Answer: 17.2
Step-by-step explanation:
Mean means average, add all of the numbers and divide by how many there are.
mean = (11+17+21+14+23)/5
=17.2
Solve these problems
The two column proof is completed as follows
Statement Reason
line RS ≅ line UT Given
line RT ≅ line UT Given
line ST ≅ line TS Reflexive property
Δ RST ≅ ΔUTS SSS congruence theorem
What is reflexive property
The refleхive propеrty is a mathеmatical propеrty thаt statеs thаt аny element is еqual to itself. I
This is to say, for аny mathеmatical object or value, х, the refleхive propеrty statеs thаt
х = х.This propеrty holds truе for all mathеmatical operаtions and is used to proof that all the sides of the triangle RST is congruent to sides of the triangle UTS. This is according to the SSS congruence theorem
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12 men or 15women can do apeice of work in 24days in how many days same work done by 8men Or 8women
If 8 men and 8 women collaborate, the task can be finished in 36 days for the former and 45 days for the latter.
Using the following formula, we can determine the total quantity of work that needs to be done.
Total work equals (people) ×(time spent).
The first scenario is as follows:
Total work is equal to (288 man-days)= (12 men x 24 days)
Total work is calculated as (15 women) x (24 days) = (360 woman-days).
Using the same approach, we can now determine how many days it would take eight people—either eight men or eight women—to do the same amount of work:
Eight men:
Total labor: (8 men)× (time required)
288 man-days = (8 men) × (time taken)
time taken = 36 days
Eight women:
total work = (8 women) × (time taken)
360 woman-days = (8 women) × (time taken)
time taken = 45 days
Thus, if 8 males collaborate, they can finish the job in 36 days, and if 8 women collaborate, they can finish the job in 45 days.
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boxplot if a person is making the minimum salary for a construction worker, they are making more than what percentage of teachers? they are making more than % of teachers.
A boxplot is a graphical representation of data distribution, and it can be used to compare the salary distributions of construction workers and teachers. To determine the percentage of teachers that a person making the minimum salary for a construction worker is making more than, we must first understand the data for both groups.
The minimum salary for a construction worker is a specific value, representing the lowest amount paid to workers in that profession. In contrast, teachers' salaries can vary based on various factors such as experience, education level, and location. Comparing these two distributions can provide insights into the relative earnings of these professions.
Using a boxplot, the minimum, first quartile, median, third quartile, and maximum values can be visually represented for each group. The comparison between the minimum salary for a construction worker and the distribution of teachers' salaries can be observed by looking at the overlap of these two boxplots.
If the minimum salary for a construction worker falls within the first quartile of teachers' salaries, it would mean that a person earning that salary is making more than 25% of teachers. If it falls within the second quartile (between the first and median), the person is making more than 50% of teachers. Similarly, if it falls within the third quartile, they would be making more than 75% of teachers.
To provide a precise percentage, the data for both groups needs to be analyzed and compared. However, without specific numbers, a definitive answer cannot be given. In summary, a boxplot can be used to visualize and compare the salary distributions of construction workers and teachers to determine what percentage of teachers a person making the minimum salary for a construction worker is earning more than.
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Find the volume of the rectangular pyramid below.
Round your answer to the nearest
tenth if necessary.
The volume of the rectangular pyramid is 130.7 cubic units.
The volume of a rectangular pyramid can be calculated as:
V = 1/3 × l × w × h
where l, w, and h are the length, width, and height of the rectangular pyramid, respectively.
Given that a rectangular pyramid with a length of 7 units, a width of 7 units, and a height of 8 units.
To find the volume of a given pyramid, substitute the values into the formula:
V = 1/3 × l × w × h
V = 1/3 × 7 × 7 × 8
V = 130.7 units³
Therefore, the volume of the rectangular pyramid is 130.7 cubic units.
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The missing figure has been attached below.
Jahlil made 14 pints of ice cream. He gave 1 quart to his neighbor and then made milkshakes with half of the remaining ice cream. How many quarts of ice cream did Jahlil use to make milkshakes?
Answer:
6 pints or 3 quarts
Step-by-step explanation:
We can represent the actions onto the ice cream with mathematical operations. First, we can represent giving 1 quart away as subtraction:
14 pints - 1 quart
= 14 pints - 2 pints
= 12 pints
So after giving 1 quart to his neighbor, Jahlil has 13 pints left. Of these 13 pints, he uses half to make milkshakes. We can represent a half by dividing by 2:
12 pints / 2
= 6 pints
= 3 quarts
So, Jahlil used 6 pints or 3 quarts of ice cream to make the milkshakes.
use mathematical induction to prove that if n people stand in a line, where n is a positive integer, and if the first person in the line is a woman and the last person in line is a man, then somewhere in the line there is a woman directly in front of a man.
In both cases, for a line of k+1 people, there exists a woman directly in front of a man. Thus, by mathematical induction, the statement holds true for all positive integers n where the first person is a woman and the last person is a man.
To prove this statement using mathematical induction, we will first establish a base case. When n = 2, there are only two people in the line, and the statement is true. The first person is a woman and the last person is a man, and therefore the woman is directly in front of the man.
Next, we assume that the statement is true for n = k, where k is a positive integer. That is, if k people stand in a line with a woman first and a man last, then there is a woman directly in front of a man somewhere in the line.
Now, we need to show that the statement is also true for n = k + 1. Suppose k + 1 people stand in a line, with a woman first and a man last. We can remove the first person (the woman) and consider the remaining k people. By our induction hypothesis, there is a woman directly in front of a man somewhere in this line of k people.
Now, we have two cases for the position of the woman and man in the line of k people:
Case 1: The woman directly in front of the man is in the first k positions. In this case, we can add the first person back into the line, and the statement is true for n = k + 1.
Case 2: The woman directly in front of the man is in the last k positions. In this case, we can remove the first person and consider the line of k people from the second person to the last person. By our induction hypothesis, there is a woman directly in front of a man somewhere in this line of k people. When we add the first person back into the line, this woman will be directly in front of the man, and the statement is again true for n = k + 1.
Therefore, by mathematical induction, we have shown that if n people stand in a line, where n is a positive integer, and if the first person in the line is a woman and the last person in line is a man, then somewhere in the line there is a woman directly in front of a man.
Mathematical induction to prove the given statement. We'll use the terms base case, induction hypothesis, and induction step.
Base case (n=2): When there are two people in the line (a woman followed by a man), it's clear that there's a woman directly in front of a man. This establishes our base case.
Induction hypothesis: Assume that for some positive integer k, if there are k people in a line with a woman at the beginning and a man at the end, there exists a woman directly in front of a man.
Induction step: Let's prove this for k+1 people. We have two cases:
1. If the second-to-last person is a woman, we can remove the last man and have a line of k people. By the induction hypothesis, there exists a woman directly in front of a man in this line, and this also holds for the k+1 people line.
2. If the second-to-last person is a man, we can remove the first woman and have a line of k people. By the induction hypothesis, there exists a woman directly in front of a man in this line. When we add back the first woman, she is now directly in front of the second-to-last man, which also satisfies the condition.
In both cases, for a line of k+1 people, there exists a woman directly in front of a man. Thus, by mathematical induction, the statement holds true for all positive integers n where the first person is a woman and the last person is a man.
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If You invest $1000 in an account with 4.5% annual interest, how much money will you have in 5 years?
The amount of money you will get if you invest $1000 in an account with 4.5% annual interest, compounded annually, in 5 years is: $1221.
How to the Future Value of an Investment with Interest Compounded Annually?If we assume that the interest is compounded annually, the formula we would apply to calculate the future value of an investment is given as:
[tex]FV = PV x (1 + r)^n[/tex] where we have:
PV = present value (the initial investment)
r = annual interest rate (as a decimal)
n = number of compounding periods
In this case, we are given the following:
PV = $1000
r = 0.045 (4.5% expressed as a decimal)
n = 5 (5 years)
Substituting these values into the formula, we would get the following:
FV = $1000 x (1 + 0.045)^5
FV = $1000 x 1.221
FV = $1221
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⚠️please help I have been stuck for forever⚠️
1. Seats in a theater are curved from the front row to the back. The front row has 15 chairs, the second has 20
and the third has 25, and so on.
a. Write a recursive rule for this series (2 points):
b. Write an explicit rule in simplified form for this series:
Show the steps to simplify. (2 points)
c. Using the explicit formula, find the number of chairs in row 9 (show work) (3 points):
d. The auditorium can hold 18 rows of chairs. Write a sigma notation for this series, and then use either series
formulas to calculate how many chairs can fit in the auditorium. Show all your work. (5 points)
There are 95 chairs in row 9.
How to solveb. Consider a_n to signify the number of chairs in the nth row with the purpose of making a recursive rule for this sequence. In terms of a_n-1, we can thus write a_n as a_n = a_n-1 + 5.
The default setting is a_1 = 15.
c. Utilizing the formula for an arithmetic series' nth term, we can devise a clear-cut rule to define this sequence: a_n = a_1 + (n - 1)d. Here, 'd' is a common difference, 'a_1' represents the initial term, and 'n' denotes the sequential number of said term.
As such, applying these variables to our scenario with 'a_1=15' and 'd=5', we conclude that:
a_n = 15 + (n - 1)5
a_n = 10n + 5
d. To find the number of chairs in row 9 using the explicit rule, we plug in n = 9
a_9 = 10(9) + 5
a_9 = 90 + 5
a_9 = 95
Therefore, there are 95 chairs in row 9.
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Find the area of the shaded segment of the circle
The area of the shaded segment of the circle is 55.9 cm².
Given that a sector in the circle having the central angle = 90° with the radius = 14 cm.
We need to find the area of the segment which is shaded.
So, for finding the same, we will find the area of the sector and then the right triangle formed by the radii and the chord.
Area of the sector = 1/4 × π × radius²
= 1/4 × 3.14 × 14²
= 153.86 cm²
Area of the triangle = 1/2 × base × height
= 1/2 × 14 × 14
= 98 cm²
Area of the segment = 153.86 - 98 ≈ 55.9 cm²
Hence, the area of the shaded segment of the circle is 55.9 cm².
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If you had 8,923,283- 12035 what would it be
Answer:
8,911,248
Step-by-step explanation:
8,923,283 - 12,035 = 8,911,248
The mean and median of a set of five positive integers is 5. What is the largest integer that can be in the set?
The largest integer that can be in the data-set is given as follows:
13.
How to obtain the mean and the median of a data-set?The mean is given by the sum of all observations divided by the number of observations.The median of a data-set is the middle element of the data-set.The data-set in this problem has five integers, hence the median of 5 is the third element, given as follows:
x, y, 5, w, z.
The mean is also of 5, meaning that the sum of the other elements is 20, that is:
x + y + w + z = 20.
z will be maximized when x = 1, y = 1 and w = 5, hence:
1 + 1 + 5 + z = 20
z = 13.
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4. Which of the following is the correct formula for the area of a triangle?
O
O
A = =acco
cos B
a² = sin² b + cos² c
1
A = -bcsin A
2
a² + b² = c²
Option c. A = 1/2 bcSin A is the area of a traingle
What is the area of a triangle using the sine ruleThe sine rule is a mathematical equation that links the side lengths of a triangle to the sines of its respective angles. The formula for determining the area in such a triangle is represented as follows:
A = (1/2) ab sin C
= (1/2) bc sin A
= (1/2) ac sin B
where a, b, and c symbolize the lengths of the triangle's sides while A, B, and C stand for the measures of their opposite angles.
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Mrs. Lee bought an ipad that was on sale for $250. She got 15% off the original cost. What was the original price
The iPad bought by Mr. Lee has the original price of $294 if he paid $250 with 15% off the iPad
Discount refers to the reduction in the original price to allure customers to buy the product.
Original price = discounted price + discount
Let the original price be x
Given, discounted price (or price after discount) = $250
Discount % = 15%
Discount % = (original price - discounted price) / original price * 100
15 = [tex]\frac{x-250}{x}*100[/tex]
15x = (x - 250) 100
3x = 20x - 5000
5000 = 17x
x = 294.1
Therefore, the original price is around $294
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What is | 62 |
IGNORE THISSSSSSSSSSSSSSSSSSS
The expression |62| is equal to 62.
We have,
The absolute value of 62 is 62, because the absolute value is a measure of distance from zero, and 62 is 62 units away from zero in the positive direction on the number line.
The absolute value of a number is defined as the distance of that number from zero on the number line, regardless of the direction.
It is always a non-negative value.
In this case, the number is 62.
If we locate 62 on the number line, we can see that it is 62 units away from zero in the positive direction, because we count 62 units to the right of zero to reach 62.
Since the absolute value of a number is always non-negative, we take the positive value of 62 and conclude that the absolute value of 62 is 62, written as |62| = 62.
Therefore,
|62| is equal to 62.
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In circle C, TL is a diameter, mICR = (3x + 5)°, and m/RCL = (x - 1)°.
The value of the equation gives mILR = 52°.
How to determine the valueSince TL is a diameter, we know that mICR + m/RCL = 90°. Substituting the given values, we get: (3x + 5)° + (x - 1)° = 90°
Simplifying the equation, we get:
4x + 4 = 90
Solving for x, we get:
x = 22
Therefore, x = 22.
2. Since TL is a diameter, we know that mICR = 90°. Substituting the value of x, we get:
mICR = (3x + 5)° = (3 × 22 + 5)° = 71°
Therefore, mICR = 71°.
3. Since TL is a diameter, we know that mILR = 180° - mICR - m/RCL. Substituting the values of x, we get:
mILR = 180° - (3x + 5)° - (x - 1)° = 180° - 4x - 4°
Substituting the value of x, we get:
mILR = 180° - (4 × 22 + 4)° = 52°
Therefore, mILR = 52°.
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HELP QUICKLY!!!!!!
A circle with equation (x + 2 )2 + (y – 3 )2 = 16 is graphed on the coordinate plane. Which represents the line of tangency at the point (–6, 3)?
A. y = –6
B. x = –6
C. x = 4
D. x = 6
Answer:
Step-by-step explanation:
Pemdas
tyler plays basketball for his local deaf school. he is leading his league in scoring, and he wants to have his grandparents, who live in a different state, follow his games. where should they look to find statistics and videos about him and his team?
Tyler's grandparents can follow his basketball games by checking the local deaf school's website or social media pages for updates on team statistics and videos. Additionally, they can search for local news outlets or sports websites that may cover the games and provide more information about Tyler and his team.
Tyler's grandparents can look for statistics and videos about him and his team on the website or social media pages of his local deaf school. They can also check with the league that his team plays in to see if they have any online resources available for following games and keeping track of player statistics. Another option is to reach out to Tyler's coach or the school's athletic department to see if they can provide any updates or links to game footage.
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A tanker truck carrying fuel initially contains 100L. More fuel is added to the tank before it goes across Canada. Every 6 minutes, 1620 L are added into the tank. A)Independent variable (x) -
b) Dependent variable (y) -
c) What is the rate of change for this linear relationship?
Independent variable (x) is time in minute, dependent variable (y) is amount of fuel in the tank in liters and the rate of change for this linear relationship is 270 L/min.
The variable that is independent in time because no matter what, it will flow. The dependent variable is amount of fuel. The rate of change of this linear relationship is the slope of the line, which is equal to the amount of fuel added per unit time, in this case, 1620 L added every 6 minutes, or 270 L/min. This indicates that the amount of gasoline in the tank rises at a constant pace of 270 liters per minute.
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(Chapter 12) If u * v = 0 and u X v = 0, then u or v = 0
Therefore, in either partial derivatives, we have u = 0 or v = 0.
The given information implies that two vectors u and v satisfy:
u * v = 0, where * denotes the dot product between vectors.
u X v = 0, where X denotes the cross product between vectors.
From the first equation, we know that the angle between u and v is either 90 degrees or 270 degrees. That is, u and v are orthogonal (perpendicular) to each other.
From the second equation, we know that the magnitude of the cross product u X v is equal to the product of the magnitudes of u and v multiplied by the sine of the angle between them. Since u and v are orthogonal, the angle between them is either 90 degrees or 270 degrees, which means that the sine of the angle is either 1 or -1. Therefore, we have:
|u X v| = |u| * |v| * sin(θ)
= 0
Since the magnitudes of u and v are non-negative, it follows that sin(θ) must be zero. This can only happen if the angle between u and v is either 0 degrees (i.e., u and v are parallel) or 180 degrees (i.e., u and v are anti-parallel).
In the case where u and v are parallel, we have:
u * v = |u| * |v| * cos(θ)
= |u|²
= 0
This implies that |u| = 0, which means that u = 0.
In the case where u and v are anti-parallel, we have:
u * v = |u| * |v| * cos(θ)
= -|u|²
= 0
This again implies that |u| = 0, which means that u = 0.
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How do you solve (2b) to the 4th power?
To solve (2b) to the 4th power, we need to raise the entire quantity of 2b to the power of 4.
This can be done by multiplying the quantity (2b) by itself 4 times using the power rule of exponents, which states that (a^m)^n = a^(m*n).
So we have:
(2b)^4 = (2b) x (2b) x (2b) x (2b)
Expanding this expression, we get:
(2b)^4 = 2 x 2 x 2 x 2 x b x b x b x b
Simplifying further, we can multiply the 2's together to get:
(2b)^4 = 16b^4
Therefore, the expression (2b) to the 4th power simplifies to 16b^4.
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Find the missing side.
36°
19
X
X <= [?]
Round to the nearest tenth.
Remember: SOHCAHTOA
Enter
Applying SOHCAHTOA, the value of the missing side in the right triangle is calculated as: x ≈ 11.2.
How to Find the Missing Side Using SOHCAHTOA?SOHCAHTOA is an acronym for the basic trigonometric ratios that can be used to find the missing side of a right triangle. It means:
SOH implies sin ∅ = opposite / hypotenuse
CAH implies cos ∅ = adjacent / hypotenuse
TOA implies tan ∅ = opposite / adjacent.
To find the missing side, x, we will apply SOH:
sin 36 = x/19
19 * sin 36 = x
x ≈ 11.2
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The US government monitors the consumption of different products.
The table shows y, the amount of ice cream consumed, in millions of pounds, for x years since 2010.
The quadratic equation that models the amount of ice cream consumed, in millions of pounds, since 2010 is shown.
y = 12(x - 6)² + 3922
Determine when the amount of ice cream consumed in the United States would be 5,650 millions of pounds.
The required, in the year 2028, the amount of ice cream consumed in the United States would be 5,650 million pounds.
We are given that y represents the amount of ice cream consumed in millions of pounds and x represents the number of years since 2010. So we want to solve for x when y = 5,650.
y = 12(x - 6)² + 3922
5,650 = 12(x - 6)² + 3922
1,728 = 12(x - 6)²
144 = (x - 6)²
Take the square root of both sides (remembering to include both the positive and negative square roots):
±12 = x - 6
x = 6 ± 12
So we have two possible solutions:
x = 6 + 12 = 18
x = 6 - 12 = -6
Since we are looking for a year since 2010, the year -6 does not make sense in this context. Therefore, the amount of ice cream consumed in the United States would be 5,650 million pounds in the year 2010 + 18 = 2028.
So the answer is that in the year 2028, the amount of ice cream consumed in the United States would be 5,650 million pounds.
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is it likely that nasa repeatedly made the same data entry errors in recording temperatures in los angeles? select the correct response that would be the most likely explanation for using the data value 999.9 in the data file you downloaded.
It is not likely that NASA repeatedly made the same data entry errors in recording temperatures in Los Angeles. The use of the value 999.9 in the data file that was downloaded is more likely a placeholder value used to indicate missing data.
This is a common practice in data collection and analysis, as it allows for the inclusion of all available data points in the analysis without compromising the accuracy of the results. It is important to note that missing data can occur for a variety of reasons, including instrument malfunctions, power outages, or human error.
In order to ensure the accuracy and reliability of the data, it is important to identify and account for missing data in the analysis. This can be done through various methods, such as imputation or exclusion of incomplete data points.
In conclusion, while data entry errors can occur, it is not likely that NASA repeatedly made the same errors in recording temperatures in Los Angeles. The use of the value 999.9 in the data file is more likely a placeholder value used to indicate missing data, and it is important to properly account for missing data in the analysis to ensure the accuracy and reliability of the results.
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Fill in the missing term on each line. Simplify any fractions.
2p + 4 = 6
2p=______ Subtract 4 from both sides ***
p=_______ Divide both sides from 2***
The expression 2p + 4 = 6 solved with the defined steps is p = 1
Solving the expression with the defined stepsFrom the question, we have the following parameters that can be used in our computation:
2p + 4 = 6
Subtract 4 from both sides
2p = 2
Divide both sides from 2
p = 1
Hence, the solution is p = 1
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which type of data visualization method can be helpful when the intention is to show relationship between time spent working out per week (in hours) and average amount of sleep an individual gets per week (in hours)?
When the intention is to show the relationship between time spent working out per week and average amount of sleep an individual gets per week, a scatter plot can be a helpful data visualization method. This type of graph uses two axes to plot data points that correspond to each variable.
In this case, the x-axis can represent time spent working out per week, while the y-axis can represent average amount of sleep an individual gets per week. Each data point will correspond to a specific individual and will be plotted based on the values of the two variables.
By using a scatter plot, it will be possible to identify any patterns or trends in the data. For example, it may become clear that individuals who spend more time working out tend to get more sleep per week. Alternatively, it may be evident that there is no relationship between the two variables, which can be equally informative.
Overall, using a scatter plot can help to visualize the relationship between time spent working out per week and average amount of sleep an individual gets per week. This can be useful in understanding the data and drawing conclusions about the relationship between these two variables.
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Q2. A rectangle has a perimeter of 90 cm. What could its area be in square metres?
Answer:
2 square meters
Step-by-step explanation:
90 centimeters = .9 meters
one side could be 5 and the other could be 40.
40+40+5+5= 90
40 x 5 = 200
remember, 1 meter= 100 centimeters, so it would be 2 meters
try that! Also, hope that helps!!
The variables x and y vary inversely. Use the given values to write an equation relating x and y. Then find y when x = 2.
x = 1
y = 2
Based on the given inverse variation between x and y, y = 1 when x = 2.
How to write an equation relating x and y?x = k/y
Where,
k = constant of proportionality
When x = 1 and y = 2
x = k/y
1 = k/2
cross product
1 × 2 = k
2 = k
So,
x = 2/y
find y when x = 2.
x = 2/y
2 = 2/y
cross product
2y = 2
divide both sides by 2
y = 2/2
y = 1
Ultimately, the value of y when x = 2 is 1.
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The following function represents the production cost f(x), in dollars, for x number of units produced by company 1:
f(x) = 0.25x2 − 8x + 600
The following table represents the production cost g(x), in dollars, for x number of units produced by company 2:
x g(x)
6 862.2
8 856.8
10 855
12 856.8
14 862.2
Based on the given information, determine which company has a lower minimum and find the minimum value.
A- g(x) at (10, 855)
B- f(x) at (16, 536)
C- g(x) at (16, 536)
D- f(x) at (10, 855)
The company that has a lower minimum and find the minimum value will be B- f(x) at (16, 536)
How to explain the informationThe following function represents the production cost f(x), in dollars, for x number of units produced by company 1:
f(16) = 0.25(16)^2 - 8(16) + 600 = 536
So, the minimum value of f(x) is 536, and it occurs at x = 16.
In conclusion, the correct option is B.
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If the true population distribution is close to the population distribution assumed in the null hypothesis, the power tends to be:_______
If the true population distribution is close to the population distribution assumed in the null hypothesis, the power tends to be lower.
Explanation:
1. True population distribution refers to the actual distribution of the population.
2. Null hypothesis is the assumption that there is no significant difference between the observed and expected data.
3. Power is the probability of correctly rejecting the null hypothesis when it is false.
So if the true population distribution is close to the null hypothesis distribution, it is less likely that significant differences will be found in the sample.
When the true population distribution is close to the null hypothesis, it becomes more difficult to detect a significant difference, leading to a lower power in the statistical test.
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T/F : In a 5 by 5. If A has two pivots, then the dimension of Nul A is 2
True. In a 5 by 5 matrix, if A has two pivots, then the dimension of the nullspace (Nul A) is 3, which can be found using the rank-nullity theorem:
rank(A) + dim(Nul A) = 5
Since A has two pivots, its rank is 2.
Suppose we have an m x n matrix A. The rank-nullity theorem states that the sum of the rank of A and the dimension of the nullspace of A is equal to the number of columns in A, that is:
rank(A) + dim(Nul A) = n
where rank(A) is the number of linearly independent rows or columns in A, and dim(Nul A) is the dimension of the nullspace of A.
Now, suppose we have a 5 x 5 matrix A that has two pivots. Since A has two pivots, it means that there are two linearly independent rows or columns in A. Therefore, the rank of A is 2. Applying the rank-nullity theorem, we get:
rank(A) + dim(Nul A) = 5
Substituting rank(A) = 2, we get:
2 + dim(Nul A) = 5
which implies that dim(Nul A) = 3. This means that the nullspace of A has dimension 3, which means that there are 3 linearly independent solutions to the equation Ax = 0.
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