability of having at most three defects in a four-hour period is approximately 0.8572.

Step 1

Find the value of s

we know that

In a Rhombus all sides are congruent

so

[tex]AB=BC=CD=DA[/tex]

[tex]AB=9s+29\\CD=10s-16[/tex]

equate AB and CD

[tex]9s+29=10s-16\\[/tex]

Combine like term

[tex]10s-9s=29+16\\[/tex]

[tex]s=45\ units[/tex]

The answer Part a) is

the value of s is [tex]45\ units[/tex]

Step 2

Find the value of side AB

[tex]AB=9s+29[/tex]

substitute the value of s

[tex]AB=9*45+29=434\ units[/tex]

Remember that the sides are congruent

[tex]AB=BC=CD=DA[/tex]

therefore

the answer Part b) is

The length of the side BC is [tex]434\ units[/tex]

The value of s is 45 and the length of side BC is; 434.

A rhombus is a plane shape of the form given I'm the attached image with all its sides equal.

Hence,

To determine the value of s;

The length of side BC = CD = 10(45) - 16 = 450 - 16

The length of side BC = 450 -16 = 434

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Answer:

five ordered pairs are (-2,1) (-1,3) (0,5) (1, 7) (2, 9)

y intercept is (0,5)

x intercept is (-2.5, 0)

Step-by-step explanation:

y=2x+5

To get 5 ordered pairs , plug in some random number for x and find out y

x           y= 2x+5

-2            2(-2) + 5= 1

-1             2(-1) + 5= 3

0             2(0) + 5= 5

1             2(1) + 5= 7

2             2(2) + 5= 9

five ordered pairs are (-2,1) (-1,3) (0,5) (1, 7) (2, 9)

When x=0 , the value of y = 5

So y intercept is (0,5)

To find x intercept we plug in 0 for y

y=2x+5

0 = 2x +5

subtract 5 from both sides

-5 = 2x

divide by 2 on both sides

x= -5/2

so x intercept is (-2.5, 0)

The standard form of the equation for the line passing through (-5, -1) and (10, -7) is y = x - 17.

The standard form of the equation for the line passing through the points (-5, -1) and (10, -7) can be found using the point-slope form given, which is y + 7 = (x - 10).

First, let's convert this equation into slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

Subtracting 7 from both sides of the equation, we get y = x - 17.

This is the standard form of the equation for the line passing through (-5, -1) and (10, -7).

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Final answer:

To show that z = (xy)/(x+y) given that 3^x = 4^y = 12^z, logarithms and the properties of exponents are used to simplify and solve for z.

Explanation:

The problem given is to show that if 3^x = 4^y = 12^z, then z = (xy)/(x+y). To solve this, we can use logarithms or equate the expressions to a common variable. Since 12 = 3 × 4, it means that 12^z = (3^z)(4^z).

Remember that the original equations given were 3^x = 12^z and 4^y = 12^z.

Substituting z into these equations we have 3^x = 3^z × 4^z and 4^y = 3^z × 4^z.

Taking logarithms, we find

[tex]log(3^x) = log(3^z) + log(4^z) and log(4^y) = log(3^z) + log(4^z).[/tex]

Applying the power rule of logarithms, the equations simplify to

[tex]x log(3) = z log(3) + z log(4)[/tex]and  [tex]y log(4) = z log(3) + z log(4).[/tex]

Dividing these equations by log(3) and log(4) respectively gives

[tex]x = z + z(log(4)/log(3))[/tex] and [tex]y = z + z(log(3)/log(4)).[/tex]

Adding these equations, we get

[tex]x + y = 2z + z[(log(4)/log(3)) + (log(3)/log(4))],[/tex]

which simplifies to x + y = 2z because [tex](log(4)/log(3)) + (log(3)/log(4)) = 1.[/tex]Finally, solving for z gives z = (xy)/(x+y) as desired.

A vector can be defined as an element of a vector space, which represents an object that has both magnitude and direction.

Assuming that; [tex]\left \{ {{x=rcos \theta} \atop {y=rsin \theta}} \right.[/tex]

For the channel:

The vector representing the flow from North to South is given by:

[tex]15 cos (\frac{3\pi}{2} ), \; 15 sin (\frac{3\pi}{2} )\\\\15(0),\; 15(-1)[/tex]

Vector = 0, -15

For the sailboat:

The vector representing the sailboat is given by:

[tex]40 cos (30 ), \; 40 sin (30)\\\\30(\frac{\sqrt{3} }{2} ),\; 30(\frac{1 }{2} )\\\\15\sqrt{3} ,\;15[/tex]

Vector = [tex]15\sqrt{3} ,\;15[/tex]

For the strong wind:

The vector representing the strong wind blowing towards northwest is given by:

[tex]30 cos (\frac{3\pi}{4} ), \; 30 sin (\frac{3\pi}{4} )\\\\30(\frac{-\sqrt{2} }{2} ),\; 30(\frac{\sqrt{2} }{2} )\\\\-15\sqrt{2},\;15\sqrt{2}[/tex]

Vector = [tex]-15\sqrt{2},\;15\sqrt{2}[/tex]

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This is a binomial probability problem where we calculate the probability of fewer than 9 successful surgeries out of 10, given a success rate of 75% for each surgery. The probabilities of 0 to 8 successful surgeries are calculated and summed up.

The question asks for the probability that fewer than 9 out of 10 surgeries are successful, given that each surgery has a 75% chance of being successful. This is a binomial probability problem since each surgery can result in either success (75%) or failure (25%), and we are dealing with a fixed number of independent trials (10 surgeries).

To solve this, we first calculate the probability of exactly 8 successes, exactly 7 successes, and so on down to 0 successes, then sum these probabilities. The binomial probability formula is P(x) = (nCx) * (p^x) * ((1-p)^(n-x)), where n is the number of trials, x is the number of successes, p is the probability of success, and nCx is the combination of n items taken x at a time.

However, calculating each probability individually and summing them can be tedious, and tools or calculators designed for binomial distributions are often used to compute this more efficiently. For a result of fewer than 9 successes, we would add the probabilities of getting 8, 7, ..., down to 0 successful surgeries.

divide them

17.16/0.624 = 27.5

27.5 is the answer

double check by multiplying 27.5 by 62.4%

27.5*.624 = 17.16

Answer:

18 boxes of wrenches and 22 boxes of hammers.

Step-by-step explanation:

515x + 285y = 11,080

x + y = 30

The probability of drawing the slips '1', '2', '3', and '4' in that order is 1/126 or approximately 0.0079.

The probability that the drawn slips are '1', '2', '3', and '4', in that order, can be calculated by considering the number of favorable outcomes and the total number of possible outcomes.

There are 9 slips in the box, so the total number of possible outcomes is 9 choose 4, which is denoted as C(9,4) or 9!/(4!(9-4)!), which simplifies to 126.

Since we want the slips to be drawn in a specific order, the number of favorable outcomes is 1, because there is only 1 way to arrange the slips in the order '1', '2', '3', and '4'.

Therefore, the probability is 1/126, which simplifies to approximately 0.0079 or 0.79%.

The value of the test statistic in this problem is approximately equal to [tex]\frac{0.83-0.9}{\sqrt{\frac{(0.9)(0.1)}{100} } }[/tex]

Aspirin also known as acetylsalicylic acid, is a medication used to reduce pain, fever, or inflammation. A test statistic is the random variable that calculated from sample data it used in a hypothesis test. You can use test statistics to determine whether to reject the null hypothesis or not. The test statistic can compare the data with what is expected under the null hypothesis

A survey (a general view, examination, or description of someone or something) claims that 9 out of 10 doctors recommend aspirin for their patients with headaches. to test this claim against the alternative that the actual proportion of doctors who recommend aspirin is less than 0.90, a random sample of 100 doctors results in 83 who indicate that they recommend aspirin. the value of the test statistic in this problem is approximately equal to [tex]\frac{0.83-0.9}{\sqrt{\frac{(0.9)(0.1)}{100} } }[/tex] where

a = People who indicate that they recommend aspirin = 0.83

b = the actual proportion of doctors who recommend aspirin = 0.9

c =  the actual proportion of doctors who not recommend aspirin = 1-0.9 = 0.1

d = a random sample = 100

[tex]\frac{a-b}{\sqrt{\frac{(b)(c)}{d} } } =\frac{0.83-0.9}{\sqrt{\frac{(0.9)(0.1)}{100} } }[/tex]

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divide

42/96 = 0.4375

0.4375*100 = 43.75 percent

 round the answer as needed.

The probability that the card chosen is not a heart is 0.75

Probability refers to a possibility that deals with the occurrence of random events.

The probability of all the events occurring need to be 1.

The formula of probability is defined as the ratio of a number of favorable outcomes to the total number of outcomes.

Sample space ={13 H + 13 D + 13 S + 13 C}

= 52 cards

The total number of cards in a deck is 52

Number of cards with hearts = 13

Therefore, P(getting one heart) = 13/52 = 1/4

P( getting NO heart) = 1-1/4 = 3/4 = 0.75

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Answer:

Step-by-step explanation:

The easiest way is to divide the number by 100, 10 and 1 in that order so:

15638/100=156.38 <- From this number you take only the integer (or the number without decimals), and that would be your hundreds, for this case 156 is the integer part, so 156 hundreds.

Next we take take the 38 we had left from the above division, and we divide it by 10.

38/10=3.8   <- we apply exactly the same steps as before but with the tens, working only with the integer, meaning 3, so you end up with 3 tens.

Last but not least, the rest, that is 8, will be your ones. In this case, just 8 ones.

Now, the combinations are infinite, if you take one from the hundreds it becomes 10 tens or 100 ones, and if you take 1 from the tens you get 10 ones. So you could have

155 hundreds (155-1), 12 tens (3+10-1), and 18 ones. Or any permutation you prefer.

For quick studies, it is easier to round down to the nearest 5 or 0, so another way to see this would be:

155 hundreds, 10 tens, and 38 ones.

The time taken to reach the maximum height = 3 s

Quadratic function is a function that has the term x²

The quadratic function forms a parabolic curve

The general formula is

where a, b, and c are real numbers and a ≠ 0.

The parabolic curve can be opened up or down determined from the value of a. If a is positive, the parabolic curve opens up and has a minimum value. If a is negative, the parabolic curve opens down and has a maximum value

So the maximum is if a <0 and the minimum if a> 0.

After t seconds, a ball height h (in feet) is given by the function h (t) = 96t-16t²

The maximum value of the function is obtained if the first derivative of the function h (t) = 0

96-32t = 0

96 = 32t

t = 3 s

The maximum height

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domain of the function

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Final answer:

The ball reaches its maximum height after 3 seconds. This is determined by finding when the derivative of the height function h(t) = 96t - 16t² equals zero and solving for t.

Explanation:

To determine the time at which the ball reaches its maximum height, we need to analyze the function h(t) = 96t - 16t². Here, h(t) represents the height of the ball after t seconds. The ball achieves its maximum height when the derivative of the height function with respect to time is zero (i.e., when its upward velocity is zero).

To find this, we can take the derivative of h(t) and set it equal to zero to find the critical points:

h'(t) = d/dt (96t - 16t²) = 96 - 32t.

Setting the derivative equal to zero:

96 - 32t = 0

32t = 96

t = 96/32 = 3 seconds.

Therefore, the ball reaches its maximum height after 3 seconds.

Answer:

A

Step-by-step explanation: