a) The probability that neither fire engine is available when needed is 0.0016. b) The probability that a fire engine is available when needed is 0.9984.
The probability of an event occurring is the likelihood that it will happen. In this case, we are looking at the probability of a specific fire engine being available when needed.
The probability that neither fire engine is available when needed can be found by multiplying the probabilities of each fire engine not being available. The probability of a specific fire engine not being available is [tex]1 - 0.96 = 0.04[/tex]. So the probability that neither is available is [tex]0.04 * 0.04 = 0.0016.[/tex]
The probability that a fire engine is available when needed can be found by subtracting the probability that neither is available from 1. So the probability that a fire engine is available when needed is[tex]1 - 0.0016 = 0.9984.[/tex]
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find the perimeter of 18.6 18.9 6.9
The perimeter of the polygon with sides of lengths 18.6, 18.9, and 6.9 is 44.4 units.
What is the perimeter?
The perimeter of a polygon is the total distance around its boundary, which is equal to the sum of the lengths of all its sides. To find the perimeter of a polygon, you simply add up the lengths of all its sides.
The perimeter of any polygon is the sum of the lengths of its sides. In this case, we have three sides of lengths 18.6, 18.9, and 6.9.
To find the perimeter, we simply add these lengths together:
Perimeter = 18.6 + 18.9 + 6.9
Perimeter = 44.4
Therefore, the perimeter of the polygon with sides of lengths 18.6, 18.9, and 6.9 is 44.4 units.
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7. Solve the system: ( 4 points ) 3a−2b=1
7a−b=17
and show all step
Answer:
a = 3, b = 4
Step-by-step explanation:
[tex]3a-2b=1\\7a-b=17\\\\[/tex]
Multiply second equation by 2
[tex]3a-2b=1\\14a-2b=34\\[/tex]
Subtract the equations
[tex]3a-2b=1\\-(14a-2b=34)\\\\-11a=-33\\a=3[/tex]
Plug back in to either original equation to solve for b
[tex]3(3)-2b=1\\9-2b=1\\-2b=-8\\b=4[/tex]
Solution:
(3,4)
coloca los números del 1 al 5 y del 7 al 11 de tal manera de que la suma de cada línea sea 18.
in pawnee, indiana, the price of a pound of bacon (x) varies from day to day according to a normal distribution with mean of $4.10 and a standard deviation of $0.16. the price of a dozen eggs (y) also varies from day to day according to a normal distribution with a mean of $1.90 and a standard deviation $0.06. assume the prices for a pound of bacon and a dozen eggs are independent. (a) find the probability that on a given day, the price of a pound of bacon is more than twice as expensive as a dozen eggs. that is, find p(x > 2y). (b) ron wants to cook breakfast for himself, so he buys 10 pounds of bacon and 6 dozen eggs. find the probability that he paid more than $50 for the bacon and eggs.
The probability that on a given day, the price of a pound of bacon is more than twice as expensive as a dozen eggs is 0.0001. The probability that Ron paid more than $50 for the bacon and eggs is 0.0028.
(a) To find the probability that on a given day, the price of a pound of bacon is more than twice as expensive as a dozen eggs, we need to use the normal distribution formula. This is given by P(x > 2y) = P(X > 2µy + 2σy) = P(Z > (2µy + 2σy - µx) / σx) = P(Z > (2(1.90) + 2(0.06) - 4.10) / 0.16) = P(Z > -4.24) = 1 - P(Z < -4.24) = 1 - 0.9999 = 0.0001
(b) To find the probability that Ron paid more than $50 for the bacon and eggs, we need to calculate the probability that the 10 pounds of bacon and 6 dozen eggs cost more than $50. This is given by P(10x + 6y > 50) = P(10X + 6µy + 6σy > 50) = P(Z > (50 - 10µx - 6µy - 6σy) / (√10σx + √6σy)) = P(Z > (50 - 10(4.10) - 6(1.90) - 6(0.06)) / (√10(0.16) + √6(0.06)) = P(Z > -2.87) = 1 - P(Z < -2.87) = 1 - 0.9972 = 0.0028.
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Help with Graded Assignment
Unit Test, Part 2: Measurement of Two-Dimensional Figures
The capacity of Leo's container is (WH x L) / 2, where W, H, and L represent the width, height, and length of the container, respectively.
To find the capacity of Leo's container, we need to calculate its volume. The volume of a rectangular prism is calculated by multiplying its length, width, and height. We can express this formula mathematically as:
Volume = Length x Width x Height
Since we do not have the exact measurements of the container, we cannot determine the exact volume.
The surface area of a rectangular prism is given by the formula:
Surface Area = 2 x (Length x Width + Width x Height + Length x Height)
Therefore, we can set the equation for surface area equal to the equation for the total surface area of the container:
Surface Area = 2 x (Length x Width + Width x Height + Length x Height) = Total Surface Area of Container
Then we have:
Total Surface Area of Container = 2LW + 2WH + 2HL Surface Area of Material = 6LW
Since the material covers all six sides, we can set the surface area of the material equal to the total surface area of the container:
6LW = 2LW + 2WH + 2HL
Simplifying this equation, we get:
4LW = 2WH + 2HL
Dividing by 2, we get:
2LW = WH + HL
Multiplying by the height, we get:
2LWH = WH² + HL²
Now, we can rearrange this equation to get the volume of the container:
Volume = Length x Width x Height = (WH x L) / 2
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Complete Question:
Leo made a container to store his camping gear. The container is in the shape of a rectangular prism. The container has material covering all six sides.
What is the capacity of the container?
what is the value of 12.5-31/2 +1 1/4?
The value for the expression is obtained as -7/4.
What is an expression?
Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation.
To evaluate the expression 12.5 - 31/2 + 1 1/4, we need to convert the mixed number 1 1/4 to an improper fraction -
1 1/4 = 5/4
Now we can substitute the values into the expression and simplify -
Substitute 1 1/4 with 5/4 -
= 12.5 - 31/2 + 1 1/4
= 12.5 - 31/2 + 5/4
Convert 12.5 to an improper fraction with a denominator of 2 -
= 25/2 - 31/2 + 5/4
Get a common denominator of 4 -
= (50)/(4) - (62)/(4) + 5/4
= 55/4 - 62/4
= -7/4
Therefore, the value is -7/4.
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The average cost of a large pizza in the United States is roughly $ 12.50, with a standard deviation of 51.50. Use that information to answer the following questions.
a) Use the Empirical Rule to make an interval that should include 99.7% of the data
b) Use the Empirical Rule to make an interval that would be considered usual
c) Find the z-score of a pizza that cost $10
a) The interval would be:Interval = $12.50 ± ($51.50 x 3)Interval = $12.50 ± $154.50 Interval = ($12.50 - $154.50, $12.50 + $154.50)Interval = (-$142.00, $179.00)
b) The interval would be:Interval = $12.50 ± ($51.50 x 1)Interval = $12.50 ± $51.50 Interval = ($-39.00, $64.00)This interval can be considered usual since it includes about 68% of the data.
c) A pizza that costs $10 is about 0.049 standard deviations below the mean.
Use the Empirical Rule to make an interval that should include 99.7% of the data.The Empirical Rule, also known as the 68-95-99.7 rule, is a statistical rule that states that for a normal distribution, about 68% of observations will fall within one standard deviation of the mean, about 95% will fall within two standard deviations, and about 99.7% will fall within three standard deviations.To make an interval that should include 99.7% of the data, we need to find the mean plus/minus three standard deviations. The mean is given as $12.50, and the standard deviation is given as $51.50. Therefore, the interval would be:Interval = $12.50 ± ($51.50 x 3)Interval = $12.50 ± $154.50Interval = ($12.50 - $154.50, $12.50 + $154.50)Interval = (-$142.00, $179.00)
Use the Empirical Rule to make an interval that would be considered usual.Again, we can use the Empirical Rule to find an interval that would be considered usual. This would be the interval that falls within one standard deviation of the mean. The mean is given as $12.50, and the standard deviation is given as $51.50. Therefore, the interval would be:Interval = $12.50 ± ($51.50 x 1)Interval = $12.50 ± $51.50 Interval = ($-39.00, $64.00)This interval can be considered usual since it includes about 68% of the data.
Find the z-score of a pizza that costs $10.The z-score is a measure of how many standard deviations a data point is from the mean. To find the z-score of a pizza that costs $10, we first need to find the difference between $10 and the mean of $12.50, and then divide that difference by the standard deviation of $51.50.z-score = (10 - 12.50) / 51.50z-score = -0.049 This means that a pizza that costs $10 is about 0.049 standard deviations below the mean.
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For each statement below, state whether the lengths, angles or both are invariant.
Translation
Reflection
Rotation
Enlargement
The cone-shaped paper cup is 2/3 full of sand. What is the volume of the part of the cone that is filled with sand?
The volume of the part of the cone that is filled with sand is 2/9 times the total volume of the cone.
To do this, we need to first determine the total volume of the cone. The formula for the volume of a cone is:
V = 1/3 x π x r² x h
Where V is the volume, r is the radius of the base of the cone, h is the height of the cone, and π is a mathematical constant (approximately equal to 3.14).
Since we know that the paper cup is cone-shaped, we can assume that it has a circular base. Let's say that the radius of the base of the cone is r and the height of the cone is h.
The total volume of the cone can be calculated using the formula above:
V = 1/3 x π x r² x h
Now we need to determine the volume of the part of the cone that is filled with sand. We know that the cup is 2/3 full of sand, so the volume of sand in the cup is 2/3 of the total volume of the cup. We can express this as:
V = 2/3 x V
We can substitute the formula for V into this equation to get:
V = 2/3 x (1/3 x π x r² x h)
Simplifying this equation, we get:
V = 2/9 x π x r² x h
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A diameter of a circle is 78 cm
WHat is the area
since it has a diameter of 78, then its radius is half that, or 39.
[tex]\textit{area of a circle}\\\\ A=\pi r^2 ~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ r=39 \end{cases}\implies A=\pi (39)^2\implies A\approx 4778.36~cm^2[/tex]
X=-3, 0, 5
y=5, 2,-3
Draw a straight line y=2 - x
The line on the graph will be a straight line passing through the points (3,5) and (-3,2).
To draw a straight line y=2 - x, first plot the given points on the graph, (3,5) and (-3,2). Draw a straight line between these two points. This line will be the graph of the equation y=2 - x. To find the equation, take the x-value of the first point, 3, and subtract it from the y-value of the same point, 5. This is the equation of the line, y=2 - x. With the equation of the line, we can now draw a line on the graph that passes through the two given points. The line will be a straight line, with a slope of -1 and a y-intercept of 2.
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Polygon ABCD with vertices at A(−4, 6), B(−2, 2), C(4, −2), D(4, 4) is dilated using a scale factor of 3/4 to create polygon A′B′C′D′. Determine the vertices of polygon A′B′C′D′.
A′(−3, 4.5), B′(−1.5, 1.5), C′(3, −1.5), D′(3, 3)
A′(−12, 18), B′(−6, 6), C′(12, −6), D′(12, 12)
A′(3, −4.5), B′(1.5, −1.5), C′(−3, 1.5), D′(−3, −3)
A′(4.5, −3), B′(1.5, −1.5), C′(−1.5, 3), D′(3, 3)
The vertices of the dilated polygon A'B'C'D' are A'(-3, 4.5), B'(-1.5, 1.5), C'(3, -1.5), and D'(3, 3). the correct option is (A) A′(−3, 4.5), B′(−1.5, 1.5), C′(3, −1.5), D′(3, 3).
What is polygon?
A polygon is a closed plane figure that is formed by three or more straight sides, which are connected end-to-end to create a closed shape.
To find the vertices of the dilated polygon, we need to apply the scale factor of 3/4 to each of the original vertices. This will result in a new set of coordinates for each vertex.
The coordinates of point A' can be found by multiplying the x-coordinate and y-coordinate of A by 3/4:
x-coordinate of A' = -4 × 3/4 = -3
y-coordinate of A' = 6 × 3/4 = 4.5
Therefore, A' has coordinates (-3, 4.5).
Using the same method, we can find the coordinates of the other vertices:
B':
x-coordinate of B' = -2 × 3/4 = -1.5
y-coordinate of B' = 2 × 3/4 = 1.5
Therefore, B' has coordinates (-1.5, 1.5).
C':
x-coordinate of C' = 4 × 3/4 = 3
y-coordinate of C' = -2 × 3/4 = -1.5
Therefore, C' has coordinates (3, -1.5).
D':
x-coordinate of D' = 4 × 3/4 = 3
y-coordinate of D' = 4 × 3/4 = 3
Therefore, D' has coordinates (3, 3).
Thus, the vertices of the dilated polygon A'B'C'D' are A'(-3, 4.5), B'(-1.5, 1.5), C'(3, -1.5), and D'(3, 3). Therefore, the correct option is (A) A′(−3, 4.5), B′(−1.5, 1.5), C′(3, −1.5), D′(3, 3).
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Find the value of the unknown in the figure below
b =(17,6x 8,9):19,7=7,95
What is the best comparison between the theoretical and experimental probability of tossing heads?
Answer: Theoretical probability is based on mathematical analysis and predictions, while experimental probability is based on actual data collected through repeated trials or experiments.
For example, the theoretical probability of tossing a fair coin and getting heads is 1/2, because there are two equally likely outcomes (heads and tails) and only one of them is heads.
On the other hand, the experimental probability of tossing heads can be found by conducting multiple trials of coin flips and recording the frequency of getting heads. As the number of trials increases, the experimental probability should approach the theoretical probability.
Therefore, the best comparison between theoretical and experimental probability of tossing heads is that the theoretical probability is based on mathematical analysis and predictions, while the experimental probability is based on actual data collected through repeated trials or experiments. The two probabilities may be similar or different, but as the number of trials increases, the experimental probability should converge to the theoretical probability.
Step-by-step explanation:
Factor the expression completely. 25 a 2 − 30 a + 9 ( 5 a − 3 ) 2 ( 5 a + 3 ) 2 ( 5 a + 3 ) ( 5 a − 3 )
The factored form of the expression 25a² - 30a + 9 is (5a - 3)².
What is the factored form of the given expression?Given the expression in the question;
25a² - 30a + 9
To factor completely, first rewrite the expression as as follows;
25a² - 30a + 9
rewrite 25a² as ( 5a )²
( 5a )² - 30a + 9
rewrite 9 as 3²
( 5a )² - 30a + 3²
Now, check that the middle term is two times the product of the numbers being squared in the first and third term.
30a = 2 × ( 5a ) × 3
Next, rewrite the polynomial;
( 5a )² - 2 × ( 5a ) × 3 + 3²
( 5a )² - 2( 5a )3 + 3²
Factor using the perfect square trinomial rule.
a² - 2ab + b² = ( a - b )²
Here; a = 5a and b = 3
Hence, we have;
(5a - 3)²
Therefore, the factored form is (5a - 3)².
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Which is an example of a suggestive survey question?
What do you like and don’t like about the CEO?
Do you own a car or a truck?
What is your favorite brand of soda?
How much credit card debt do you owe?
The answer of the given question based of a suggestive survey the answer is "What do you like and don’t like about the CEO?"
Who is CEO?The CEO or Chief Executive Officer is highest-ranking executive in company who is responsible for making major corporate decisions, managing overall operations, and leading organization towards achieving its goals and the objectives. The CEO reports to board of directors and is responsible for success or failure of company.
The suggestive survey question is: "What do you like and don’t like about the CEO?"
This question is suggestive because it implies that the respondent should have opinions about the CEO, and it encourages the respondent to share both positive and negative opinions. This can bias the responses towards certain answers, rather than allowing the respondent to express their own thoughts freely.
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Statistics grades: In a statistics class of 42 students, there were 11 men and 31 women. Six of the men and seven of the women received an A in the course. A student is chosen at random from the class.
Give all answers as decimals rounded to 4 digits after the decimal point
(a) Find the probability that the student is a woman.
(b) Find the probability that the student received an A.
(c) Find the probability that the student is a woman or received an A.
(d) Find the probability that the student did not receive an A.
A is 0.8218
A is 0.6905.
(a) The probability that the student is a woman.The number of women in the class is 31 while the total number of students is 42. The probability of a randomly selected student being a woman is given as:P(woman) = (Number of women in the class) / (Total number of students in the class)P(woman) = 31 / 42P(woman) = 0.7381(b) The probability that the student received an A.The number of men and women that received an A is given as follows:Men that received an A = 6Women that received an A = 7The total number of students that received an A is:Total number of students that received an A = (Number of men that received an A) + (Number of women that received an A)Total number of students that received an A = 6 + 7Total number of students that received an A = 13The probability of a randomly selected student receiving an A is given as:P(A) = (Number of students that received an A) / (Total number of students in the class)P(A) = 13 / 42P(A) = 0.3095(c) The probability that the student is a woman or received an A.The probability of the student being a woman or received an A is given as:P(woman or A) = P(woman) + P(A) - P(woman and A)From parts (a) and (b), we know that:P(woman) = 0.7381P(A) = 0.3095Now, we have to determine the probability that the student is a woman and received an A.The number of women that received an A = 7 while the total number of women in the class is 31.The probability of a woman receiving an A is given as:P(woman and A) = (Number of women that received an A) / (Total number of women in the class)P(woman and A) = 7 / 31P(woman and A) = 0.2258Therefore:P(woman or A) = P(woman) + P(A) - P(woman and A)P(woman or A) = 0.7381 + 0.3095 - 0.2258P(woman or A) = 0.8218(d) The probability that the student did not receive an A.The probability that the student did not receive an A is given as:P(not A) = 1 - P(A)From part (b), we know that:P(A) = 0.3095Therefore:P(not A) = 1 - P(A)P(not A) = 1 - 0.3095P(not A) = 0.6905Therefore, the answers are given as follows:(a) The probability that the student is a woman is 0.7381.(b) The probability that the student received an A is 0.3095.(c) The probability that the student is a woman or received an A is 0.8218.(d) The probability that the student did not receive an A is 0.6905.
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Find the interquartile range (IQR) of the data set.
2. 6, 3, 2004. 9, 5, 5, 6, 6, 7. 9, 8, 2008. 2
The interquartile range (IQR) of the given data set is 4, calculated by subtracting the third quartile (Q3) from the first quartile (Q1).
An indicator of a data set's statistical dispersion is the interquartile range (IQR). The third quartile (Q3) is subtracted from the first quartile to compute it (Q1). The middle 50% of the data's spread is measured by the IQR.In the given data set, the data points are 2, 3, 5, 5, 6, 6, 7, 8, 9, 9, 2004, 2008. We first need to arrange the data points in ascending order, which will be 2, 3, 5, 5, 6, 6, 7, 8, 9, 9, 2004, 2008. To calculate the IQR, we first need to calculate the first quartile and the third quartile. The first quartile is the median of the lower half of the data set, which is 5. The third quartile is the median of the upper half of the data set, which is 9. The IQR is calculated by subtracting the third quartile (Q3) from the first quartile (Q1). Therefore, the IQR of the given data set is 4 (9 - 5 = 4).
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A statistics course has 19 students in which 82.0% passed. At
90% confidence, the lower confidence limit for the proportion
students passing with 3 decimal places is
At 90% confidence, the lower confidence limit for the proportion students passing with 3 decimal places is 0.563.
To find the lower confidence limit for the proportion of students passing, we need to use the formula for confidence interval of a proportion:
Lower limit = sample proportion - z*standard error
where z is the z-score associated with the confidence level, and standard error is the standard deviation of the sampling distribution of the proportion.
First, let's calculate the sample proportion of students passing:
proportion passing = 82.0% = 0.820
Next, we need to calculate the standard error:
standard error = sqrt(p*(1-p)/n)
where p is the sample proportion, and n is the sample size.
standard error = sqrt(0.820*0.180/19) = 0.154
Now we need to find the z-score associated with the 90% confidence level. We can use a standard normal distribution table or a calculator to find that the z-score is 1.645.
Finally, we can plug in the values to find the lower confidence limit:
Lower limit = 0.820 - 1.645*0.154 = 0.563
Rounding to 3 decimal places, the lower confidence limit for the proportion of students passing is 0.563.
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Twice differentiable function
Therefore, there exists a value c in the interval (4, 7) such that f'(c) = -1, as required.
What makes a function unique?If any vertical line drawn may cross the graph at no more than one point, the graph is a function. If there is any spot on the graph where a vertical line can cross it at two or more points, the graph is not a function.
By the mean value theorem, there exists a point c in the interval (4, 7) such that:
f'(c) = (f(7) - f(4))/(7 - 4)
Substituting the given values, we get:
f'(c) = (4 - 7)/(7 - 4) = -1/3
Now, we need to show that there exists a value c in the interval (4, 7) such that f'(c) = -1.
Let g(x) = f(x) + x. Then, g(4) = f(4) + 4 = 11 and g(7) = f(7) + 7 = 11.
By Rolle's theorem, there exists a point c in the interval (4, 7) such that g'(c) = 0.
Now, g'(c) = f'(c) + 1, so we have:
f'(c) + 1 = 0
f'(c) = -1
Therefore, there exists a value c in the interval (4, 7) such that f'(c) = -1, as required.
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Using the information in the table, what is one way you can find the number of cups in 8 gallons?
There are 64 cups in 8 gallons.
To find the number of cups in 8 gallons, you can use the conversion rate of 8 cups per gallon. This means that the formula to find the number of cups in 8 gallons is:
Cups = 8 x Gallons
Thus, the calculation to find the number of cups in 8 gallons is:
Cups = 8 x 8
Cups = 64
Therefore, there are 64 cups in 8 gallons. This calculation shows that for every gallon, there are 8 cups. To find the number of cups in any number of gallons, you can multiply the number of gallons by 8. This calculation can be seen in the formula:
Cups = 8 x Gallons
By using this formula, you can easily find the number of cups in any number of gallons.
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Find the volume of the cone in cubic feet. Use 3.14 for π and round your answer to the nearest tenth.
HELPPPPPPPPP
Answer:
third option
Step-by-step explanation:
Volume of a cone = V = πr²(h/3)
Where,
r = radiush = heightHere we are given the following :
Height = 54m Radius = 38m π = 3.14Note that we are asked to find the volume in square feet so we must first convert the height and radius into feet
Converting given values
Height = 54 * 3.2808 = 177.1632 ftRadius = 38 * 3.2808 = 124.6704 ftπ = 3.14By plugging these values into the formula we acquire
V = (3.14)(124.6704²)(177.1632/3)
==> evaluate exponent
V = (3.14)(15542.7086362)(177.1632/3)
==> divide 177.1632 by 3
V = (3.14)(15542.7086362)(59.0544)
==> multiply 3.14 and 15542.7086362
V = 48804.1051175(59.0544)
==> multiply 48804.1051175 and 59.0544
V = 2882097.1 ft² (rounded to the nearest tenth)
The volume of the cone is 2882097.1 so the correct answer choice is the third option.
Important note : Its important to note that we must convert meters to feet before finding the volume. If we try to convert after finding the volume, our answer will be incorrect.
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PLEASE HELP WITH THE PROBLEM !!
Answer:
the answer is 18
Step-by-step explanation:
volume of triangular prism =l×w×h
9×2×1=18
Please answer the Bonus question.
Answer:
FG = 25.6
Step-by-step explanation:
ABC to DEC = 5:2
DEC to FEG = 1:4
so,
ABC to FEG = 5/2 × 1/4 = 5:8
that means the scaling factor from small to large is 8/5.
therefore,
EG = BC × 8/5 = 10×8/5 = 2×8 = 16
the side length ratio for ABC tells us
AB:BC:AC = 5 : 2 1/2 : 4
BC = 10
so,
AB:BC = 5 : 2 1/2 = 5 : 2.5
AB:10 = 5 : 2.5
2.5AB = 50
AB = 50/2.5 = 20
AC:AB = 4:5
AC:20 = 4:5
AC = 20×4/5 = 80/5 = 16
FG = AC × 8/5 = 16 × 8/5 = 128/5 = 25.6
A golf coach bought g golf balls. The balls came in 7 packages. Write an expression that shows how many golf balls were in each package.
Answer:
g/7
Step-by-step explanation:
Answer:G/7
Step-by-step explanation:G/7
In a standard normal distribution, an area of \( 0.3974 \) on the right corresponds to a z-score of:
A. -0.26 B. -0.92
C. 2.60 D. 0.26
Option A -026 is the correct answer
The standard normal distribution represents a normal distribution with a mean of 0 and a standard deviation of 1. It is also known as the z-distribution.What is the z-score formula?The z-score formula is: z = (x - μ) / σ where z is the z-score, x is the raw score, μ is the mean, and σ is the standard deviation.In a standard normal distribution, an area of 0.3974 on the right corresponds to a z-score of -0.26. This is option A. Therefore, option A is the correct answer.
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Find a polynomial \( f(x) \) of degree 4 with leading coefficient 1 such that both \( -4 \) and 2 are zeros of multiplicity 2 . \[ f(x)=x \]
Polynomial with a degree 4 and leading coefficient 1 is to be found. Also, -4 and 2 are the zeros of multiplicity 2.
Given the roots, the polynomial can be expressed in factored form as:
\[f(x) = (x + 4)^2(x - 2)^2\]
Expanding this:
\begin{align*}
f(x) &= (x + 4)^2(x - 2)^2\\
&= (x^2 + 8x + 16)(x^2 - 4x + 4)\\
&= x^4 + 4x^3 - 4x^3 - 32x^2 + 16x^2 + 128x + 64\\
&= x^4 - 16x^2 + 128x + 64
\end{align*}
Thus, the polynomial is:
\[f(x) = x^4 - 16x^2 + 128x + 64\]Therefore, the answer is \[f(x) = x^4 - 16x^2 + 128x + 64\].
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Let \( A=\left[\begin{array}{rrr}-2 & -5 & -13 \\ 4 & 5 & 11 \\ 1 & 2 & 6\end{array}\right] \) Find the third column of \( A^{-1} \) without computing the other two columns. How can the third column o
The third column of the inverse of the matrix, A = [tex]\begin{bmatrix}-2 & -5 & -13 \\4 &5 &11 \\1 &2 &6 \\\end{bmatrix}[/tex], obtained using the product of A and the third column of A⁻¹ is; [tex]\begin{bmatrix} 1 \\ -3\\ 1 \\\end{bmatrix}[/tex]
What is an inverse of a matrix?The product of a matrix and the inverse of the matrix is the multiplicative identity.
The specified matrix can be presented as follows;
A = [tex]\begin{bmatrix}-2 & -5 & -13 \\4 & 5 & 11 \\1 &2 & 6 \\\end{bmatrix}[/tex]
The third columns of the inverse of the matrix, A, A⁻¹, can be obtained without computing the other two columns, as follows;
The inverse of the matrix A, A⁻¹ is the solution of the following matrix equation;
Ax = [tex]e_i[/tex]
Where;
[tex]e_i[/tex] = The column i of the identity matrix
x = The third column of the inverse matrix
Therefore, for the third column, we get;
[tex]e_i[/tex] = [tex]e_3[/tex] = [tex]\begin{bmatrix}0 \\ 0\\1\\\end{bmatrix}[/tex]
Ax = [tex]e_3[/tex] = [tex]\begin{bmatrix}0 \\ 0\\1\\\end{bmatrix}[/tex]
Therefore;
x = [tex]e_i[/tex]/X, which using a graphing calculator, indicates;
[tex]x = \frac{\begin{bmatrix}0 \\ 0\\1\\\end{bmatrix}}{\begin{bmatrix}-2 & -5 & -13 \\4 & 5 & 11 \\1 &2 & 6 \\\end{bmatrix}} = \begin{bmatrix}1 \\ -3\\1\\\end{bmatrix}[/tex]
The third column of the inverse matrix is; [tex]\begin{bmatrix}1 \\ -3\\1\\\end{bmatrix}[/tex]Learn more on the inverse of a matrix here: https://brainly.com/question/20663530
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Solve the following equations . a² = 64. 2x-2 = 16. 3y² = 27
Answer:
-3
Step-by-step explanation:
a² = 64
Taking the square root of both sides, we get:
a = ±√64
a = ±8
Therefore, the solutions to the equation a² = 64 are a = 8 and a = -8.
2x-2 = 16
Adding 2 to both sides, we get:
2x = 18
Dividing both sides by 2, we get:
x = 9
Therefore, the solution to the equation 2x-2 = 16 is x = 9.
3y² = 27
Dividing both sides by 3, we get:
y² = 9
Taking the square root of both sides, we get:
y = ±√9
y = ±3
Therefore, the solutions to the equation 3y² = 27 are y = 3 and y = -3.
Solve for Vtotal help me out please TT TT
Using Pythagorean theorem to calculate the distance of the charges, the electric potential is -93V
What is the electric potentialThe electric potential at point P can be calculated using the formula:
[tex]V=\frac{1}{4\pi\epsilon_0}\frac{q_1}{r_1}+\frac{1}{4\pi\epsilon_0}\frac{q_2}{r_2}[/tex]
where [tex]\epsilon_0[/tex] is the vacuum permittivity, q1 and q2 are the charges, and r1and r2 are the distances from the charges to point P.
To find r1 and r2, we can use the Pythagorean theorem:
[tex]r_1=\sqrt{(0.1\text{ m})^2+(0.15\text{ m})^2}=0.18\text{ m}[/tex]
[tex]r_2=\sqrt{(0.1\text{ m})^2+(0.25\text{ m})^2}=0.27\text{ m}[/tex]
Substituting the values into the formula, we get:
[tex]V=\frac{1}{4\pi\epsilon_0}\frac{-5.0\times10^{-6}\text{ C}}{0.18\text{ m}}+\frac{1}{4\pi\epsilon_0}\frac{5.0\times10^{-6}\text{ C}}{0.27\text{ m}}[/tex]
Using the value of vacuum permittivity
[tex]\epsilon_0=8.85\times10^{-12}\text{ F/m}[/tex] we get:
[tex]V=-267\text{ V}+174\text{ V}= -93\text{ V}[/tex]
Therefore, the electric potential at point P is -93 V.
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