There is not enough evidence to conclude that the percent at her school is less than 40%.
How to reach the conclusion of the hypothesis tested?The conclusion of the hypothesis test is reached depending on the confidence interval as follows:
Upper bound below 40%: enough evidence to conclude that the percent at her school is less than 40%.Upper bound above 40%: not enough evidence to conclude that the percent at her school is less than 40%.The upper bound of the interval is given as follows:
0.4240 = 42.40%.
Hence not enough evidence to conclude that the percent at her school is less than 40%.
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Identify the kind of sample that is described. A sports columnist contacts a random sample of 3 owners, a random sample of 6 players, and a random sample of 20 fans to ask their opinion about the potential basketball lockout. the sample is a (Choose one) A) convenience B) voluntary response C) simple random D) stratified E) cluster F) systematic
A cluster sample is a type of sampling technique wherein clusters of subjects that are similar in certain characteristics are selected for the study. In this case, the sample includes 3 owners, 6 players, and 20 fans, all of whom have something in common - an interest in basketball. Therefore, the sample described is a cluster sample.
A cluster sample is a type of sampling technique wherein clusters of subjects that are similar in certain characteristics are selected for the study. In this case, the sample includes 3 owners, 6 players, and 20 fans, all of whom have something in common - an interest in basketball. Therefore, the sample described is a cluster sample. This type of sample is helpful when the researcher is interested in specifically examining a certain group of people and their opinions. A cluster sample is relatively easy to implement and can be used to quickly obtain a large amount of data in a short period of time. It also eliminates the need to identify and select individuals from a population. By using a cluster sample, the sports columnist was able to quickly get feedback from a variety of individuals who may have different opinions on the potential basketball lockout.
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A hypothetical university has two departments, A and B. There are 2,000 male applicants, of whom half apply to each department. There are 1,100 female applicants: 100 apply to department A and 1,000 apply to department B. Department A admits 60% of the men who apply and 60% of the women. Department B admits 30% of the men who apply and 30% of the women. "For each department, the percentage of men admitted equals the percentage of women admitted,; this must be so for both departments together." True or False
The statement regarding the percentages of men and women admitted is true.
How to obtain the percentages?A percentage is obtained similarly to a proportion, with the division of the number of desired outcomes by the number of total outcomes, and then multiplied by 100%.
The number of applicants for each department is given as follows:
Department A: 1000 male and 100 female.Department B: 1000 male and 1000 female.The percentages of admitted people for each department is 50% men and 50% women, meaning that the statement in this problem is true.
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If packages of shredded cheese in the refrigerator case are lined up in 7 rows with 8 packages of shredded cheese in each row, how many packages of shredded cheese are there all together?
Using multiplication operation, the total number of packages of shredded cheese is 56.
What is multiplication operation?Multiplication operation is one of the four basic mathematical operations, including subtraction, division, and addition.
Multiplication operation has the following parts: multiplicand (the number being multiplied), the multiplier (the number doing the multiplication), the equal symbol, and the product (the result of the multiplication operation)
The number of rows = 7
The number of packages of shredded cheese per row = 8
The total number of packages = 56 (7 x 8)
Thus, the number of packages in the refrigerator is 56 shredded cheese.
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On January 1st, the temperature in St Petersburg was -7°C and the temperature in Rome was 15°C. The temperature in Reigate was exactly halfway between St Petersburg and Rome. What was the temperature in Reigate?
To find the temperature in Reigate, which is halfway between St. Petersburg and Rome, we need to find the average of the two temperatures.
Average = (Temperature1 + Temperature2) / 2
Average = (-7 + 15) / 2 = 8 / 2 = 4
So, the temperature in Reigate was 4°C.
Answer:
4°C
Step-by-step explanation:
Try drawing it on a number line and counting 11 from -7° C and 11 from 15°C you'll end up with 4°C as the middle
Select the number that round to 387.4 when rounded to the nearest tenth.
A. 387.461
B. 387.344
C. 387.309
D. 387.352
E. 387.779
Answer:
D
Step-by-step explanation:
When rounding, the number 5 is rounded up. So the number 387.352 will be rounded up 387.4. Other options are not suitable. Correct answer is "D"
If you think my answer is the best, please mark it as the Brainliest.
Thank you! :))
Review: geometry are the lines represented by the equations at right parallel? support your reasoning with convincing evidence. homework help
Yes, the lines represented by the equations at right are parallel. This can be seen by examining the slopes of the equations. The slope of the first equation is -3/5 and the slope of the second equation is -3/5. Since the slopes are equal, the lines are parallel.
To determine if two lines are parallel, we need to compare their slopes. The equation of the first line is y = -3x/5 and the equation of the second line is y = 5x/3. To calculate the slope of the first line, we need to take the coefficient of x (-3/5) and divide it by the coefficient of y (1).
The resulting slope is -3/5. To calculate the slope of the second line, we need to take the coefficient of x (5/3) and divide it by the coefficient of y (1). The resulting slope is also -3/5. Since the slopes of both lines are equal, the lines are parallel.
The equation of the first line is y = -3x/5 and the equation of the second line is y = 5x/3. To calculate the slope of the first line, we need to take the coefficient of x (-3/5) and divide it by the coefficient of y (1).
The resulting slope is -3/5. To calculate the slope of the second line, we need to take the coefficient of x (5/3) and divide it by the coefficient of y (1). The resulting slope is also -3/5. Since the slopes of both lines are equal, the lines are parallel.
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The following scenario represents a proportional relationship.
Jerome's dry cleaning fee was $19.50 for 3 pairs of pants.
What is the constant of proportionality?
Answer:
6.5
Step-by-step explanation:
Proportional relationship:
[tex]y \propto x \implies y=kx[/tex]
where k is the (non-zero) constant of proportionality.
To determine the constant of proportionality, substitute the given values into the equation and solve for k:
x = 3y = 19.50[tex]\implies 19.50=3k[/tex]
[tex]\implies k=6.5[/tex]
Therefore, the constant of proportionality is 6.5.
what are tangible items sold by retailers called
Answer:
A tangible asset is an asset that has physical substance. Examples include inventory, a building, rolling stock, manufacturing equipment or machinery, and office furniture. There are two types of tangible assets: inventory and fixed assets.
Step-by-step explanation:
Which of the following statements is/are true? I. If f'(x) exists and is nonzero for all x, then f(1) is not equal f(0). II.If f is differentiable for all x and f(-1) = f (1), then there is a number c, such that |c| < 1 and f'(c) = 0. III. If f'(c) = ), then f has a local maximum or minimum at x = c.
The evaluation of the values of the derivative of the function f, using Lagrange's theorem indicates that the true statements is the following option;
I and III anlyWhat does the Lagrange's Theorem highlights?Lagrange Theorem states that there is a point, c between points a and b of an arc such the slope at c is the same as the average slope of the arc between points a and b.
The evaluation of the statements in the question are as follows;
I. If f'(x) exists and is nonzero for all x, then f(1) is not equal to f(0)
If f'(x) exists and the value is non zero, then the slope is not infinite, and the sign of the slope of the function f(x) never changes, such that the as the input (x) value increases the output value decreases, when the slope is negative, or increases when the slope of f(x) is positive, therefore;
The value of f(1) is either larger than or lesser than the value of f(0), which proves that f(1) is not equal to f(0).
The first statement, I, is therefore, trueII. If f is differentiable for all x and f(-1) = f(1), then there is a number c, such that |c| < 1 and f'(c) = 0
The condition if f is differentiable for all x, then according to Lagrange's, mean value theorem, we get;
When f(|c| < 1) < f(1)
f'(c) < (f(1) - f(-1))/(1 - (-1)) = (f(1) - f(-1))/(2)
f'(c) < (f(1) - f(-1))/(1 - (-1)) = (f(1) - f(-1))/(2) = 0
f'(c) < 0
Similarly, when f(|c| < 1) > f(1), then, f'(c) < 0
Therefore, f'(c) ≠ 0
The statement II is not trueIII. If f'(c) = 0, then f has a local maximum or minimum at x = c.
The slope of the function f at point x = c is f'(c)
A local maximum or minimum is a point after which the slope of the graph changes sign, and the point at which the slope is zero as the change in the output variable is zero
Therefore, if f'(c) = 0, the graph has a local maximum or minimum at the point f = c
The true statements from the specified options, and based on the possible options in the question, obtained from a similar question posted online are;
I and III onlyThe possible question options are;
I and II only
I and III only
I only
II only
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use the following for the first two problems. Listed are 32 ages for Academy Award winning best actors in order from smallest to largest. (Round your answers to the nearest whole number.)
18; 18; 21; 22; 25; 26; 27; 29; 30; 31; 31; 33; 36; 37; 37; 41; 42; 47; 52; 55; 57; 58; 62; 64; 67; 69; 71; 72; 73; 74; 76; 77
(a) Find the percentile of 37. 14/32= .4375
44 th percentile
(b) Find the percentile of 71 27/32=.8437?
I get the 1st one but can't seem to get the 2nd, i come up with 27/32 but the computer doesn't agree can anyone help?
Similarly, 27 out of the 32 ages are equal to or less than 71, so the percentile of 71 is 27/32, which is equal to 84.375%. This means that 84.375% of the Academy. Award winning best actors are 71 years old or younger.
The percentile of a number is the percentage of numbers that are equal to or less than that number. In this case, 27 out of the 32 ages are equal to or less than 71, so the percentile of 71 is 27/32, which is equal to 84.375%. This means that 84.375% of the Academy Award winning best actors are 71 years old or younger.
The percentile of a number is the percentage of numbers that are equal to or less than that number. For example, among the 32 Academy Award winning best actors, 37 is the 14th smallest number, so the percentile of 37 is 14/32, which is equal to 43.75%. This means that 43.75% of the Academy Award winning best actors are 37 years old or younger. Similarly, 27 out of the 32 ages are equal to or less than 71, so the percentile of 71 is 27/32, which is equal to 84.375%. This means that 84.375% of the Academy Award winning best actors are 71 years old or younger.
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4x-y=3 6y=4+2x solve by substitution
Using the substitution method the equation 4x - y = 3 and 6y = 4 + 2x is solved to be
x = 1y = 1How to solve the equation using substitution methodIn this method, the equation 4x - y = 3 will first be written in terms of y
4x - y = 3
y = 4x - 3
substituting y = 4x - 3 into 6y = 4 + 2x
6y = 4 + 2x
6(4x - 3) = 4 + 2x
24x - 18 = 4 + 2x
collecting like terms
24x - 2x = 18 + 4
22x = 22
x = 1
substituting x = 1 into y = 4x - 3
y = 4x - 3
y = 4(1) - 3
y = 4 - 3
y = 1
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A coffee shop is having a sale on prepackaged coffee and tea. Yesterday they sold 34
packages of coffee and 27 packages of tea, for which customers paid a total of $420. The day
before, 41 packages of coffee and 45 packages of tea was sold, which brought in a total of
$606. How much does each package cost?
Polynomial equations are those created by combining variables, exponents, and coefficients.The cost of coffee is 8 and the cost of tea is 4.
How much does each package cost?The higher exponent, known as the degree of the equation, might have several exponents.
When there is at least one algebraic term with at least one variable and all of the exponents are integers that are equal to or larger than zero, the equation is said to be a polynomial equation.
Coffee cost = Rs 8
Tea cost = Rs 4
Let the price of coffee be X
and the price of Tea be y , then
45 x + 41 y = 524
20 x + 11y = 204
y = 524 - 45 x /41=
= 20 x + 11( 524 -45 x)/41)
= 204
820 x + 5764- 495 x = 204* 41
325x = 2600
x = 8 y = 524 - 8* 45/45
y = 4
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Polynomial equations are those created by combining variables, exponents, and coefficients. The cost of coffee is 8 and the cost of tea is 4.
How much does each package cost?The higher exponent, known as the degree of the equation, might have several exponents.
When there is at least one algebraic term with at least one variable and all of the exponents are integers that are equal to or larger than zero, the equation is said to be a polynomial equation.
Coffee cost = Rs 8
Tea cost = Rs 4
Let the price of coffee be X
and the price of Tea be y , then
45 x + 41 y = 524
20 x + 11y = 204
y = 524 - 45 x /41=
= 20 x + 11( 524 -45 x)/41)
= 204
820 x + 5764- 495 x = 204* 41
325x = 2600
x = 8 y = 524 - 8* 45/45
y = 4
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Based on the table, do the data support the use of a normal model to approximate population characteristics?
A. Yes, because the sum of the relative frequencies is 1.00.
B. Yes, because the distribution of relative frequencies is very close to the empirical rule for normal models.
с. No, because the values are negative and normal models are used for positive values.
D. No, because the distribution of relative frequencies is very far from the empirical rule for normal models.
E. No, because the sample size and the population parameters are not known.
Yes, because the distribution of relative frequencies is very close to the empirical rule for normal models.
Interval Relative Frequency
−0.34≤x<−0.31 0.02
−0.31≤x<−0.28 0.15
−0.28≤x<−0.25 0.33
−0.25≤x<−0.22 0.36
−0.22≤x<−0.19 0.11
−0.19≤x<−0.16 0.03
A standard deviation (or σ) is a measure of how dispersed the data is in relation to the mean. Low standard deviation means data are clustered around the mean, and high standard deviation indicates data are more spread out.
According to the question,
The values that are one standard deviation away are:
0.33 + 0.36 = 0.69
As per the empirical rule, the value will be: 0.68
Consequently, with two standard deviations, the quantities have always been 0.95, and according to the empirical rule. So the empirical rule followed by the data.
Therefore the correct option is (B).
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The complete question includes :
The mean and standard deviation of the sample data collected on continuous variable x, x are −0.25 and 0.03, respectively. The following table shows the relative frequencies of the data in the given intervals.
the length of a rectangular platform is 2 feet longer than three times its width. The area of the platform is 56 square feet. Write a polynomial that represents the area of the platform
Answer
The area of the platform :
56 = (3x+2) × x
Explanation (+ solving for x)
Let the width of Rectangular Platform is x feet.
Then,
Length of Rectangular Platform is (3x+2) feet
Area of Rectangular Platform = Length × width
[tex]56=(3x+2) × x \\ 56 = 3x² + 2x \\ 3x² +2x-56= 0 \\ x = \frac{ -2±\sqrt{ {2}^{2} - 4 \times 3 \times ( - 56)} }{2 \times 3} \\ = \frac{-2± \sqrt{4 + 672} }{6} \\ = \frac{ - 2±26}{6} \\ = - \frac{28}{6} or \: 4 [/tex]
Since x can't be negative,
Value of x is 4 feet.
So, width of Platform= 4
feet Length of Platform = 14 feet
Find the product of the binomial factors using the appropriate special product (difference of two squares, square of a binomial sum, or square of a binomial difference).
(x−5)2
The product of the binomial factors is x² - 10x + 25.
What is Binomials?Binomials are kind of polynomials which consist of only two terms.
The given binomial factor is x - 5
We have to find the product of this binomial with itself, (x - 5)².
We have to use the special product of square of a binomial difference, which is defined as,
(a - b)² = a² - 2ab + b²
The given product of the binomial factors is (x - 5)².
(x - 5)² = x² - (2 × x × 5) + 5²
= x² - 10x + 25
Hence the product of the binomial factors is x² - 10x + 25.
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find the acute angles between the curves at their points of intersection. (the angle between two curves is the angle between their tangent lines at the point of intersection. give your answers in degrees, rounding to one decimal place. enter your answers as a comma-separated list.) y
The acute angles between two curves at their points of intersection are the angles between the tangent lines of the two curves at those points. The angles should be rounded to one decimal place and given in a comma-separated list.
The acute angles between two curves at their points of intersection can be found by calculating the angles between the tangent lines of the two curves at those points. To calculate this angle you will need to calculate the gradient of the tangent lines at each point of intersection. To do this, you need to take the derivative of each curve's equation at each point of intersection. Once you have the gradients, you can then use the equation for the angle between two lines to calculate the angle between the two tangent lines. Then, the angles should be rounded to one decimal place and given in a comma-separated list.
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ASAPPPPPPPPPPPPPP!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answer: 5 11/12 not sure
A farmer has a rectangular field with dimensions 200m (north) by 150m (east). A sheep is put on a 20 meter leash whose end is tied to a pole. (Hint: Draw the situation!) (a) Assume the south west corner of the field is the origin of a coordinate system. The pole is placed at 40 meters east and 20 meters north from this origin. Will the sheep be able to eat grass from the neighbor's field? If so, where? (b) Same situation as in (a). But the pole is now put 180m north and 160m east. Will the sheep be able to eat grass from the neighbor's field? If so, where?
a) The sheep will not be able to eat grass from the neighbor's field. The leash is 20 meters long and is tied to a pole at the point (40, 20) in the field. The farthest point the sheep can reach is a point on the circle with radius 20 centered at (40,20), which is completely contained within the rectangular field.
b) The sheep will be able to eat grass from the neighbor's field. The leash is 20 meters long and is tied to a pole at the point (180,160) in the field. The farthest point the sheep can reach is a point on the circle with radius 20 centered at (180,160), which extends beyond the east and north boundaries of the rectangular field. Specifically, the sheep can reach the area on the northeast corner of the field.
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Here is a probability scale: 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 If I flip a coin, place a cross on the probability scale to show the probability of getting a head.
Answer:
0.5
Step-by-step explanation:
If you flip a coin, the probability of getting a head is 0.5 or 0.5 on the probability scale.
It is worth noting that when a coin is fair, the probability of getting a head or a tail is always 0.5 or 50%.
So in the probability scale you should place a cross on the point 0.5 which indicates the probability of getting a head.
PLS HELP FOR REAL!!! WHAT IS THE ANSWER FOR THE 2nd ONE!! I NEED AN EXPLANATION AND ANSWER FOR IT! HUGE POINTS IF ANSWER!
Answer:
76ft^3
Step-by-step explanation:
Volume of a cube is Length x Width x Height, basically: 4 x 4 x 4 which is 64 ft^3. Add that to the small box gives you 64 + 12 or just 76 ft^3
Find the indicated real nth root(s) of a.
n = 2, a = 400
Root: ?
Answer:
20 or +-20
Step-by-step explanation:
They are asking for an nth root. This is generic-speak for square root, third root, fourth root, etc.
Since n is 2, they are talking about 2th root...really 2nd root. But we commonly call that square root.
Square root of 400 is 20. (That's the principal square root because its positive) They asked for real roots, so, technically -20 is also a square root of 400.
20^2 = 400 and
(-20)^2 = 400
So square root of 400 is +- 20.
Or if this is an intro class, then just 20 is probably a good enough answer.
The new bookcase for your future dorm room has the dimensions of 24 in. wide by 12 in. deep by 36 in.
tall. If one average textbook has the dimensions of 8.5 in wide by 11 inches long by 1.5 in. tall, how
many books could you fit on your bookshelf with no shelving?
On solving the provided question, we can say that since it is a cuboid books could you fit on your bookshelf = 24*12*36/8.5*11*1.5 = 10368/140.25 = 73.9251336898 = 73 books
what is cuboid?A cuboid is a solid or three-dimensional form in geometry. A cuboid is a convex polyhedron having 8 vertices, 12 sides, and 6 rectangular faces. Another name for a cuboid is a cuboid. A cube is a cuboid with six square faces. Boxes include things like books and bricks. The following are the key variations between cubic and cubic: In contrast to a cube, which has rectangular faces, a cube has six equally sized square faces. Even though the cube and cuboid structures have a similar appearance, they differ in some ways in terms of side length, diagonal, and area.
since it is a cuboid
books could you fit on your bookshelf = 24*12*36/8.5*11*1.5 = 10368/140.25 = 73.9251336898 = 73 books
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Lana has 4 bags with the same amount of marbles in them, totaling 14 marbles. Markus has 4 bags with the same amount of marbles in them, totaling 20 marbles. How many more marbles does Markus have in each bag?
Answer:
Lana: 4 / 14 = 3.5
Markus: 4 / 20 = 5
I'm not completely sure how they can have the same amount of marbles for both of them but... Markus does have 2.5 more than Lana in each bag.
Hope this helps <3
Step-by-step explanation:
Let's call the number of marbles in each bag "x". We can set up two equations based on the given information:
4x = 14 (for Lana)
4x = 20 (for Markus)
Solving for x in each equation, we get:
x = 14/4 = 3.5 (for Lana)
x = 20/4 = 5 (for Markus)
So Markus has 5 marbles in each bag, which is 1.5 more marbles per bag than Lana.
Sai buys $12,500 worth of music equipment. He finances his purchase
at a 2.5% simple interest rate and sets up a 24-month payment plan.
How much interest will Sai pay?
The amount of interest Sai will pay is $312.50.
What is intrest?Interest is the cost associated with borrowing money, or the return earned for lending money. It is typically expressed as a percentage of the principal amount borrowed or lent, and is usually paid periodically over the life of the loan. Interest can also be earned from investments, such as savings accounts, bonds, and other financial instruments. The rate of interest charged or earned varies depending on the nature of the loan or investment, the risk associated with the transaction, and the creditworthiness of the borrower or investor.
This is calculated by multiplying the principal amount of $12,500 by the interest rate of 2.5%, which equals 0.025. Then, the 0.025 is multiplied by the loan period of 24 months to come up with the total interest. Therefore, $12,500 x 0.025 x 24 = $312.50.
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the number of hours of sunlight in a particular location s(t) can be modeled by the function where t is the number of days after january 1st. (that is, t
After solving the function S(t) = 12 + 2 sin [(2π)/(365)t], the amount of sunlight becomes 12 hours for the first time is obtained as option B: t = 365/2.
What is a function?
In mathematics, a function is a unique arrangement of the inputs (also referred to as the domain) and their outputs (sometimes referred to as the codomain), where each input has exactly one output and the output can be linked to its input.
The given function is -
S(t) = 12 + 2 sin [(2π)/(365)t]
Here, t is the number of days after 1st January.
S(t) is the number of hours of sunlight in a particular location.
In order to figure out when the amount of sunlight becomes 12 hours for the first time, substitute the value of S(t) = 12 and solve the function for t -
S(t) = 12 + 2 sin [(2π)/(365)t]
12 = 12 + 2 sin [(2π)/(365)t]
Simplifying the equation to get the result -
0 = 2 sin [(2π)/(365)t]
0 = sin [(2π)/(365)t]
π = (2π)/(365)t
Simplify the equation further -
365π = 2πt
t = 365π/2π
t = 365/2
Therefore, the value for t is obtained as t = 365/2.
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The number of hours of sunlight in a particular location S(t) can be modeled by the function S(t) = 12 + 2 sin [(2π)/(365)t], where t is the number of days after January 1st. (That is, t = 0 means January 1st.) After how many days will there be 12 hours of sunlight for the first time during the year?
365 over 4 days
365 over 2 days
1095 over 4 days
1095 over 2 days
Mr.
Comey is selling books. He sells hardcover books for $4 each and paperback books for $2
each. If he sold 8 books and made $26, how many of each type did Mr. Comey sell?
Answer:
5 hardcover and 3 paperback books
Step-by-step explanation:
h=hardcover books
p=paperback books
h + p = 8 ==> since the number of hardcover and paperback books add up
to 8 total books
4h + 2p = 26 ==> since the cost of each hardcover book is $4 and
the cost of each paperback book is $2, and the
total cost of all 8 books bought is $26
(h + p = 8) * 2 ==> multiply the equation by 2 to get 2p from p.
2h + 2p = 16 ==> simplify
4h + 2p = 26
- (2h + 2p = 16)
4h-2h + 2p-2p = 26-16 ==> subtract the two equations to isolate h
2h + 0 = 10 ==> simplify
2h = 10
h = 5 ==> divide both sides by 2
5 + p = 8 ==> substitute h=5 into the equation h + p = 8
p = 3 ==> subtract 5 on both sides to isolate p
Hence, 5 hardcover and 3 paperback books were sold.
Use the four intervals shown. I. (0, 2) II. (0, 2]III. [0, 2) IV. [0, 2]Based on the extreme value theorem, on which interval must the function f(x) = xhave both a maximum and minimum value?O IO IIO IIIO IV
Each interval (a, a), [a, a), and (a, a] represents the empty set, whereas [a, a] denotes the singleton set {a}
What do you meant by interval?
The empty set is represented by the intervals (a, a), [a, a), and (a, a, while the singleton set an is denoted by [a, a]. All four notations are often taken to denote the empty set when a > b. Both notations may be used in conjunction with other mathematical parentheses and brackets.For instance, in analytic geometry, linear algebra, and set theory, the notation (a, b) is frequently used to represent an ordered pair, the coordinates of a point or vector, or (sometimes) a complex number.Bourbaki created the notation]a, b[to signify the open interval for this reason. For ordered pairings, the notation [a, b] is also occasionally used, particularly in computer science.a ) [0, 2]
The empty set is represented by the intervals (a, a), [a, a), and (a, a, while the singleton set an is denoted by [a, a].
b ) iv
If a function f(x) is continuous on a closed interval [ a, b], then f(x) has both a maximum and minimum value on [ a, b].
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The center of the equilateral triangle below is at the origin. Which of the following rotational symmetries apply to the equilateral triangle? Rotation Applies to the figure?
Rotational symmetry of 120 about the origin
Yes/No
Rotational symmetry of 180 about the origin
Yes/No
Yes. An equilateral triangle has rotational symmetry of 120 and 180 degrees about the origin.
What is triangle?A triangle is a three-sided polygon, with three angles and three sides. It is one of the most basic and fundamental shapes in geometry, and is used in many different areas of math and science. Triangles are also often used in architecture, art, and other areas of design. Triangles come in many different forms, such as equilateral, isosceles, and scalene, and can be classified by their angles, sides, or both.
This means that the triangle can be rotated by 120 degrees or 180 degrees around the origin and still maintain its shape.
Yes, rotational symmetry of 120 about the origin applies to the equilateral triangle. This means that the triangle can be rotated by 120 degrees about the origin and the resulting shape will be exactly the same as the original triangle.
No, rotational symmetry of 180 about the origin does not apply to the equilateral triangle. This is because an equilateral triangle has rotational symmetry of 60 degrees about the origin, not 180 degrees.
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Among all the vectors in Rn whose components add up to 1, find the vector of minimal length. In the case n=2, explain your solution geometrically.
The unit circle has a radius of 1, so it is the shortest distance from the origin to (x1,x2).
The vector of minimal length in Rn whose components add up to 1 is called the unit vector. The unit vector is calculated by dividing each component of the vector by the sum of all components. Thus, in Rn, the unit vector is given by (x1/s, x2/s, ..., xn/s), where s is the sum of all components.
In the case n=2, the unit vector is given by (x1/s,x2/s). Geometrically, this vector is the vector from the origin to the point (x1,x2) on the unit circle. This vector has minimal length because the unit circle has a radius of 1, so it is the shortest distance from the origin to (x1,x2).
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-7t 42t the position of an object moving along a line is given by the function . find the average velocity of the object over the following intervals.
(a) Average velocity over [1,5] is -21. (b) Average velocity over [1,4] is -14. (c) Average velocity over [1,3] is -7. (d) Average velocity over [1, 1+h] is -7t + 70 where h is any real number.
Calculating average velocity requires the calculation of the change in position divided by the change in time.
For a: The change in position is -70 and the change in time is 4. So, the average velocity is -70/4 = -17.5.
For b: The change in position is -56 and the change in time is 3. So, the average velocity is -56/3 = -18.67.
For c: The change in position is -35 and the change in time is 2. So, the average velocity is -35/2 = -17.5.
For d: The change in position is -7t and the change in time is h. So, the average velocity is -7t/h = -7t + 70.
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A complete question: The position of an object moving along a line is given by the function s(t) = -7t^2 +70t. Find the average velocity of the object over the following intervals.
(a) [1,5]
(b) [1,4]
(c) [1,3]
(d) [1, 1+h] where h > 0 is any real number
