Answer: Let the probability that hunter P hits the target be denoted by "p", and the probability that hunter Q hits the target be denoted by "q".
(a) The probability that they both miss the target is given by:
(1-p)*(1-q)
(b) If the target is hit, then there are three possible outcomes:
(i) Only hunter P hits it: The probability of this event is given by:
p*(1-q)
(ii) Only hunter Q hits it: The probability of this event is given by:
(1-p)*q
(iii) Both hunters hit the target: The probability of this event is given by:
p*q
Note that the sum of the probabilities in (i), (ii), and (iii) is equal to 1, since one of these three events must occur if the target is hit.
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Juro circles of radii 5cm and 3cm intersect at two points and the distance between their centres is 4cm. Find the length of the common chord
By using Pythagorean theorem, the length of the common chord is approximately 4.58 cm.
What is the Pythagorean theorem?
The Pythagorean theorem is a fundamental result in Euclidean geometry that describes the relationship between the three sides of a right-angled triangle.
In equation form, this can be written as:
[tex]c^2 = a^2 + b^2[/tex]
where c is the length of the hypotenuse, and a and b are the lengths of the other two sides (also known as the legs) of the right-angled triangle.
Let O1 and O2 be the centers of the circles with radii 5cm and 3cm, respectively, and let A and B be the points of intersection of the circles.
Let O1 and O2 be the centers of the circles with radii 5 cm and 3 cm, respectively, and let A and B be the points of intersection of the circles. Then, by the Pythagorean theorem, we have:
[tex]$$\begin{aligned} AB^2 &= AO_1^2 - BO_1^2 \ &= (5^2 - 2^2) , \text{cm}^2 \ &= 21 , \text{cm}^2 \end{aligned}$$[/tex]
Therefore, the length of the common chord AB is:
[tex]$$AB = \sqrt{21} \approx 4.58 , \text{cm}$$[/tex]
Thus, the length of the common chord is approximately 4.58 cm.
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6. All of the students in a classroom list their birthdays.
a. Is the birthdate, b, a function of the student, s? Explain your reasoning.
b. Is the student, s, a function of the birthdate, b? Explain your reasoning.
Answer:
is the birthdate
Step-by-step explanation:
because very student in the classroom need to give their day to be recorded
Use the normal approximation to the binomial to find the probabilities for the specific values of X. N=50, p=. 8, X=44
The probability of X = 44 is approximately 0.087 using the normal approximation to the binomial.
The normal approximation to the binomial is a technique used to estimate probabilities for large binomial distributions when calculating by hand becomes impractical. In this case, we have N = 50, p = 0.8, and we want to find the probability of X = 44.
To use the normal approximation, we need to first check if the binomial distribution is approximately normal. This can be done by checking if the conditions np >= 10 and n(1-p) >= 10 are met. In our case, np = 50 x 0.8 = 40 and n(1-p) = 50 x 0.2 = 10, so the conditions are met.
Next, we calculate the mean and standard deviation of the normal distribution using the formulas μ = np and σ = sqrt(np(1-p)). In our case, μ = 40 and σ = sqrt(50 x 0.8 x 0.2) ≈ 2.83.
Finally, we use the normal distribution with mean μ and standard deviation σ to find the probability of X = 44. We need to standardize X using the formula Z = (X - μ) / σ, which gives Z = (44 - 40) / 2.83 ≈ 1.41.
Using a standard normal table or calculator, we find that the probability of Z being less than 1.41 is approximately 0.921. This means that the probability of X being less than or equal to 44 is approximately 0.921.
Therefore, the probability of X being exactly 44 is approximately the difference between the probability of X being less than or equal to 44 and the probability of X being less than or equal to 43. Using a continuity correction, we adjust 43.5 to 43, which gives us:
P(X = 44) ≈ P(43.5 < X < 44.5) ≈ P(Z < (44.5 - 40) / 2.83) - P(Z < (43.5 - 40) / 2.83) ≈ 0.087.
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Question is on photo.
Therefore , the solution of the given problem of triangle comes out to be x = Tan⁻¹(15/8) .
A triangle is what exactly?Because a triangle has two or so more extra parts, it is a polygon. It has a straightforward rectangular shape. Only two of a triangle's three sides—A and B—can differentiate it from a regular triangle. Euclidean geometry produces a single area rather than a cube when boundaries are still not perfectly collinear. Triangles are defined by their three sides and three angles. Angles are formed when a quadrilateral's three sides meet. There are 180 degrees of sides on a triangle.
Here,
Given :
=> AS =15 and AN = 8
So,
We have to find m∠N = x
=> Tanx = 15/8
=> x = Tan⁻¹(15/8)
Therefore , the solution of the given problem of triangle comes out to be x = Tan⁻¹(15/8) .
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#2. Chelsey has 18 coins all nickels and quarters. The dollar value of her coins is $2.10. Her brother
Alex has 1.5 times the number of nickels Chelsey has but only 2/3 of the number of quarters. What is the
dollar value of Alex's coins?
Let's use variables to represent the number of nickels and quarters Chelsey has.
Let x be the number of nickels that Chelsey has, and let y be the number of quarters that she has. From the problem statement, we know that:
x + y = 18 (Chelsey has 18 coins in total)
0.05x + 0.25y = 2.10 (the total value of her coins is $2.10)
We can use these equations to solve for x and y.
First, we can solve for x in terms of y from the first equation:
x = 18 - y
Substituting this expression for x into the second equation, we get:
0.05(18 - y) + 0.25y = 2.10
Simplifying this equation, we get:
0.9 - 0.05y + 0.25y = 2.10
0.2y = 1.2
y=6
So Chelsey has 6 quarters. Substituting this value of y back into the first equation, we get:
x + 6 = 18
x = 12
So Chelsey has 12 nickels.
Now we can move on to Alex's coins. We know that he has 1.5 times the number of nickels that Chelsey has, which is:
1.5 * 12 = 18
And he has 2/3 of the number of quarters that Chelsey has, which is:
2/3 * 6 = 4
Therefore, Alex has 18 nickels and 4 quarters.
The dollar value of Alex's coins is:
0.05(18) + 0.25(4) = 0.90 + 1.00 = $1.90
So the dollar value of Alex's coins is $1.90.
What is the volume of the triangular prism below?
Answer:
210 [tex]cm^3[/tex]
Step-by-step explanation:
The formula of solving the volume of a triangular prism is:
volume = (height x base x length)^3 / 2
First, put in the numbers:
V = [tex]\frac{(2cm*7cm*30cm)^3}{2}[/tex]
V = [tex]\frac{(14cm*30cm)^3}{2}[/tex]
V = [tex]\frac{420^3}{2}[/tex]
V = [tex]210[/tex] [tex]cm^3[/tex]
Select the correct answer.
A sport statistician gathered data on the number of points scored in each game by high school basketball players across the country. She found
a population mean of 20. 15 and a standard deviation of 3. 7. Each sample size was 20 players. By the central limit theorem, which interval do
68% of the sample means fall within?
A. 19. 32 and 20. 98
B. 17. 67 and 22. 63
C. 19. 96 and 20. 34
D. 18. 50 and 21. 81
68% of the sample means fall within 19.32 and 20.98.
By the central limit theorem, we know that the distribution of sample means will approach a normal distribution with a mean equal to the population mean (20.15) and a standard deviation equal to the population standard deviation divided by the square root of the sample size (3.7 / sqrt(20) ≈ 0.828).
To find the interval within which 68% of the sample means fall, we need to find the z-scores corresponding to the 16th and 84th percentiles of the normal distribution (which is the range that encompasses 68% of the data in a normal distribution).
Using a z-table or a calculator, we can find that the z-score for the 16th percentile is approximately -1.0 and the z-score for the 84th percentile is approximately 1.0.
Therefore, the interval within which 68% of the sample means fall is:
20.15 - 1.0(0.828) = 19.32
to
20.15 + 1.0(0.828) = 20.98
So the answer is option A: 19.32 and 20.98.
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What is the largest even number of 5,2,4,3
Answer:
The largest even number of 5243 is 14.
step by step:- 5+2+4+3
=14
so that ,14 is the largest even number.
Suppose that there are three factories that manufacture light-bulbs. For factory i, every manufactured light-bulb (independently) has a chance of being defective with a probability pi ,p1 =0.05,p2 =0.1,p3=0.3. Initially, I thought that when I order a box of light-bulbs it is equally likely to come from any of the three factories. Upon receiving the box I found 8 out of 100 to be defective. What is my posterior probability that the box came from factoryi,i∈{1,2,3}?
The posterior probability that the box of light-bulbs came from factory 1 is 0.2111, from factory 2 is 0.5219, and from factory 3 is 0.2669.
Bayes' theorem formula can be used to answer this question.The formula is as follows:P(A|B) = P(B|A) P(A) / P(B)Here, A is the event that the box of light bulbs came from a particular factory (i.e., A = {i | i ∈ {1,2,3}}), and B is the event that 8 out of 100 bulbs in the box are defective.
First, we need to find the probability of observing 8 defective light-bulbs out of 100 for each of the three factories. The probability of observing k defective light-bulbs out of n total light-bulbs is given by the binomial distribution: P(k) = n! / (k!(n-k)!) * pk * (1-p)n-k
Factory 1: p1 = 0.05n = 100P(8) = 100! / (8!(100-8)!) * (0.05)8 * (1-0.05)100-8 = 0.0993Factory 2:p2 = 0.1n = 100P(8) = 100! / (8!(100-8)!) * (0.1)8 * (1-0.1)100-8 = 0.2452Factory 3:p3 = 0.3n = 100P(8) = 100! / (8!(100-8)!) * (0.3)8 * (1-0.3)100-8 = 0.1041
The sum of these probabilities gives the marginal likelihood:P(B) = P(8|1)P(1) + P(8|2)P(2) + P(8|3)P(3) = 0.0993 * 1/3 + 0.2452 * 1/3 + 0.1041 * 1/3 = 0.1495 Using Bayes' theorem, we can now calculate the posterior probabilities:P(1|8) = P(8|1) P(1) / P(B) = 0.0993 * 1/3 / 0.1495 = 0.2111P(2|8) = P(8|2) P(2) / P(B) = 0.2452 * 1/3 / 0.1495 = 0.5219P(3|8) = P(8|3) P(3) / P(B) = 0.1041 * 1/3 / 0.1495 = 0.2669
Therefore, the posterior probability that the box of light-bulbs came from factory 1 is 0.2111, from factory 2 is 0.5219, and from factory 3 is 0.2669.
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Sophia who took the Graduate Record Examination (GRE) scored 160 on the Verbal Reasoning section and 157 on the Quantitative Reasoning section. The mean score for Verbal Reasoning section for all test takers was 151 with a standard deviation of 7, and the mean score for the Quantitative Reasoning was 153 with a standard deviation of 7.67. Suppose that both distributions are nearly normal.
(a) Write down the short-hand for these two normal distributions.
(b) What is Sophia’s Z-score on the Verbal Reasoning section? On the Quantitative Reasoning
section? Draw a standard normal distribution curve and mark these two Z-scores.
(c) What do these Z-scores tell you?
(d) Relative to others, which section did she do better on?
(e) Find her percentile scores for the two exams.
(f) What percent of the test takers did better than her on the Verbal Reasoning section? On the
Quantitative Reasoning section?
(g) Explain why simply comparing raw scores from the two sections could lead to an incorrect
conclusion as to which section a student did better on.
(h) If the distributions of the scores on these exams are not nearly normal, would your answers to
parts (b) - (f) change? Explain your reasoning.
Short-hand for these two normal distributions: N(151, 7) for Verbal Reasoning and N(153, 7.67) for Quantitative Reasoning, z-score is 1 and 0.52 and she performed better in the verbal reasoning section.
Sophia’s Z-score on the Verbal Reasoning section is (160-151)/7 = 1, and her Z-score on the Quantitative Reasoning section is (157-153)/7.67 = 0.52. The Z-scores are marked on the standard normal distribution curve as shown below.
The Z-scores tell us that Sophia scored higher than the mean score of other test takers in the Verbal Reasoning section (1 is to the right of the mean, 0) and lower than the mean score of other test takers in the Quantitative Reasoning section (0.52 is to the left of the mean, 0).
Relative to others, Sophia did better on the Verbal Reasoning section.
Sophia’s percentile score for the Verbal Reasoning section is 84% and for the Quantitative Reasoning section is 64%.
Approximately 84% of the test takers did better than Sophia on the Verbal Reasoning section, and 64% of the test takers did better than Sophia on the Quantitative Reasoning section.
Comparing raw scores from the two sections can lead to an incorrect conclusion as to which section a student did better on because raw scores do not take into account the number of students who scored higher or lower than the student in question. For example, a student may have scored higher than another student in one section but scored lower than many other students in that same section.
If the distributions of the scores on these exams are not nearly normal, then my answers would change. This is because a non-normal distribution would not follow the standard normal distribution curve, and thus the Z-scores and percentiles would be different.
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3. In a translation, the
remains the same.
A concrete sidewalk is being put around a square park. If the diagonal line across the center of a square park is 24 feet, what is the exact length of each side?
By answering the presented question, we may conclude that As a result, Pythagorean theorem each side of the square park is exactly [tex]12\sqrt(2)[/tex]feet long.
what is Pythagorean theorem?The Pythagorean Theorem, generally known as Pythagorean Theorem, is the foundational Euclidean arithmetic that links the three points of a right triangle. This rule states that the area of a cube only with hypotenuse side is equal to the total of the areas of triangles that have both two sides. The Pythagorean Theorem states that the square that spans the hypotenuse of a right triangle opposite the right angle is the total of all the squares that span its vertices. It is sometimes represented in broad algebraic notation as a2 + b2 = c2.
Let us solve this problem using the Pythagorean theorem. The diagonal in a square is the hypotenuse of a right triangle, with each side of the square acting as one of its legs. The Pythagorean theorem asserts that the sum of the squares of the two shorter sides of a right triangle equals the square of the hypotenuse.
x² + x² = 24²
2x² = 576
x² = 288
[tex]x = \sqrt (288)\\x = \sqrt(144 * 2)\\x = 12\sqrt (2)\\[/tex]
As a result, each side of the square park is exactly [tex]12\sqrt(2)[/tex]feet long.
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3) When 100 coins are tossed find the probability that exactly 35 will be heads (numerical answer followed by Matlab code), assuming the number of experiments is 100000.
Write a matlab code please
This program creates [tex]N=100000[/tex] trials in which [tex]n=100[/tex] coins are tossed, and it counts the trials in which precisely [tex]k=35[/tex] heads are thrown. This code's output will be the calculated probability of receiving [tex]35[/tex] heads.
What are examples and probability?It is based on the possibility that something will materialize. The fundamental underpinning of maximum possible is just the explanation of likelihood. For instance, while flipping a coin, there is a 12-percent probability that it will land on its head.
How should a novice compute probability?Determine the number of alternative ways to roll a 4 as well as multiply it by the overall number of outcomes to determine the likelihood of the event occurring.
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A random sample of 50 BCTC students is asked, "Would you rather speak all languages or speak to animals?" After pondering the question carefully, 30 of the students say they would rather speak to animals. What is the low end of a 95% confidence interval for the proportion of all BCTC students who say they would rather speak to animals? Enter your answer as a decimal rounded to 4 decimal places.
The low end of a 95% confidence interval for the proportion of all BCTC students who say they would rather speak to animals is 0.3964.
The low end of a 95% confidence interval for the proportion of all BCTC students who say they would rather speak to animals is 0.3964.What is confidence interval?A confidence interval is a range of values, calculated from a data sample, that is used to estimate an unknown population parameter.The formula for the margin of error:margin of error = z* (standard deviation / sqrt(n))Wherez* = critical valueStandard deviation (σ) = Sample standard deviationn = Sample sizeHow to calculate the low end of a 95% confidence interval for the proportion of all BCTC students who say they would rather speak to animals?The formula for calculating the confidence interval for a proportion is:p ± z* (sqrt((p(1-p))/n))Here,Sample proportion (p) = 30/50 = 0.6Sample size (n) = 50Critical value for a 95% confidence interval (z*) = 1.96Put these values in the above formula to calculate the low end of the confidence interval:0.6 ± 1.96 * sqrt((0.6(1-0.6))/50)0.6 - 1.96 * 0.1143 = 0.3964 (rounded to 4 decimal places)Therefore, the low end of a 95% confidence interval for the proportion of all BCTC students who say they would rather speak to animals is 0.3964.
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for which of the following functions can we use the intermediate value theorem to prove the existence of roots in the indicated interval? i. f(x)
The intermediate value theorem can be used to prove the existence of roots in the indicated interval for a function that is continuous on the interval.
The intermediate value theorem states that if a function is continuous on an indicated interval [a,b], and if f(a) and f(b) have opposite signs, then there must be at least one root in the interval [a,b].
Therefore, for the given function f(x), we can use the intermediate value theorem to prove the existence of roots in the indicated interval if the function is continuous on the interval and if f(a) and f(b) have opposite signs.
In order to determine if the function is continuous on the indicated interval, we need to check if there are any discontinuities or breaks in the function on the interval. If there are no discontinuities or breaks, then the function is continuous on the indicated interval.
Next, we need to check if f(a) and f(b) have opposite signs. If f(a) and f(b) have opposite signs, then there must be at least one root in the indicated interval.
In conclusion, we can use the intermediate value theorem to prove the existence of roots in the indicated interval for a function that is continuous on the interval and if f(a) and f(b) have opposite signs.
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Jina opened a savings account with $500 and was paid simple interest at an annual rate of 3%. When Jina closed the account, she was paid $30 in interest. How long was the account open for, in years?
We can conclude after answering the provided question that Therefore, interest the account was open for 2 years.
what is interest ?In mathematics, interest is the amount of money earned or owed on an initial investment or loan. You can use either simple or compound interest. Simple interest is calculated as a percentage of the original amount, whereas compound interest is calculated on the principal amount plus any previously earned interest. If you invest $100 at a 5% annual simple interest rate, you will earn $5 in interest every year for three years, for a total of $15.
We know that the amount of interest earned, I, is given by the formula:
[tex]I = P * r * t\\P = $500\\r = 0.03 \\I = $30\\t = I / (P * r)\\t = $30 / ($500 * 0.03)\\t = 2 years[/tex]
Therefore, the account was open for 2 years.
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Sum to infinity:-
[tex]1 + \frac{3}{4} + \frac{7}{16} + \frac{15}{64} + ...[/tex]
When summed to infinity, the series would be 2/3.
How to sum to infinity ?Let's analyze the pattern of the numerators first:
1, 3, 7, 15, ...
We can see that the numerators are increasing in powers of 2, minus 1:
1 = 2¹ - 1
3 = 2² - 1
7 = 2³ - 1
15 = 2⁴ - 1
The denominators are increasing powers of 4:
4 = 2²
16 = 2⁴
64 = 2⁶
Now, we can rewrite the series as:
1 + (2^1 - 1) / 2^2 + (2^2 - 1) / 2^4 + (2^3 - 1) / 2^6 + ...
To find the sum to infinity, we can rewrite the series as a single summation:
∑[(2^n - 1) / 2^(2n)] for n = 0 to infinity
To evaluate this sum, we can split it into two separate summations:
∑[2^n / 2^(2n)] - ∑[1 / 2^(2n)] for n = 0 to infinity
Now, subtract the second summation from the first:
S = S1 - S2 = 2 - 4/3 = (6 - 4) / 3 = 2/3
So, the sum to infinity for this series is 2/3.
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use the distributive property or simplify to match the equivalent expressions 4(100-3)
Step-by-step explanation:
4 ( 100 -3) = 4 x 100 - 4 x 3
= 400 - 12 = 388
A dairy farmer milks his two cows every day. He determined the chance that he gets anywhere between 12 and 14 gallons of milk in one day is around 32%. Identify the method of probability the farmer used to reach this conclusion. Select the correct answer below: theoretical relative frequency
The dairy farmer used the relative frequency method of probability to reach his conclusion.
Relative frequency is a method of calculating probability that is based on the observation of how often an event occurs in a sample. The farmer likely observed how often he gets between 12 and 14 gallons of milk in a day and used that data to calculate the probability of it happening.
In contrast, theoretical probability is based on the assumption that all possible outcomes are equally likely. It is calculated by dividing the number of desired outcomes by the total number of possible outcomes.
Therefore, the correct answer is relative frequency.
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The following are NOT examples of using the "Distributive Property"
Example 1:4(x-2)=4•x-4•2
Example 2:4•26
4•(20+6)=4•20+4•6
A:True
B:False
False, Example 1:4(x-2)=4•x-4•2 and Example 2:4•26 and Example 3: 4•(20+6)=4•20+4•6 are NOT examples of using the "Distributive Property".
The Distributive Property is a fundamental property of algebra that is used to simplify expressions involving multiplication and addition. It states that the product of a number and the sum or difference of two or more numbers is equal to the sum or difference of the products of that number and each of the numbers in the sum or difference.
Example 1 uses the Distributive Property correctly, showing how 4 multiplied by the difference of x and 2 is equal to the difference of 4 times x and 4 times 2.Example 2 is not an example of using the Distributive Property because it is just a multiplication of two numbers, which does not involve addition or subtraction.Example 3 uses the Distributive Property correctly, showing how 4 multiplied by the sum of 20 and 6 is equal to the sum of 4 times 20 and 4 times 6.Learn more about the Distributive Property at
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Destanye replaces the light bulb
in the hall closet every 6 months
and replaces the air filter every 4
months. She just replaced both
items this month. After how many
months will she replace both the
light bulb and the air filter?
Select all that apply.
After 12 months she replace both the light bulb and the air filter
We can also use the concept of the Least Common Multiple (LCM) to find the answer to the problem.
The LCM of two numbers is the smallest number that is divisible by both of them. To find the LCM of 6 and 4, we can list the multiples of each number and look for the smallest multiple that is common to both lists. Alternatively, we can use the prime factorization of each number and multiply the highest powers of the common prime factors.
Prime factorization of 6: 2 x 3
Prime factorization of 4: 2^2
To find the LCM, we multiply the highest powers of each prime factor:
LCM = 2^2 x 3 = 12
Therefore, the LCM of 6 and 4 is 12. This means that the next time Destanye will replace both the light bulb and the air filter is in 12 months.
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(2 A football team tries to move the ball forward as many yards as possible on each play, but sometimes they end up behind where they started. The distances, in yards, that a team moves on its first five plays are 2, - 1, 4, 3, and - 5. A positive number indicates moving the ball forward, and a negative number indicates moving the ball backward.
(1)
Which number in the list is the greatest?
(2)
What is a better question to ask to find out which play went the farthest from where the team started?
(3)
The coach considers any play that moves the team more than 4 yards from where they started a "big play. Which play(s) are big plays?
4 is bigger than the other numbers in the list, it is the biggest number. 2. "What is the absolute value of the largest distance travelled on a single play?" is a better question,
What is the distinction between vector quantities' magnitude and direction?The size or amount of a vector quantity is referred to as its magnitude, and its direction is referred to as its orientation or angle. For instance, the magnitude of the vector quantity of velocity refers to the speed, or how quickly an item is travelling, whereas the direction refers to the direction in which the object is going. While direction is often represented by a vector, which includes both magnitude and direction, magnitude is typically represented by a scalar, which is a single integer that reflects the magnitude of the quantity.
We only need to compare the numbers against one another to determine which is the highest number in the list. As 4 is bigger than the other numbers in the list, it is the biggest number.
(2) "What is the absolute value of the largest distance travelled on a single play?" is a better question to ask in order to determine which play took the team the furthest from its starting point. Regardless of the direction, the absolute number will show us the size of the distance travelled.
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shawls has 3 times as many stickers as abigail they have a total of 12 stickers how many stickers does shayla have
Syal =3x Abigail
Syal+Abigail =12
(3 x abigail)+1 Abigail =12 Abigail
Abigail =12:4 = 3 striker
Syal =3x3= 9 striker
△ABC and △DEF are similar triangles. Find BC.
Answer:
5 units
Step-by-step explanation:
We know that Triangles ABC and DEF are similar, therefore their values will be Equal
So , x + 7 + x - 4 = 12 + 5
2x + 3 = 15
2x = 15-3
2x = 12
x = 6
So in Triangle ABC , AC = x + 7 = 6 + 7 = 13
Therefore, by Pythagorean Therom,
BC^2 + AB^2 = AC^2
BC^2 + 144 = 169
BC^2 = 169 -144 = 25
BC = [tex]\sqrt{25} = 5[/tex]
Hope it helps.
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Here is another triangle similar to DEF found in the lesson section labeled “Shrinking Triangles”.
• Label the triangle D”E”F”.
• What is the scale factor from triangle DEF to triangle D”E”F”?
• What are the coordinates of F”? Explain how you know.
• What are cos(D”), sin(D”), and tan(D”)?
The scale factor of dilation is 0.075 and the coordinates of F" are (0.9, 0.375)
Define dilation?A thing must be scaled down or altered during the dilation process. It is a transformation that reduces or enlarges the objects using the supplied scale factor. Pre-image refers to the original figure, whereas image refers to the new figure obtained following dilatation.
Label the triangle D” E” F”.
The label of the triangle is added as an attachment.
From question, we have
DE = 12 units
Then, we have
D"E" = 0.9 units
Using the above, we have the following:
Scale factor = D"E"/DE
Scale factor = 0.9/12
Scale factor = 0.075
Hence, the scale factor of dilation is 0.075.
The coordinates of F"
This is calculated as
F = Scale factor * F
So, we have
F = 0.075 * (12, 5)
F = (0.9, 0.375)
The trigonometry ratios the sine, cosine and tangent are calculated as follow:
sin(D") = EF/DF
cos(D") = DE/DF
tan(D") = EF/DE
So, we have
sin(D") = 0.375/1 = 0.375
cos(D") = 0.9/1 = 0.9
tan(D") = 0.375/0.9 = 0.416
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A child has three bags of fruits in which Bag 1 has 5 apples and 3 oranges, Bag 2 has 4 apples and 5 oranges, and Bag 3 has 2 apples and 3 oranges. One fruit is drawn at random from one of the bags. Calculate the probability that the chosen fruit was an orange and was drawn from Bag 2
The probability that the chosen fruit was an orange and was drawn from Bag 2 is approximately 0.269 or 26.9%.
There are three bags of fruits with different numbers of apples and oranges in each bag. We need to calculate the probability that the fruit drawn is an orange and was drawn from Bag 2.
We can use Bayes' theorem to find the conditional probability of an event, given that another event has already occurred. Let O be the event that an orange is drawn and B2 be the event that the fruit is drawn from Bag 2.
Using Bayes' theorem, we have:
P(B2|O) = P(O|B2) * P(B2) / P(O)
We need to calculate P(O|B2), P(B2), and P(O) to find P(B2|O).
P(O|B2) is the probability that an orange is drawn given that the fruit is drawn from Bag 2. This can be calculated as:
P(O|B2) = Number of oranges in Bag 2 / Total number of fruits in Bag 2
= 5 / (4 + 5)
= 5/9
P(B2) is the probability that the fruit is drawn from Bag 2, without any information about the color of the fruit. As all three bags are equally likely to be chosen, we have:
P(B2) = 1/3
P(O) is the probability that an orange is drawn, without any information about the bag it was drawn from. This can be calculated as the weighted average of the probability of drawing an orange from each bag, using the probabilities of choosing each bag. We have:
P(O) = P(O|B1) * P(B1) + P(O|B2) * P(B2) + P(O|B3) * P(B3)
= (3/8) * (1/3) + (5/9) * (1/3) + (3/5) * (1/3)
= 31/135
Substituting the calculated values into Bayes' theorem, we get:
P(B2|O) = P(O|B2) * P(B2) / P(O)
= (5/9) * (1/3) / (31/135)
= 25/93
≈ 0.269
Therefore, the probability that the chosen fruit was an orange and was drawn from Bag 2 is approximately 0.269 or 26.9%.
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. Determine the number of ways in which 2023 can be written as a1 + a2 + a3 + · · · + an where n ≥ 2 is a positive integer, a1 ≤ a2 ≤ a3 ≤ · · · ≤ an are positive integers, and a1 ≥ an−1
Given that, 2023 can be written as a1 + a2 + a3 + · · · + an where n ≥ 2 is a positive integer, a1 ≤ a2 ≤ a3 ≤ · · · ≤ an are positive integers, and a1 ≥ an−1. We need to determine the number of ways in which 2023 can be written in such a manner.
Let's apply the formula for finding the number of ways to represent an integer, where each term in the sum is an increasing sequence of positive integers.
In general, the formula for the number of ways to represent n is the number of partitions of n into an increasing sequence of positive integers. Let p(n) denote the number of partitions of n into an increasing sequence of positive integers.
Then, the number of ways to represent n as an increasing sequence of positive integers is given by p(n) - 1, as we cannot use the representation where n is the only term in the sum.
If we find p(2023), we can find the number of ways to represent 2023 as an increasing sequence of positive integers. Therefore, p(2023) - 1 is the required number of ways to represent 2023.
Let's calculate p(2023). The easiest way to calculate p(2023) is by generating functions. Since we are looking for an increasing sequence, we can use the formula for the generating function for partitions into distinct parts, which is:
(∑n=0∞xn)/(1−x)=1+x+x2+x3+x4+⋯.
We can replace x^n in the numerator with a generating function for the sum of the partitions of n into increasing parts to get the desired generating function.
(∑n=0∞x(n2+n)/2)/(1−x)=1+x+2x2+3x3+4x4+⋯.
The numerator in the generating function is (∑n=0∞x(n2+n)/2)=(∑n=0∞x(n2/2+n/2))=(∑n=0∞x(n/2)2+(1/4)(n+1/2))=(∑n=0∞(x1/4)n(n+1)+(x1/4)n+1/2)/2. The numerator is a sum of two geometric series, so we can simplify it.
(∑n=0∞(x1/4)n(n+1)+(x1/4)n+1/2)/2=(1/(1−x/4))^2+(x/2(1−x/4)^2).
(x/2(1−x/4)^2)=x(1/(1−x/4))^2−(x/2)/(1−x/4).
Therefore,
(∑n=0∞xn(n+1)/2)/(1−x)=p(0)+p(1)x+p(2)x2+p(3)x3+⋯=((1/(1−x/4))^2)−(x/2(1−x/4)^2).
The coefficient of x^2023 is x^2023−2−x/2(1−x/4)^2. Since x^2023−2=691104804/4^2, x^2022^2=−2022/4, and x^2021^2=303805/16, the required number of ways to represent 2023
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OA and OB are opposite rays. If x = 42 degree, what is the value of y?
Answer:
69
Step-by-step explanation:
2y + x = 180 ( Linear Pair)
As x = 42
2y + 42 = 180
2y = 180- 42
2y = 138
y = 138/2
y = 69
Patel is solving 8x2 + 16x + 3 = 0. Which steps could he use to solve the quadratic equation? Select three options. 8(x2 + 2x + 1) = –3 + 8 x = –1 Plus or minus StartRoot StartFraction 5 Over 8 EndFraction EndRoot x = –1 Plus or minus StartRoot StartFraction 4 Over 8 EndFraction EndRoot 8(x2 + 2x + 1) = 3 + 1 8(x2 + 2x) = –3
The steps he can use are [tex]x = -\frac{-16 \± \sqrt{16\² - 4 \cdot 8 \cdot 3}}{16}[/tex], (x + 1)² = 5/8 and 8(x² + 2x + 1) = -3 + 8
How to determine the steps he can useFrom the question, we have the following parameters that can be used in our computation:
8x² + 16x + 3 = 0
Assuming he uses the quadratic formula method, which is represented as
[tex]x = -\frac{-b \± \sqrt{b\² - 4ac}}{2a}[/tex]
So, we have
[tex]x = -\frac{-16 \± \sqrt{16\² - 4 \cdot 8 \cdot 3}}{2 \cdot 8}[/tex]
[tex]x = -\frac{-16 \± \sqrt{16\² - 4 \cdot 8 \cdot 3}}{16}[/tex]
If he uses completing the square, then we have
8x² + 16x = -3
So, we have
x² + 2x = -3/8
x² + 2x + 1 = -3/8 + 1
So, we have
(x + 1)² = 5/8
If he factorizes, then the expression is
8(x² + 2x + 1) = -3 + 8
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The probability that Carmella will pay for the tickets is greatest, approximately __%, if Ben selects his tile using method __
The probability that Carmella will pay for the tickets is greatest, approximately 50%, if Ben selects his tile randomly.
Now, let's calculate the probability of Carmella paying for the tickets based on the different methods that Ben can use to select his tile.
If Ben selects his tile randomly, then the probability of him selecting an even-numbered tile is 3/6, or 1/2. This is because there are three even-numbered tiles (2, 4, and 6) and six total tiles. Similarly, the probability of Ben selecting an odd-numbered tile is also 1/2, as there are three odd-numbered tiles (1, 3, and 5).
So, the probability of Carmella paying for the tickets if Ben selects his tile randomly is 1/2, or approximately 50%.
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