(b) The value of f^(-1) is g.
(c) The value of f ◦ f ◦ f = f ◦ (f ◦ f) is f((f ◦ f)(x)) = f((x * (x-1))*((x * (x-1)) - 1)).
According to the question
(a) To prove that f is bijective, we need to show that it is both injective (one-to-one) and surjective (onto).
Injectivity: Let x, y ∈ a such that f(x) = f(y). Then, x * (x-1) = y * (y-1). Multiplying both sides by x-1, we get x = y, meaning that f is injective.
Surjectivity: Let z ∈ a. Then, z = z * (z-1) / (z-1), which means that there exists an x ∈ a such that f(x) = z. This shows that f is surjective.
Therefore, f is bijective.
(b) To find the inverse of f, we need to find a function g: a → a such that f(g(x)) = x and g(f(x)) = x for all x ∈ a.
Consider the function g: a → a defined by g(x) = x / (x-1) for all x ∈ a.
Let x ∈ a. Then, f(g(x)) = f(x / (x-1)) = (x / (x-1)) * ((x / (x-1)) - 1) = x. Similarly, g(f(x)) = g(x * (x-1)) = x * (x-1) / (x * (x-1) - 1) = x. This shows that f and g are inverses.
Therefore, f^(-1) = g.
(c) To find f ◦ f ◦ f, we first evaluate f ◦ f and then evaluate f of the result.
Let x ∈ a. Then, (f ◦ f)(x) = f(f(x)) = f(x * (x-1)) = (x * (x-1)) * ((x * (x-1)) - 1). Hence, f ◦ f ◦ f = f ◦ (f ◦ f) = f((f ◦ f)(x)) = f((x * (x-1)) * ((x * (x-1)) - 1)).
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25 Points for the answer to this question
The measure of angle B is m∠B = 71°
How to solveIf ABCD is a kite, then ∠A and ∠C must be equal to each other.
∠A = ∠C, so ∠C = 87°.
ABCD is a quadrilateral, so all of its interior angles must add up to 360°. We know 3 of the 4 angles, so we'll add them up and subtract the answer from 360.
∠A + ∠C + ∠D ⇒ 87 + 87 + 115 = 289
360 - 289 = 71
m∠B = 71°
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given f(x)=2*sqrt(9-x), what is the x coordinate of the point on the curve that is closest to the origin
As a result, the x-coordinate of the point on the curve closest to the origin is 2.
What is coordinate?Coordinates are a pair of integers that are used to locate a point or object in a two-dimensional plane. A point's location on a 2D plane is defined by two integers called the x-coordinate and the y-coordinate. The distance of a point from the y-axis scaled with the x-axis is known as its abscissa or x coordinate. The ordinate is the distance of a point from the x-axis scaled with the y-axis. Coordinates are the abscissa and ordinate combined. To find the coordinates of a point in a coordinate system, do the inverse. Start at the point and trace a vertical line up or down to the x-axis. There you have your x-coordinate.
Here,
The x-coordinate of the point on the curve that is closest to the origin can be found by minimizing the distance between the point on the curve and the origin. The distance between two points (x1, y1) and (x2, y2) is given by the Pythagorean theorem:
d = √((x2 - x1)^2 + (y2 - y1)^2)
For the origin and a point on the curve, we have x1 = y1 = 0, so the distance between the origin and a point (x, y) on the curve is:
d = √(x^2 + y^2)
Using the equation of the curve, we have y = 2 * √(9 - x), so the distance can be expressed as:
d = √(x^2 + (2 * √(9 - x))^2)
= √(x^2 + 4(9 - x))
= √(x^2 + 36 - 4x)
To find the x-coordinate of the point that minimizes d, we can use calculus. Setting the derivative of d with respect to x equal to zero and solving for x, we obtain:
d' / dx = 2x - 4 = 0
x = 2
So the x-coordinate of the point on the curve that is closest to the origin is 2.
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What is the difference between minimum and infimum?
The terms minimum and infimum are both related to the concept of lower bounds in mathematics. A minimum is the smallest value in a set of numbers, whereas an infimum is the greatest lower bound of a set.
A minimum is a value that is less than or equal to all other values in the set, while an infimum is a value that is less than or equal to all other values in the set, but not necessarily the smallest value.
For example, if we consider the set {2, 4, 6}, the minimum is 2 and the infimum is also 2. However, if we consider the set {2, 4, 6, 8}, the minimum is 2 and the infimum is 4. In the latter set, 4 is the greatest lower bound, meaning that all other values in the set are greater than or equal to 4.
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A pyramid and a prism both have heights 8.2 cm and congruent hexagonal bases with area 22. 3 cm cube. What is the ratio of their volumes?
The ratio of their volumes is 3: 1
What is the volume ratio between them?If two solids are identical, their volume ratio is equal to the cube of their corresponding side ratio. (It should be noted that volume is a 3-D measurement, not a “length” measurement. When the base and height of a pyramid and a prism are the same, their volumes are always in the ratio of 1:3Volume of prism.
A triangular prism and a triangular pyramid with identical bases and heights with the volume of the prism being three times that of the pyramid. Students should use manipulatives such as water, rice, or beans to physically represent this connection.
The volume of a prism =Base Area× Height and Volume of Pyramid = 1/3 × base area × Height
Given:
Height of both pyramid and a prism is 8.2 cm.
Area of base is 22.3 cm cube.
Volume of Prism = 22.3 × 8.2
Volume of Pyramid = 1/3 × 22.3 × 8.2
Ratio of volume of prism and volume of pyramid = (22.3 × 8.2) : 1/3 × (22.3 × 8.2)
Ratio of volume of prism and volume of pyramid = 1: 1/3
Ratio of volume of prism and volume of pyramid = 3: 1
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Please put in correct order
Steps are:
Step: 1
Given,
AE ≅ EB
∠DAE ≅ ∠CBE
Step: 2
congruent angles added to congruent angles form congruent angles
∠DAB ≅ ∠CBA
Step: 3
corresponding parts of congruent Triangles are congruent
DA ≅ CB
Step 4:
Vertical angles are congruent
∠DEA ≅ ∠CEB
Step 5:
Reflexive property
AE ≅ EB
Step 6:
In a triangles, angles opposite of congruent sides are congruent
∠EAB ≅ ∠EBA
Step 7:
SAS property
ΔDAB ≅ ΔCBA
Step 8:
ASA property
ΔDEA ≅ ΔCEB
What is congruent angles?Angles which are equivalent with one another are referred to as congruent angles. As a result, these angles have the same measure. In general, not all congruent angles are supplementary angles. Angles must be additional in order to add up to 180°. Due to the fact that they have the same measure and add up to 180, only right angles and supplementary angles are congruent.
Correct order is as follows:
Step: 1
Given,
AE ≅ EB
∠DAE ≅ ∠CBE
Step: 2
congruent angles added to congruent angles form congruent angles
∠DAB ≅ ∠CBA
Step: 3
corresponding parts of congruent Triangles are congruent
DA ≅ CB
Step 4:
Vertical angles are congruent
∠DEA ≅ ∠CEB
Step 5:
Reflexive property
AE ≅ EB
Step 6:
In a triangles, angles opposite of congruent sides are congruent
∠EAB ≅ ∠EBA
Step 7:
SAS property
ΔDAB ≅ ΔCBA
Step 8:
ASA property
ΔDEA ≅ ΔCEB
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The consumer expenditure, in dollars, for a commodity is the product of its market price, p, and the number of units demanded, x. Suppose that for a certain commodity, the consumer expenditure is given by
15,000p − 125p^2.
a)Factor this in order to find an expression for the number of units demanded. (Simplify your answer completely.)
Answer:
Step-by-step explanation:
Trade in the Vedic Age led to
the rise of kingdoms because of the importance of controlling land and long-distance travel routes
the building of an extensive system of roads because of the difficult terrain
the creation of a large banking class because of the need to exchange different currencies
the development of a centralized government that controlled the economic decisions of its people
Trade in the Vedic Age led to: A, the rise of kingdoms because of the importance of controlling land and long-distance travel routes
What happened in the Vedic Age?The Vedic Age was a period between 1500 BC and 600 BC. This period was characterized by the next major civilization that occurred in ancient India following the decline of the Indus Valley Civilization by the year 1400 BC. The Vedas were very composed in this age.
Thus, we can say that in this vedic age;
- Indian civilization saw an unprecedented increase.
- Kingdoms were formed that gained riches and prominence.
These kingdoms mostly formed because increased trade in the area gave rise to the need to be controlled and profited from.
In conclusion, option A is correct.
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Answer:
Trade in the Vedic Age led to: A, the rise of kingdoms because of the importance of controlling land and long-distance travel routes
Step-by-step explanation:
just did the quiz lol
Archimedes went to sleep beside a big rock. He wanted to get up at 7 77 AM, but the alarm clock was yet to be invented! He decided to sleep at the spot where the rock's shadow should end when it's 7 77 AM so as to be awakened by the direct sunlight. Archimedes knew that at 7 77 AM, the sunlight reaches the ground at an angle of 3 1 ∘ 31 ∘ 31, degree. The rock beside which he slept was 5 55 meters tall. How far from the rock did Archimedes go to sleep? Round your final answer to the nearest hundredth
The distance from the rock was 11.26 meters.
The formula used to calculate the distance is:
d = h/tan(α)
Where d is the distance, h is the height of the rock, and α is the angle of the sun at 7 77 am.
Using this formula, the distance from the rock can be calculated as follows:
d = 5.55/tan(31∘)
d = 11.26 meters
Rounding the answer to the nearest hundredth, the distance from the rock is 11.26 meters.
Archimedes knew that the sunlight reaches the ground at an angle of 3 1 ∘ 31∘ 31, degree at 7 77 AM. He also knew that the rock beside him was 5 55 meters tall. Using this information, he was able to calculate the distance he needed to go in order to be awakened by the direct sunlight. The formula used to calculate the distance is d = h/tan(α), where d is the distance, h is the height of the rock, and α is the angle of the sun. After plugging in the values, the distance from the rock was calculated to be 11.26 meters. Rounding the answer to the nearest hundredth, the distance from the rock was 11.26 meters.
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Match each set of values of a function to the pair of points that satisfy the function.
After matching each set of values of a function to the pair of points that satisfy the function, we get
1-42-23-14-3What is a Function?A function from a set X to a set Y assigns to each element of X exactly one element of Y.
Given the set of data
(4,3) and (6.7) rate of change = 2, and initial value = -5
k= 7-3/6-4 = 2, y=2x-5
(2,10) and (5, 19) rate of change=3, and initial value = 4
k= 19-10/5-2 = 9/3 = 3, y=3x+4
(2,5) and (5.2)→ rate of change = -1, and initial value = 7.
k= 2-5/5-2 = -3/3 = -1, y= -x+7
(2.4) and (4,3) →→ rate of change = -0.5, and initial value =5
k= 3-4/4-2 = -1/2 =-0.5, y=-0.5x+5
A function has two sets of values: the set of input values and the set of output values. The set of input values makes up the function's domain. The set of output values makes up the function's range.
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Drag the tiles to the correct boxes to complete the pairs. Match each set of values of a function to the pair of points that satisfy the function. rate of change = -1, and initial value = 7 rate of change = 2, and initial value = -5 rate of change = -0.5, and initial value = 5 rate of change = 3, and initial value = 4 (4, 3) and (6, 7) arrowRight (2, 10) and (5, 19) arrowRight (2, 5) and (5, 2) arrowRight (2, 4) and (4, 3) arrowRight
. if y1(x) and y2(x) are solutions of y″ p(x)y′ q(x)y = r1(x) and y″ p(x)y′ q(x)y = r2(x), show that y(x) = y1(x) y2(x) is a solution of y″ p(x)y′ q(x)y = r1(x) r2(x)
We can prove this using the product rule. Let y(x) = y1(x) y2(x). Then:
y'(x) = y1'(x) y2(x) + y1(x) y2'(x)
y''(x) = y1''(x) y2(x) + 2y1'(x) y2'(x) + y1(x) y2''(x)
Substituting into the equation yields:
y'' p(x)y' q(x)y = [y1''(x) y2(x) + 2y1'(x) y2'(x) + y1(x) y2''(x)] p(x) [y1'(x) y2(x) + y1(x) y2'(x)] q(x) [y1(x) y2(x)]
= [y1''(x) p(x)y1'(x) + 2y1'(x) p(x)y2'(x) + y1(x) y2''(x) p(x)] q(x) [y1(x) y2(x)]
= [r1(x) + r2(x)] q(x) [y1(x) y2(x)]
= r1(x)r2(x) q(x) [y1(x) y2(x)]
Since this is equal to y″ p(x)y′ q(x)y, it follows that y(x) = y1(x) y2(x) is a solution of y″ p(x)y′ q(x)y = r1(x) r2(x). We can prove this using the product rule. Let y(x) = y1(x) y2(x). Then:
y'(x) = y1'(x) y2(x) + y1(x) y2'(x)
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a landscape architect is planning a flower garden. she decides to arrange the garden in rows, and for each row, the number of plants will increase by 3. she has a total of 70 plants and plans for 5 rows. how many plants will be on the first row? responses
The number of plants in the first row is 8, which corresponds to option "b. 8 plants".
The concept used in this problem is arithmetic sequence or arithmetic progression. An arithmetic sequence is a sequence of numbers in which each term is the previous term plus a constant. In this problem, the constant is 3, and the number of plants in each row increases by 3. By using the formula for the sum of an arithmetic series, the total number of plants in all the rows can be calculated.
Let's assume that the first row has x number of plants.
The total number of plants in all rows is 70, so we can write an equation using x to represent the number of plants in each row:
x + (x + 3) + (x + 6) + (x + 9) + (x + 12) = 70
5x + 30 = 70
5x = 40
x = 8
So, the number of plants in the first row is 8, which corresponds to option "b. 8 plants".
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The number of plants in the first row is 8, which corresponds to the option "b. 8 plants".
The concept used in this problem is an arithmetic sequence or arithmetic progression. An arithmetic sequence is a sequence of numbers in which each term is the previous term plus a constant. In this problem, the constant is 3, and the number of plants in each row increases by 3. By using the formula for the sum of an arithmetic series, the total number of plants in all the rows can be calculated.
Let's assume that the first row has x number of plants.
The total number of plants in all rows is 70, so we can write an equation using x to represent the number of plants in each row:
x + (x + 3) + (x + 6) + (x + 9) + (x + 12) = 70
5x + 30 = 70
5x = 40
x = 8
So, the number of plants in the first row is 8, which corresponds to the option "b. 8 plants".
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Judy uses 7.5 pints of white paint and blue paint to paint her bedroom walls.
3/4 of this amount is white paint, and the rest is blue paint. How many pints of blue paint did she use to paint her bedroom walls?
Step-by-step explanation:
3/4*7.5
5.625
7.5-5.625
1.875
of blue pints
how many acres in a square mile
One square mile contains 640 acres. 640 acres in a square mile.
What does acres mean?The size of an acre is equal to 0.4047 hectares (4,047 square metres). The term "acre" has its origins in the usual area that could be ploughed in a single day with a yoke of oxen pushing a wooden plough. It is derived from Middle English "aker" (from Old English "aecer") and is similar to Latin "ager" ("field").The circumference of the two-acre plot, if it were two square acres placed side by side, would be 1252.26 feet, or almost four and a half times around, or a mile. The circumference of the two-acre area, which is 50 feet by 1742.4 feet, is 3584.8 feet, or around one and a half times around, or a mile.To learn more about acres refer to:
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Bill need to build a rectangular harp pen. The pen mut have a perimeter of 24 meter
The cost of the fence used to make the rectangular sheep pen is £57.6. The result is obtained by multiplying the perimeter by the fencing price.
How to calculate the cost for fencing a building?The cost for fencing a building can be calculated by multiplying the price of a certain size of fence by the perimeter or the length side to be fenced.
Bill builds a rectangular sheep pen.
Perimeter, P = 24 mFencing cost = £1.20 per half meterFind the total cost of the fence!
The total cost of the fence will be
= P × fencing cost
= 24 m × £1.20 / ½ m
= 24 × £2.40
= £57.6
Hence, the fence to make the sheep pen will cost £57.6.
Your question is incomplete. But most probably your full question was
Bill needs to build a rectangular sheep pen. The pen must have a perimeter of 24 m. Every half meter of fencing costs him £1.20. Work out the cost of the fence used to make the sheep pen!
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What is the difference between sound and unsound deductive reasoning? Explain your answer in 3 - 4 sentences.
The sound and unsound deductive reasoning is differentiated below.
What is deductive reasoning?Deductive reasoning is a logical approach where you go from general ideas to specific conclusions.
Given is the difference between sound and unsound deductive reasoning.
Sound deductive reasoning -
If an argument is both valid and has all true premises, we will say that the argument is sound.
Unsound deductive reasoning -
An argument is unsound if it either has a false premise, or is invalid.
Therefore, the sound and unsound deductive reasoning is differentiated above.
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3/5x - 1/3x = 42 find x
The value of x in the given equation, 3/5x - 1/3x = 42, is x = 157.5
Calculating the value of xFrom the question, we are to determine the value of x in the given equation.
The given equation is
3/5x - 1/3x = 42
To determine the value of x,
First, we will eliminate the denominators.
Multiply through by 15
15 × 3/5x - 15 × 1/3x = 15 × 42
3 × 3x - 5 × x = 630
9x - 5x = 630
4x = 630
Divide both sides by 4
4x/4 = 630/4
x = 157.5
Hence, the value of x is 157.5
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Vector A -2.00 +1.00 y and vector B -3.00 x + 4.00 V.What is vector =....?A) 1.00x+5.00yB) -100x+-3.00yC) 1.00x+3.00yD) -100x+3.00yE) -100x+-5.00y
The vector C is 1.00 x + 3.00 y
Therefore, the correct answer is C) 1.00 x + 3.00 y.
In mathematics and computer science, a vector is an ordered list of numbers that can be used to represent things like position, direction, and magnitude. Vectors can be added and subtracted, and they can be multiplied by scalars (numbers).
To subtract two vectors, we need to subtract the corresponding components. So, the x-component of vector C is
(-3.00 x) - (-2.00 x) = -3.00 + 2.00
= 1.00
and the y-component of vector C is
(4.00 y) - (1.00 y) = 4.00 - 1.00
= 3.00.
Therefore, vector C = 1.00 x + 3.00 y.
--The question is incomplete, answering to the question below--
"Vector A = -2.00x + 1.00y and vector B = -3.00x + 4.00y .What is vector C = B - A?
A) 1.00x+5.00y
B) -100x+-3.00y
C) 1.00x+3.00y
D) -100x+3.00y
E) -100x+-5.00y"
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If you can buy one can of pineapple chunks for $2 then how many can you buy with $10?
(round to the nearest whole number)
Magnesium has an HCP crystal structure, a c/a ratio of 1. 624, and a density of 1. 74g/cm^3. Compute the atomic radius for Mg
The c/a ratio is divided by the density of the material and then multiplied by the square root of two thirds. This gives the atomic radius of the magnesium atom.
The atomic radius of an atom can be calculated by using the c/a ratio and the density of the material. The c/a ratio is the ratio between the lattice parameter c and lattice parameter a. The density is the amount of mass per unit volume of a substance. The formula for calculating the atomic radius of magnesium is
Atomic Radius= (c/a)√((2/3)*density)
Using the provided data for magnesium, the c/a ratio is 1.624 and the density is 1.74 g/cm^3. Plugging these values into the formula above gives an atomic radius of 1.6708 angstroms. To calculate the atomic radius, the c/a ratio of the crystal structure for magnesium is used, as well as the density of the material. The c/a ratio is divided by the density of the material and then multiplied by the square root of two thirds. This gives the atomic radius of the magnesium atom.
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what is 2 times 2 Thank you
Answer:
2 • 2 = 4 because you multiply numbers by adding groups of a number!
Answer:
Step-by-step explanation:
2x2=4
a smaller star shape has a diameter of 3.6 and a length of x. the same shape but bigger has a diameter of 5.76 and a length of 8.32. find x
Therefore , the solution of the given problem of ratio comes out to be
x has length of 5.2 .
It establishes a ratio.The simple formula "a / b" can be used to create a pair of similar variables "a" and "b," where "b" may also be greater than zero. One ratio is created by combining two ratios. If there were only one man and three women, the ratio would be 1:1. There are 1/4 boys and 3/4 girls in the group. A separation between a or more objects, a numeral, or a piece of a component's volume are all examples of parts.
Here,
Given :
A figure of star shape
having diameter of 3.6 and length x
and other star having same shape but
having diameter 5.76 and length 8.32
Since , both the figures are same ,
Thus , there length ratio will also be same.
=> 8.32/5.76 = x / 3.6
=> 832/576 = 10x/36
=> 832/576 * 36 /10 = x
=> x = 1.44 * 3.6 =5.2
Therefore , the solution of the given problem of ratio comes out to be
x has length of 5.2 .
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if all else remains the same, what happens to a confidence interval if the sample standard deviation gets smaller?
The confidence interval narrows as the sample standard deviation gets smaller.
The width of a confidence interval depends on the sample size, the confidence level, and the sample standard deviation. The formula for a confidence interval is given as (sample mean - z-score * (sample standard deviation/square root of sample size)) to (sample mean + z-score * (sample standard deviation/square root of sample size)), where z-score depends on the confidence level.
Since the sample standard deviation is in the denominator of this formula, a smaller sample standard deviation results in a smaller value in the denominator and thus a narrower interval.
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Two teams need to raise $100 each for their end-of-the-year field trips. Team A wants to sell
popcorn at the Spring Fling Carnival, and Team B wants to sell cotton candy. It costs $15 to rent
a popcorn machine and it costs $25 to rent a cotton candy maker. The cost of additional
supplies for the popcorn is $0.05 per bag. The additional cost for the cotton candy is $0.10 per
stick. Team A will sell the bags of popcorn for $0.50 each. Team B will sell the cotton candy for
$0.75 per stick
question:
5: What is the least number of items Team A will need to sell to avoid losing money? pls explain!!!
Answer:
approximately 256 popcorn bags
Step-by-step explanation:
the cost of popcorn machine is $15. So we should add it to 100$
Then it is 115$. This is the amount that team A need
The cost of additional supplies for the popcorn is $0.05 per bag & team A will sell the bags of popcorn for $0.50 each. then the profit that the team A can gain by selling a bag is,
[tex]0.50 - 0.05 = 0.45[/tex]
Then,
[tex] \frac{115}{0.45} = 255.55[/tex]
what is 23.9*square root of 0.7124 divided by 3.877*52.18
Answer:
271.498934853
Step-by-step explanation:
Answer:
The answer is 0.039.
Step-by-step explanation:
If the equation of a line-of-best fit is y = -3x +4, what must be true about the scatter plot?
explain why an elevation less than -5 feet represents a distance from sea level greater than 5 feet
The elevation represents a distance from the sea level greater than 5 feet, because the original description specifies a distance greater than 5 feet below sea level.
What is Vector and Scalar quantity?Physical quantities like mass and electric charge are examples of scalar quantities because they only have magnitudes. In contrast, a vector quantity is a physical quantity like force or weight that has both magnitudes and directions.
The elevation is the height or distance above sea level and is expressed as a distance.
Given that it contains both magnitude (distance) and direction, elevation in the previous description is a vector (above sea level).
Distance is a scalar quantity with only one magnitude that describes the area travelled by a moving object.
A height of less than 5 feet denotes a location that is less than 5 feet above sea level, which is equivalent to a location that is more than 5 feet below sea level.
Greater than 5 feet below sea level = less than -5 feet above sea level, therefore;
A distance from (below) sea level of more than 5 feet is represented by an elevation of less than -5 feet.
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For which values of t is each set linearly independent?
(a) S = {(t, 1, 1), (1, t, 1), (1, 1, t)}
(b) S = {(t, 1, 1), (1, 0, 1), (1, 1, 3t)}
S = {(t, 1, 1), (1, 0, 1), (1, 1, 3t)} is linearly independent of t.
Therefore the answer is b) S = {(t, 1, 1), (1, 0, 1), (1, 1, 3t)}.
(a) S is linearly independent for all values of t if the following equation has a non-trivial solution:
a(t, 1, 1) + b(1, t, 1) + c(1, 1, t) = (0, 0, 0).
Expanding this equation, we get:
at + b + c = 0
a + bt + c = 0
a + b + ct = 0.
Solving this system of linear equations, we find that a = b = c = 0. This implies that any linear combination of the vectors in S results in the zero vector, so S is linearly dependent for all values of t.
(b) S is linearly independent if the following equation has a non-trivial solution:
a(t, 1, 1) + b(1, 0, 1) + c(1, 1, 3t) = (0, 0, 0).
Expanding this equation, we get:
at + b + 3ct = 0
b = 0
a + c = 0.
Since a and c cannot both be zero, this system has no trivial solution, which means that S is linearly independent for all values of t.
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. Find the area for the following figure.
20.3 cm
6.2 cm
The area of the triangle is 62.93 cm²
Area of the triangle:Triangle is a 2-dimensional closed figure that is formed by 3 sides. The area of the triangle is the region that is bounded by the three sides of the triangle. The formula for the area of the triangle is given by
Area of triangle = (1/2) × base × height
Here we have
a triangle upside down
The base of the triangle = 20.3 cm
Height of the triangle = 6.2 cm
From the formula,
Area of the triangle = 1/2 base × height
Area of the triangle = (1/2) × 20.3 × 6.2
= 20.3 × 3.1
= 62.93 cm²
Therefore,
The area of the triangle is 62.93 cm²
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[Coin Changing] Let An = {a1,a2,...,an} be a set of distinct coin types (e.g., a1 = 50 cents, a2 = 25 cents, a3 = 10 cents, etc). Note that ai may be any.
// Returns an array C[1..n] of the number of coins of each value, where
// the sum from 1 to n of C[i]*A[i] will equal t.
Greedy-Coins( t, A )
Define array C as an array of length n with all values zero for i = 1 to n
C[i] = t / A[i] // Note use INTEGER arithmetic here
t = C[i] mod A[i] // Note using integer modulo here to get the remainder
end fo ... ro
return C
An easy counterexample would be to use V = ( 5, 3, 1 ), and t= 9. The greedy algorithm will return values ( 1, 0, 4), but a little intuition shows that the array ( 0, 3, 0 ) will use fewer coins.
c) If we consider the array A = ( kn-1, kn-2, ..., k0), and k >0, then we can see that for any k < (n-1), there will be at most (k-1) coins of that value; if there had been sufficient \"value\" remaining to select k coins o that value, the greedy algorithm would have selected one more coin of the next coin up. Another way to look at this is that for any selection by the greedy algorithm from ki ... k0, the value MUST be less than ki+1, since ki+1 is by definition a multiple of k.
Complete question:
Let An = { a1, a2, ..., an } be a finite set of distinct cointypes (e.g., a1= 50 cents, a2= 25 cents, a3= 10 cents etc.). We assume each ai is an integer and that a1 > a2 > ... > an. Each type is available in unlimited quantity. The coin-changing problem is to make up an exact amount C using a minimum total number of coins. C is an integer > 0.
(a) Explain that if an != 1 then there exists a finite set of coin types and a C for which there is no solution to the coin-changing problem.
(b) When an = 1 a greedy solution to the problem will make a change by using the coin types in the order a1, a2, ..., an. When coin type ai is being considered, as many coins of this type as possible will be given. Write an algorithm based on this strategy.
(c) Give a counterexample to show that the algorithm in (b)doesn\'t necessarily generate solutions that use the minimum total number of coins
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3 find the distance from the point (1; 2; 3) to the line that contains the two points (1; 3; 2) and (5;
The distance from the point for the line is (0; -1; 1).
A line is a set of points that lie on the same straight path, extending in opposite directions without end.
To find the distance from the point (1; 2; 3) to the line that contains the two points (1; 3; 2) and (5; 7; 4), we can use the vector formula.
First, we need to find the direction vector of the line by subtracting the first point from the second point:
=> (5; 7; 4) - (1; 3; 2) = (4; 4; 2).
Next, we need to find a vector that connects the point to the line, which is
=> (1; 2; 3) - (1; 3; 2) = (0; -1; 1).
Finally, the magnitude of this cross product is the distance from the point to the line.
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