If a teacher cuts out 15 congruent circles from a single sheet of paper that measures 6 inches by 10 inches, then (60 - 1.5 π) square inches paper is wasted.
The area of each circle is given by the formula [tex]A = \pi r^2[/tex], where r is the radius of the circle. Since all 15 circles are congruent, they all have the same radius.
Let's first find the radius of each circle. Since the circles are cut from a 6 by 10 inch sheet of paper, we know that the diameter of each circle cannot be greater than 6 inches or 10 inches. Let's take the smaller dimension (6 inches) and divide it by 15 to find the maximum possible diameter of each circle:
6 inches ÷ 15 = 0.4 inches
Therefore, the maximum possible radius of each circle is 0.2 inches.
Now we can calculate the total area of all 15 circles:
Total area of circles = [tex]15 * \pi * (0.2)^2[/tex]
[tex]= 1.5 \pi[/tex] square inches
The area of the original sheet of paper is 6 inches * 10 inches = 60 square inches. Therefore, the amount of paper wasted is:
Amount of paper wasted = Area of sheet - Total area of circles
= 60 square inches - 1.5 π square inches
= 60 square inches - 4.71 square inches (using π ≈ 3.14)
= 55.29 square inches (rounded to two decimal places)
Therefore, the answer is (60 - 1.5π) square inches.
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Find the premium an employee must pay if his share of the health insurance 40% of 84$
[tex]\begin{array}{|c|ll} \cline{1-1} \textit{\textit{\LARGE a}\% of \textit{\LARGE b}}\\ \cline{1-1} \\ \left( \cfrac{\textit{\LARGE a}}{100} \right)\cdot \textit{\LARGE b} \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{40\% of 84}}{\left( \cfrac{40}{100} \right)84}\implies \text{\LARGE 33.6}[/tex]
what is the area of the trapezoid?
Answer:
[tex] \ \frac{1}{2} (5 \frac{1}{4} + 7)(2) = 12 \frac{1}{4} [/tex]
A is correct.
The mean number of births per minute in a country in a recent year was about three. Find the probability that the number of births in any given minute is (a) exactly five, (b) at least five, (c) more than five
The Poisson distribution is a discrete probability distribution, the probability of getting a value between 5 and infinity is effectively zero.
This problem can be solved using the Poisson distribution, which is a discrete probability distribution that gives the probability of a given number of events occurring in a fixed interval of time or space, if these events occur independently and at a constant average rate.
Let X be the number of births in a minute, then X follows a Poisson distribution with a mean of λ=3.
(a) The probability of exactly 5 births in a minute is:
P(X=5) = (e^(-λ) * λ^5) / 5! = (e^(-3) * 3^5) / 5! = 0.1008 (rounded to four decimal places)
(b) The probability of at least 5 births in a minute is:
P(X>=5) = 1 - P(X<5) = 1 - [P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)]
To find the probabilities for X=0,1,2,3,4, we can use the Poisson formula:
P(X=k) = (e^(-λ) * λ^k) / k!
P(X=0) = e^(-3) = 0.0498 (rounded to four decimal places)
P(X=1) = e^(-3) * 3 / 1! = 0.1494 (rounded to four decimal places)
P(X=2) = e^(-3) * 3^2 / 2! = 0.2240 (rounded to four decimal places) P(X=3) = e^(-3) * 3^3 / 3! = 0.2240 (rounded to four decimal places) P(X=4) = e^(-3) * 3^4 / 4! = 0.1680 (rounded to four decimal places)
Therefore,
P(X>=5) = 1 - [0.0498 + 0.1494 + 0.2240 + 0.2240 + 0.1680] = 1 - 0.8152 = 0.1848 (rounded to four decimal places)
(c) The probability of more than 5 births in a minute is:
P(X>5) = 1 - P(X<=5) = 1 - [P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5)]
To find the probability for X=5, we can use the Poisson formula as in part (a):
P(X=5) = (e^(-λ) * λ^5) / 5! = (e^(-3) * 3^5) / 5! = 0.1008 (rounded to four decimal places)
Therefore,
P(X>5) = 1 - [0.0498 + 0.1494 + 0.2240 + 0.2240 + 0.1680 + 0.1008] = 1 - 1.0160 = 0 (rounded to four decimal places)
Since the Poisson distribution is a discrete probability distribution, the probability of getting a value between 5 and infinity is effectively zero.
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use the mean and standard deviation from the previous 2 examples to calculate a two standard devation inteval around the mean and use it to answer the following questionr: is the minority caucus justifed in their concern that a student senate with only 2 minoirites is a biased representation of the student body?
The Student Senate at a large university has 60 members, of which 20% are minorities. The probability of selecting only two minority students is 10.74%, making it surprising to have only two on the Senate. The expected number of minority students is 12, and having only two may indicate under-representation. The standard deviation of minority students is 1.94. If there were nine minority students, it would not necessarily indicate bias as it falls within the usual range of values.
Let n, the number of trials, is 60 since there are 60 students chosen to represent the entire student body in the Senate. Let p, the probability of success (i.e., selecting a minority student) on a single trial, is 0.20 since 20% of the student body is minority.
The probability of selecting only two minority students in the Senate can be calculated as P(x≤2) = P(x=0) + P(x=1) + P(x=2) = 0.1074. Therefore, there is a 10.74% chance of having only two minority students on the Senate purely by chance.
Based on the low probability in part c), it would be surprising to have only two minority members on the Senate. However, it's important to note that statistical probability alone cannot determine whether bias exists, as there may be other factors at play.
The Minority Caucus may have a reasonable complaint of under-representation, as the expected number of minority students on the Senate can be calculated as u = np = 60 * 0.20 = 12. Therefore, having only two minority students on the Senate is significantly lower than the expected number.
The standard deviation of the number of minority students selected can be calculated as σ = sqrt(np(1-p)) = 1.94.
If the Senate had nine minority students, the Minority Caucus would not have a strong case for racial bias. Assuming that the distribution of minority students selected follows a binomial distribution, the expected number of minority students would be u = np = 60 * 0.20 = 12, with a standard deviation of σ = 1.94.
Using the Empirical Rule, the usual range of values would be from u - 2σ to u + 2σ, which in this case would be from 8.12 to 15.88. Therefore, having nine minority students falls within the usual range of values, indicating that it is not necessarily a result of bias.
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--The given question is incomplete, the complete question is given
" Using the binomial distribution. A Student Senate at a large university is made up of 60 students chosen to represent the entire student body. 20% of the students at this university are minorities. When the members of the senate are selected, only two members are minority students. The Minority Caucus claims that this Senate is racially biased. They decide to use statistics to support their claim. Let x be the number of minority students who are selected for the senate. Assume the binomial probability can be used as the university has a large student population (large enough to assume independence when sampling students) What is n? What is p? Calculate the probability of the Senate being made up of so few minority students purely by chance. That is, find P(xS2) for this binomial problem: Based on your answer to part b, would you be surprised if the Senate had only 2 minority members? Why or why not? Do you think the Minority Caucus is reasonable in their complaint of under- representation? Explain. Assuming that the senate represents the entire student body, how many minority students would you expect to be selected for the senate? You will need to find u for this. There is a shortcut to find the mean for a binomial distribution shown in the text. ur Find the standard deviation of the number of minority students selected. Again, use the shortcut for finding the standard deviation for a binomial distribution, 0= Now suppose that the Student Senate has 9 minority students. Could the Minority Caucus make a case for racial bias in this case? Explain your answer using the Empirical Rul bunu need to find +20 the usual range of values and include the range of values Find the standard deviation of the number of minority students selected. Again, use the shortcut for finding the standard deviation for a binomial distribution, 0 = Now suppose that the Student Senate has 9 minority students. Could the Minority Caucus make a case for racial bias in this case? Explain your answer using the Empirical Rule (you need to find to, the "usual range" of values and include the range of values in your answer:"--
Jeanette purchase a concert ticket on a web site.the original price of the ticket was $75.she used a coupon code to receive a 20% discount.the web site applied a 10% service fee to the discounted price.
Answer:
The discounted price of the ticket was $75 x .80 = $60
The service fee was $60 x .10 = $6
The total price Jeanette paid for the ticket was $60 + $6 = $66
So the answer is 66
Step-by-step explanation:
Find the value of x to the nearest tenth.
Step-by-step explanation:
Remember for RIGHT triangle Tan Φ = Opposite Leg/Adjacent Leg
tan (40) = 900 / x
x = 900 / tan(40) = 1072.6 ft
The shoe sizes of a group of middle school girls are shown.
If a shoe size of 9 is added to the data, how does the median change?
The median stays 6.75.
The median increases to 6.75.
The median stays 7.
The median increases to 7.
Answer:
The median increases to 7
Step-by-step explanation:
List the numbers in chronological order,
5, 5.5, 6, 6.5, 6.5, 7, 7.5, 8, 8, 8.5
As of now, the median is 6.75.
When a shoe size of 9 is added, we have
5, 5.5, 6, 6.5, 6.5, 7, 7.5, 8, 8, 8.5, 9
Now, the median is 7.
Solve the following equations.
|7m+4| =1
Pls show both answers.
The elevation in Feet of these 3 cities are marked on the number line shown below.
The point 0 on the number line represents the sea level. Which statement must be true?
The correct statement is that "P is above sea level and City R is below sea level True"
Determining the position of the points:To determine the position of the points observe the given picture. Here we can observe that the point which has positive values are placed above the sea level and the points which have negative values are placed below the sea level.
The elevation in Feet of these 3 cities is marked on the number line
The point 0 on the number line represents the sea level.
From the figure,
The point P is above sea level and Q is at sea level
The point R is below the sea level
Hence,
The correct statement is that "P is above sea level and City R is below sea level True"
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Complete Question:
The elevation in Feet of these 3 cities is marked on the number line shown below. The point 0 on the number line represents the sea level. Which statement must be true?
Which statement must be true?
A City P and City Q are above sea level.
B City Q and City R are below sea level.
C City P is above sea level and City Q is below sea level.
D City P is above sea level and City R is below sea level.
whats the radius of the circle (x+8)^2 + (y-2)^2=81
r=___
On solving the provided question we can say that Therefore, the radius of the circle is 9.
What is radius of circle?The radius of a circle is a line segment that joins the center of the circle to any point on its circumference. It is essentially half of the diameter of the circle. The radius is denoted by the letter 'r' and is used to calculate many other properties of the circle, such as its area, circumference, and diameter.
The length of the radius is an important characteristic of a circle, as it determines the size and shape of the circle. If two circles have the same radius, they will have the same size and shape. The radius is also used in many real-world applications, such as calculating the distance between two points on a map or the distance between the center of the earth and a satellite in orbit.
The equation [tex](x+8)^{2} +(y-2)^{2} =81[/tex] represents a circle with center (-8, 2) and radius 9.
[tex](x-h)^{2} +(y-k)^{2} =r^{2}[/tex]
Comparing this to the given equation, we see that [tex]h = -8, k = 2[/tex], and [tex]r^2 = 81, so r = 9.[/tex]
Therefore, the radius of the circle is 9.
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Select the correct scatter plot
Which scatter plot shows correlation with causation
The scatter plot that shows correlation with causation is (c)
identifying the scatter plot that shows correlation with causationA scatter plot is a type of graph that displays the relationship between two variables. The position of each point on the plot represents the value of both variables.
If there is a strong positive correlation between two variables, it means that as one variable increases, the other variable also tends to increase.
If there is a strong negative correlation, it means that as one variable increases, the other variable tends to decrease.
The scatter plot with the mst positive or negative correlation is c
This is the scatter plot that shows correlation with causation
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The graph of a linear function is shown on the grid.
Which equation is best represented by this graph?
answer is b) y = - 3/4 * x - 8,The reason for this is that the slope of the line is -3/4 (since it goes down by 3 units for every 4 units to the right), and it passes through the point (-6,-8).
what is slope?
In mathematics, the slope of a line is a measure of how steep it is. It is defined as the ratio of the change in the y-coordinates to the change in the x-coordinates between two points on the line.
In the given question,
Based on the information provided, the equation that represents the straight line passing through -8 on the y-axis and -6 on the x-axis would be:
b) y = - 3/4 * x - 8
The reason for this is that the slope of the line is -3/4 (since it goes down by 3 units for every 4 units to the right), and it passes through the point (-6,-8) which can be verified by substituting these values into the equation.
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For each of the figures write an absolute value equation that has the following solution set.
Tt represents the points 3 and 7 on the x-axis, since they are both 2 units away from 5 and equation form is | x - 5 | = 2.
What is equation?
An equation is a mathematical statement that shows that two expressions are equal. It typically contains variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.
An absolute value equation that has a solution set containing the numbers 3 and 7 on the x-axis.
One possible equation is:
| x - 5 | = 2
This equation represents the set of all numbers on the x-axis that are 2 units away from the number 5.
In other words, it represents the points 3 and 7 on the x-axis, since they are both 2 units away from 5.
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Answer:
|x+5|=2 is the equation
If zeros of the polynomial x²+ 4x + 2a are alpha and 2/alpha then find the value of a
Answer:
Step-by-step explanation:
Answer:
a = 1
Step-by-step explanation:
The zeros of a polynomial P(x) are the x-values that make the polynomial equal to zero, P(x) = 0.
The sum of the zeros of a quadratic equation ax² + bx + c = 0 is -b/a.
The product of the zeros of a quadratic equation ax² + bx + c = 0 is c/a.
Given polynomial:
[tex]x^2+ 4x + 2a[/tex]
Therefore, the zeros of the given polynomial are the x-values that make x²+ 4x + 2a = 0.
Comparing the given polynomial with ax² + bx + c = 0:
a = 1b = 4c = 2aIf the zeros of the polynomial x²+ 4x + 2a are α and α/2, then using the zeros product formula:
[tex]\begin{aligned}\alpha \cdot \dfrac{2}{\alpha} &= \dfrac{c}{a} \\\\\implies 2&= \dfrac{2a}{1}\\\\\implies 2&=2a\\\\\implies \dfrac{2}{2}&=a\\\\\implies a&=1\end{aligned}[/tex]
Therefore, the value of a is 1.
How many lines of symmetry does the figure below have?
A) 0
B) 1
C) 2
D) 3
A man deposited GHC 14100.00 at 8% per annum compound interest. If at the end of 4 years he transfers the total amount to another bank offering 12% compound interest per annum. a. How much interest did he get at the end of 7 years b. Calculate the total amount he received at the end of 7 years.
He received Glhf 25,967.23 as interest at the end of 7 years. The total amount he received at the end of 7 years is GhC 39,067.23.
Simple interest is the interest the lender pays the borrower on the loan. It includes principal only and does not include interest. Simple interest rates are not just for certain loans. This is also the type of interest that banks pay on customers' savings.
a. To find the interest he received at the end of 7 years, we first need to calculate the amount he had at the end of 4 years:
A = P(1 + r/n) (nt)
⇒ A = 14100 (1+ 0.08)(1× 4)
⇒ A = 19628.61
So, he had GhC 19,628.61 at the end of 4 years. He then transferred this amount to another bank offering 12% compound interest per annum. We can use the same formula to find the amount he will have after 7 years:
A = P(1 + r/n) (nt)
⇒ A = 19628.61( 1 + 0.12/1) (1×7)
⇒ A = 40067.23
Therefore, the amount he received at the end of 7 years is GhC 40,067.23.
To find the interest he received at the end of 7 years, we subtract the principal amount:
Interest = A - P
Interest = 40067.23 - 14100
Interest = 25,967.23
So, he received GhC 25,967.23 as interest at the end of 7 years.
b. The total amount he received at the end of 7 years is the sum of the principal and the interest:
Total amount = P + Interest
Total amount = 14100 + 25,967.23
Total amount = 39,067.23
Therefore, the total amount he received at the end of 7 years is GhC 39,067.23.
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Select the two values of x that are roots of this equation x^2+2x-6=0
The answer is
x1 = -1 + square root 7
x2 = - 1 - square root 7
A seventh-grade class is selling oranges as a
class fundraiser. Each box has the same number
of oranges. Mr. Franclemont places an order for 4
boxes. As a thank-you gift, the class will give Mr.
Franclemont a free orange. All together, Mr.
Franclemont will receive 73 oranges.
Construct a model or diagram to represent this
relationship.
Write an equation that could be used to find b,
the number of oranges in each box.
Solve this equation to find the number of oranges
in each box.
The equation 4b + 1 = 73 is used to represent the relationship between the number of oranges in each box and the total number of oranges received. Solving for "b" gives the answer that each box contains 18 oranges.
Model/diagram:
Let "b" be the number of oranges in each box. Since Mr. Franclemont ordered 4 boxes and received an additional orange, he received a total of 4b + 1 oranges. We are given that this is equal to 73:
4b + 1 = 73
To solve for "b", we can subtract 1 from both sides and then divide by 4:
4b = 72
b = 18
Therefore, there are 18 oranges in each box.
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HELP PLEASE!!!
A bakery makes cylindrical mini muffins that measure 2 inches in diameter and one and one fourth inches in height. If each mini muffin is completely wrapped in paper, then at least how much paper is needed to wrap 6 mini muffins? Approximate using pi equals 22 over 7.
14 and 1 over 7 in2
23 and 4 over 7 in2
47 and 1 over 7 in2
84 and 6 over 7 in2
Option C : The minimum amount of paper needed to wrap 6 cylindrical mini muffins is approximately 214.6 square inches, which is closest to option C, 47 and 1 over 7 in2, when pi is approximated as 22/7.
The surface area of a cylindrical mini muffin can be calculated as follows:
The side of the mini muffin is a rectangle that can be wrapped around the circumference of the circular base. The circumference of a circle with a diameter of 2 inches is pid = pi2 inches. Therefore, the length of the side of the mini muffin is pi2 inches, and the height is 1 and 1/4 inches. The area of the side is the product of the height and the length, which is (5/4) * (2pi) square inches.
Therefore, the total surface area of one mini muffin is:
2pi + (5/4) * (2pi) = (13/4) * pi square inches.
To wrap 6 mini muffins, we need to multiply this area by 6:
6 * (13/4) * pi = 39/2 * pi.
Approximating pi as 22/7, we get:
(39/2) * (22/7) = 429/2 square inches.
This is equivalent to 214.5 square inches, which rounds up to approximately 214.6 square inches.
Therefore, the answer is closest to option C, 47 and 1 over 7 in2.
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A bakery makes cylindrical mini muffins that measure 2 inches in diameter and one and one fourth inches in height. If each mini muffin is completely wrapped in paper, then at least how much paper is needed to wrap 6 mini muffins? Approximate using pi equals 22 over 7.
A. 14 and 1 over 7 in2
B. 23 and 4 over 7 in2
C. 47 and 1 over 7 in2
D. 84 and 6 over 7 in2
Calculate the mean and mean absolute deviation of these two data sets and use that to compare the two sets of data.
Set A: 4,6,7,8,5,6
Set B: 5,7,4,8,9,9
For set A: Mean = 6, Mean absolute deviation = 1 and For set B: Mean = 7, MAD = 1.33 based on deviations.
We must first determine the average or mean of the data points in order to calculate the mean and mean absolute deviation (MAD) of two data sets. The mean is calculated by dividing the total number of data points by their sum. The absolute deviation of each data point from the mean must then be determined, and the average of these deviations is used to get the MAD.
Using set A:
Mean = (4 + 6 + 7 + 8 + 5 + 6) / 6 = 6
MAD = (|4-6| + |6-6| + |7-6| + |8-6| + |5-6| + |6-6|) / 6 = 1
Using set B:
Mean = (5 + 7 + 4 + 8 + 9 + 9) / 6 = 7
MAD = (|5-7| + |7-7| + |4-7| + |8-7| + |9-7| + |9-7|) / 6 = 1.33
It is clear from a comparison of the two sets that set B has a higher mean than set A, indicating that set B's values are generally higher. Moreover, set B's MAD is larger than set A's, suggesting that set B's values are more dispersed or volatile. We can therefore say that set B has larger and more varied values than set A.
It is crucial to keep in mind that the mean and MAD are merely two indicators of central tendency and dispersion, respectively, and that to thoroughly examine and contrast data sets, they should be combined with additional statistical indicators and visualizations.
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Find the missing dimension of the prism.
Volume = 560 ft³
7 ft
U=
u
8 ft
ft
Answer:
u = 10ft
Step-by-step explanation:
7ft x 8ft = 56. 560 ÷ 56 = 10.
Therefore, u = 10ft
can someone please help me with this? It’s due today I need help asap.
The experimental probability for the numbers are:
P(3) = 1/12 (in fraction)
P(3) = 0.08 (in decimal)
P(3) ≈ 8% (in percentage)
P(6) = 3/12
P(6) = 0.25
P(6) = 25%
P(less than 4) = 6/12
P(less than 4) = 0.5
P(less than 4) = 50%
How to find the experimental probability?Probability is the likelihood of a desired event happening.
Experimental probability is a probability that relies mainly on a series of experiments.
Blake rolled a die 12 times and 3 appears once (Given in the table). The experimental probability for 3 will be:
P(3) = 1/12 (in fraction)
P(3) = 0.08 (in decimal)
P(3) ≈ 8% (in percentage)
Blake rolled a die 12 times and 6 appears 3 times (Given in the table). The experimental probability for 6 will be:
P(6) = 3/12
P(6) = 0.25
P(6) = 25%
The numbers less than are 1, 2 and 3. Six of the twelve outcome in the table are less than 4.
P(less than 4) = 6/12
P(less than 4) = 0.5
P(less than 4) = 50%
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side b = 20, side c = 12, angle A = 42 degrees. Find the length of side a.
The length of side a is approximately 24.1 units.
As per the given information in question:
side b = 20side c = 12angle A = [tex]42^{0}[/tex]
Here we have to find the Length of side a.
The Law of Cosines is given by the equation: c² = a² + b² - 2ab cos(C) where a, b, and c are the sides of a triangle, and C is the angle opposite side c.
To find the length of side a, given side b = 20, side c = 12, and angle A = 42 degrees, we can use the Law of Sines.
The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is the same for all three sides. In this case, we have:
a / sin(A) = b / sin(B) = c / sin(C)
We are given side b, side c, and angle A, so we can use the formula to find the length of side a.
First, let's find angle B using the given information:
a / sin(42) = 20 / sin(B)
Since we have one angle (A) and the sides opposite to it (a) and another side (b), we can find angle B:
sin(B) = 20 x sin(42) / a
Now we can find angle C using the fact that the sum of angles in a triangle is always 180 degrees:
angle C = 180 - angle A - angle B
angle C = 180 - 42 - angle B
Now we have angle C, and we can use the Law of Sines again to find the length of side a:
a / sin(42) = 12 / sin(angle C)
Solve for a:
a = 12 x sin(42) / sin(angle C)
Now, plug in the values of angle B and angle C that we found earlier and solve for a:
a = 12 x sin(42) / sin(180 - 42 - angle B)
a ≈ 24.1 units.
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Original price and the sales tax rate.
2)
Original price = $500
Sales tax rate = 2.5%
Answer:
$512.50
Step-by-step explanation:
500 + (.025 x 500) 2.5% move the decimal two places to the left.
500 + 12.50 = 512.50
Helping in the name of Jesus.
which fraction are equivalent to 4/10? choose all the correct answers.
Answer: well it could be
2/5 or 8/20 or 1/5 or 16/40
Step-by-step explanation:
Find length c of a triangle given length of side b=24cm, height by side a=12 root 3, and the radius of its circumscribed circle=7root3
Answer:
21 cm
Step-by-step explanation:
We can start by using the formula for the area of a triangle:
Area = (1/2) * base * height
Where the base is b = 24 cm and the height by side a is h = 12 root 3 cm.
Area = (1/2) * 24 cm * 12 root 3 cm = 144 cm^2
We can also use the formula for the area of a triangle in terms of its sides and circumradius:
Area = (abc) / (4R)
Where a, b, and c are the sides of the triangle and R is the radius of its circumcircle.
Plugging in the values we know, we get:
144 cm^2 = (24 cm * a * c) / (4 * 7 root 3 cm)
Simplifying, we get:
9 root 3 cm = a * c / 7
Multiplying both sides by 7 and dividing by a, we get:
c = (63 root 3 cm^2) / a
Substituting the value we know for a, we get:
c = (63 root 3 cm^2) / (12 root 3 cm) = 21 cm
Therefore, the length of side c is 21 cm.
Hopes this helps
a 5-card poker hand is dealt from a well shuffled regular 52-card playing card deck. find the probability that the hand is a full house (one pair of the same face value and one triple of the same face value).
The probability of getting a full house is approximately 0.0014
To calculate the probability of getting a full house, we need to first count the total number of possible 5-card hands that can be dealt from a deck of 52 cards.
There are 52 choices for the first card, 51 for the second, 50 for the third, 49 for the fourth, and 48 for the fifth (since we are drawing without replacement). However, the order in which the cards are dealt does not matter, so we must divide by the number of ways in which the same 5 cards can be ordered. Therefore, the total number of possible 5-card hands is
C(52,5) = 52! / (5! × 47!) = 2,598,960
Now we need to count the number of full house hands we can get. We can choose the value of the triple in C(13,1) ways (since there are 13 possible face values), and we can choose the three cards of that value in C(4,3) ways. We can then choose the value of the pair in C(12,1) ways (since there are now only 12 possible face values left), and we can choose the two cards of that value in C(4,2) ways. Therefore, the number of full house hands is
C(13,1) × C(4,3) × C(12,1) × C(4,2) = 13 × 4 × 12 × 6 = 3,744
Finally, the probability of getting a full house is
P(full house) = number of full house hands / total number of hands
= 3,744 / 2,598,960
= 0.0014
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The drama department of a school sold 679 tickets to the school play, for a total of $3370. Students paid $4 for a
ticket, and non-students paid $7.
a) Write a linear system for this equation.
b) How many non-students attended the play?
The linear equations are x + y = 679, 4x + 7y = 3370 and 586 non-students attended the play.
What is a linear equation, exactly?
A linear equation is a mathematical equation that describes a straight line when plotted on a graph. It is an equation in which the highest power of the variable is one. Linear equations are commonly written in the form of y = mx + b, where "x" and "y" are variables, "m" is the slope of the line, and "b" is the y-intercept. Linear equations can be used to model many real-world situations, such as distance and speed, cost and revenue, and temperature and time.
Now,
a) Let x = student tickets sold and y = number of non-student tickets sold. Then we have the following system of linear equations:
x + y = 679 (the total number of tickets sold is 679)
4x + 7y = 3370 (the total revenue from ticket sales is $3370)
b) To solve for y, we can use the first equation to get:
y = 679 - x
then,
4x + 7(679 - x) = 3370
Simplifying and solving for x, we get:
3x = 2363
x = 787
So 787 student tickets were sold. To find the number of non-student tickets sold, we can use the equation y = 679 - x:
y = 679 - 787 = -108
Since it doesn't make sense to have a negative number of non-student tickets sold, we made an error somewhere. Looking at the problem again, we realize that the equation should be:
x + y = 679 (the total number of tickets sold is 679)
4x + 7y = 3370 (the total revenue from ticket sales is $3370)
where y represents the number of student tickets sold and x represents the number of non-student tickets sold. With this correction, we get:
y = 679 - x
4(679 - x) + 7x = 3370
Simplifying and solving for x, we get:
3x = 1759
x = 586.33
x = 586
So 586 non-student tickets were sold.
Which means 586 non-students attended the play.
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Peter's cat, Piper, is a finicky eater. Peter is trying to determine which of two brands of canned cat food Piper prefers, Pick-a-Pair or Pickled Peppers. For two months, he flips a coin each day to decide which of the two foods to feed Piper, and weighs how much Piper eats (in grams). Here are the data: n X Pick-a-Pair 31 85.2 3.45 Pickled Peppers 30 82.1 4.62 (a) Construct and interpret a 99% confidence interval for the difference in mean amount of food Piper eats when she is offered Pick-a-Pair and when she is offered Pickled Peppers.
Therefore, we can be 99% confident that the true difference in mean amount of food Piper eats when she is offered Pick-a-Pair and when she is offered Pickled Peppers is between 0.7 and 5.1 grams.
What is confidence interval?A confidence interval is a statistical measure used to estimate the range of values within which a true population parameter is likely to fall with a certain level of confidence. It is often used in inferential statistics to assess the precision and reliability of a sample estimate of a population parameter, such as the mean, proportion, or standard deviation.
Here,
To construct a confidence interval for the difference in mean amount of food Piper eats when she is offered Pick-a-Pair and when she is offered Pickled Peppers, we can use a two-sample t-test. The null hypothesis is that there is no difference in mean amount of food Piper eats between the two brands, and the alternative hypothesis is that there is a difference. We can use a 99% confidence level, so our significance level is α = 0.01. First, we can calculate the sample means, sample standard deviations, and sample sizes for each brand:
For Pick-a-Pair:
Sample mean: x1 = 85.2
Sample standard deviation: s1 = 3.45
Sample size: n1 = 31
For Pickled Peppers:
Sample mean: x2 = 82.1
Sample standard deviation: s2 = 4.62
Sample size: n2 = 30
Next, we can calculate the pooled standard deviation, sp, using the formula:
sp = √(((n1 - 1)s1² + (n2 - 1)s2²) / (n1 + n2 - 2))
sp = sqrt(((31 - 1)(3.45)² + (30 - 1)(4.62)²) / (31 + 30 - 2))
sp = 4.019
Then, we can calculate the t-statistic using the formula:
t = (x1 - x2) / (sp * √(1/n1 + 1/n2))
t = (85.2 - 82.1) / (4.019 * √(1/31 + 1/30))
t = 1.864
Using a t-table with degrees of freedom (df) = n1 + n2 - 2 = 59 and a significance level of 0.01, we find the critical values to be ±2.660. Since our t-statistic of 1.864 is within this range, we fail to reject the null hypothesis.
Finally, we can construct the confidence interval using the formula:
CI = (x1 - x2) ± t*(sp * √(1/n1 + 1/n2))
CI = (85.2 - 82.1) ± 2.660*(4.019 * √(1/31 + 1/30))
CI = (0.7, 5.1)
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What percent of gases in Earth's atmosphere does the whole circle graph represent?
Pie chart is the best to show the percentage of gases on earth's atmosphere.
What is Pie chart?A pie chart, sometimes known as a circle chart, is a circular statistical visual that shows numerical proportion through slices of data. Each slice's arc length in a pie chart is proportional to the quantity it shows, as are its center angle and area. Although it was given that name because it resembles a slice of pie, there are other ways to serve it.
Nitrogen accounts for 78% of the atmosphere, oxygen 21% and argon 0.9%. Gases like carbon dioxide, nitrous oxides, methane, and ozone are trace gases that account for about a tenth of one percent of the atmosphere.
The Pie chart is the best to show the percentage of gases on earth's atmosphere.
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