determine whether the set s spans r2. if the set does not span r2, then give a geometric description of the subspace that it does span. s = {(−1, 4), (4, −1), (3, 3)}

Caroline should cut the fabric into 63 inches wide each

A word problem is a math problem written out as a short story or scenario. Basically, it describes a realistic problem and asks you to imagine how you would solve it using math.

For example if a number is added to 20 and gives 50. what is the numbers

x+20 = 50

x = 50-20

x = 30

The total width of the pieces of fabrics = 90 + 36 = 126 inches

To cut the pieces into two equal part with maximum width, we divide the total width by 2

= 126/2

= 63 inches

therefore the fabric should be cut into 63 inches wide each

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Note that the hint to demonstrating the above proof (that ΔABC ≅ ΔADE) is to compare the slopes of DE and BC. See the computation below.

The slope or gradient of a line in mathematics is a quantity that specifies both the direction and the steepness of the line.

Comparing both slopes, we have:

Slope of DE = 2-1/4-5 = 1/-1 = -1

Slope of BC = 3-1/6-8 = 2/-2 = -1

Since the slope of DE and BC are the same an they are not intersection, it means they are parrallel lines.

Since DE || BC

In ΔABC and ΔADE

∠DAE = ∠BAC

∠ADE ≈ ∠ABC (Correspoinding Angle)

∠AED = ∠ACB (Corresponding Angles)

Thus,  ΔABC ≅ ΔADE.

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Full Question:

Given A(2,1), B(6,3), C(8,1), D(4,2) and E(5,1), shown below, prove that ΔABC ≅ ΔADE

The domain of the function f(x) =  {- 3, - 1, 0 2, 4}.

Suppose we have an ordered pair (x, y) then the domain of the function is the set of values of x and the range is the set of values of y for which x  is defined.

Given, The graph represents a relation where x represents the independent variable and y represents the dependent variable.

The ordered pairs are given by, (- 3, 4), (- 1, 1), (0, - 2), (2, 0), and (4, - 1).

We know the domain of the function are all the x coordinates.

Therefore, The domain of the function f(x) = {- 3, - 1, 0 2, 4}.

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The measure of the acute base angle is 69°

The measure of the obtuse base angle is 111°

It is a two-dimensional figure which has three sides and the sum of the three angles is equal to 180 degrees.

We have,

Isosceles triangles mean two sides are equal and the angles opposite to the equal sides are also equal.

Now,

Isosceles trapezoid

Vertex angle = 42

This means,

The other two angles will be the same.

x + x + 42 = 180

2x = 180 - 42

2x = 138

x = 69

Now,

The opposite angles of an isosceles trapezoid are supplementary.

So,

Obtuse base angle + Acute base angle = 180

Obtuse base angle + 69 = 180

Obtuse base angle = 180 - 69 = 111

Thus,

Acute base angle = 69°

Obtuse base angle = 111°

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The equation of the tangent line to the graph of the function f(x)=x 1x−4 at the point (0,−14) is 4x + y +14 = 0.

To find the tangent equation we need slope and intercept for the equation.

To find slope we need to differentiate the given function as the differentiation at a given point gives us the slope :

firstly differentiate the given function :

    f(x)=x(x−4)

         = x^2 - 4x

f'(x) = 2x - 4

f'(x) at (0,-14) = 2(0) - 4  

                     = -4

Hence , the slope is -4

Use the formula to find the equation of the tangent line that passes through a point (x1, y1) with the slope m, that is, y-y1=m(x-x1).

y-y1=m(x-x1) at a point (0,−14)

y - (-14) = -4 (x-0)

y+14 = -4x

therefore, 4x + y +14 = 0 is the required equation of the tangent.

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The height of the water tank given the volume, length and width is 40 cm

Volume of the water tank= 40 liters

Length of the water tank= 50 cm

Width of the water tank= 20 cm

Height of the water tank= x

convert liters to centimeters

1 liter= 1000 cm

40 liters= 40,000 cm

Volume of the water tank= length × width × height

40,000 = 50 × 20 × x

40,000 = 1000x

divide both sides by 1000

x = 40,000 / 1000

x = 40 cm

Therefore, the height of the water tank is 40 cm

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If the diameter of the pool is 24ft, the area is 452.16 ft²

Here we have a circular pool and we want to find its area.

Remember that the area of a circle of radius R is given by:

A = pi*R²

Where pi = 3.14

Here we want to define the diameter as a value between 12 and 24 ft, so I will use 24 ft.

The radius is half of that, then we have:

R = 24ft/2 = 12ft

Then the area of the pool is:

A = 3.14*(12 ft)² = 452.16 ft²

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The investment in 2% rate of interest was $2800 and the investment in 3% rate of interest was $3200.

Let the amount invested in 2% interest rate be x. So, the remaining invested amount = (6000 - x). The simple interest will be calculated by the formula -

S.I. = P×R×T/100, where S.I. is simple interest, P is principal, R is rate of interest and T is time.

Also, the sum of interest in each is $152. So, finding the individual interest first.

SI = x × 2 × 1/100

SI = 2x/100

SI = (6000 - x) × 3 × 1/100

SI = (18000 - 3x)/100

Finding the value of x.

2x/100 + (18000 - 3x)/100 = 152

Multiplying the equation with 100

2x + 18000 - 3x = 15200

3x - 2x = 18000 - 15200

x = 2800

Investment in 3% rate = 6000 - 2800

Investment in 3% rate = $3200

Thus, the investment was $2800 and $3200.

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The volume of the prism not occupied by the sphere is 704.69 cubic in.

Given that, the radius of the sphere=4 in. and the dimensions of a rectangular prism length=9 in. width=8 in. and height 10 in.

We need to find what is the volume of the prism not occupied by the sphere.

The formula to find the volume of the rectangular prism is Volume=Length×Width×Height.

The formula to find the volume of the sphere is Volume=4/3 πr³.

Now, the volume of the rectangular prism=9×8×10=720 cubic in.

The volume of the sphere=4/3×3.14×4³

V=15.31 cubic in.

Then, the volume of the prism not occupied by the sphere=The volume of the rectangular prism-The volume of the sphere

The volume of prism=720-15.31=704.69 cubic in.

Therefore, the volume of the prism not occupied by the sphere is 704.69 cubic in.

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The geometric relation between the vectors v and w are:

The vectors are in the same direction.

Now, According to the question:

Geometrical Interpretation of Scalar Product

From the scalar product formula, we have a.b = |a| |b| cos θ = |a| proj→b(→a) proj b → ( a → ) , that is, the scalar product of vectors a and b is equal to the magnitude of vector a times the projection of a onto vector b.

Vectors describe movement with both direction and magnitude. They can be added or subtracted to produce resultant vectors. The scalar product can be used to find the angle between vectors.

Now, vw =-||v|| ||w||

a.b = |a| |b| cos θ = |a| proj→b(→a) proj b → ( a → )

vw =-||v|| ||w|| cos θ = |v| proj →w(→v) proj w → ( v → )

Hence, The vectors are in the same direction.

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Poisson distribution is a discrete probability distribution that describes the number of events that occur randomly in a fixed interval of time or space.

a. The probability that the number of drivers will be at most 10 is calculated as the cumulative distribution function of Poisson distribution at 10, which is

 [tex]P(X \leq 10) = \sum_{i}=0^{10 }e^{-\mu} *\frac{\mu^{i }}{i!}[/tex]

where i ranges from 0 to 10.

b. The probability that the number of drivers will exceed 20 is calculated as

[tex]P(X > 20) = 1 - P(X \leq 20) = 1 - \sum_i=0^{20} e^{(-\mu)} *\frac{ \mu^i }{i!}[/tex]

where i ranges from 0 to 20.

c. The probability that the number of drivers will be between 10 and 20, inclusive is calculated as

[tex]P(10 \leq X \leq 20) = \sum_i=10^{20} e^{(-\mu)} *\frac{\mu^i}{i! }[/tex]

where i ranges from 10 to 20. The probability that the number of drivers will be strictly between 10 and 20 is calculated as

[tex]P(10 < X < 20) = \sum_i=10^{19} e^{(-\mu)} * \frac{\mu^i }{i!}[/tex]

where i range from 10 to 19.

d. The standard deviation of Poisson distribution with parameter μ = 20 is [tex]\sqrt{(\mu)} = \sqrt{(20)} = 2\sqrt5[/tex]

The probability that the number of drivers will be within 2 standard deviations of the mean value is calculated as

[tex]P(\mu - 2 * \sqrt{(\mu)}\leq X\leq \mu + 2 * \sqrt{(\mu)}) = \sum_i=16^{24} e^{(-\mu)} * \frac{\mu^i}{i!}[/tex]

where i ranges from 16 to 24.

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The complete question is :

Suppose that the number of drivers who travel between a particular origin and destination during a designated time period has a Poisson distribution with parameter μ =20 (suggested in the article “Dynamic Ride Sharing: Theory and Practice,” J. of Transp. Engr., 1997: 308–312). What is the probability that the number of drivers will

a. Be at most 10?

b. Exceed 20?

c. Be between 10 and 20, inclusive? Be strictly between 10 and 20?

d. Be within 2 standard deviations of the mean value?

The horizontal distance jade she covered is approximately 2.0 km.

Let x be the horizontal distance,

Given,

Viola drives 2 kilometers up a hill that makes an angle of 9 with the horizontal.

Thus, the angle between the horizontal and the direction of his walking = 9°, and The angle between the horizontal and vertical = 90°

By making the diagram of the given situation,

The angle between vertical and the direction of his walking = 180°(9+90°)=180° - 99° = 81°

By the law of sine,

[tex]\frac{sin90}{2}=\frac{sin 81}{x}[/tex]

x*sin 90°=2*sin 81°

x= 1.97537668119 ≅ 2.0

Hence, the horizontal distance jade she covered is approximately 2.0 km.

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The central angle of the sector to the nearest tenth of a square inch is 473.2 sq.

To find the area of the sector, we can use the formula:

Area = (θ/360) * π * r²

where θ is the central angle of the sector in degrees, and r is the radius. To find the central angle, we can use the formula:

Arc length = (θ/360) * 2 * π * r

So,

Arc length = (θ/360) * 2 * π * 12 = 9

θ/360 * 2 * π * 12 = 9

θ/360 = 9 / (2 * π * 12)

θ = (9 / (2 * π * 12)) * 360

Approximating π as 3.14, we get:

θ = (9 / (2 * 3.14 * 12)) * 360 = 75.39

To the nearest tenth of a degree, this is 75.4 degrees.

Next, we can use the first formula to find the area:

Area = (75.4 / 360) * 3.14 * 12² = 473.2 sq in

To the nearest tenth of a square inch, this is 473.2 sq in.

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The central angle of the sector to the nearest tenth of a square inch is 473.2 sq.

To find the area of the sector, we can use the formula:

Area = (θ/360) * π * r²

where θ is the central angle of the sector in degrees, and r is the radius. To find the central angle, we can use the formula:

Arc length = (θ/360) * 2 * π * r

So,

Arc length = (θ/360) * 2 * π * 12 = 9

θ/360 * 2 * π * 12 = 9

θ/360 = 9 / (2 * π * 12)

θ = (9 / (2 * π * 12)) * 360

Approximating π as 3.14, we get:

θ = (9 / (2 * 3.14 * 12)) * 360 = 75.39

To the nearest tenth of a degree, this is 75.4 degrees.

Next, we can use the first formula to find the area:

Area = (75.4 / 360) * 3.14 * 12² = 473.2 sq in

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The correct conversion factors to indicate the number of feet in one mile is: 1 mi = 5280 ft.

To convert miles to feet, we must use the equality: 1 mi = 5280 ft.

Using the table of equalities, we can see that 1 mi = 5280 ft, so the correct conversion factor to indicate the number of feet in one mile is 1 mi = 5280 ft.

To convert miles to feet, we use the conversion factor of 1 mile being equal to 5280 feet. This means that for every 1 mile, there are 5280 feet. To convert a specific number of miles to feet, we multiply the number of miles by the conversion factor of 5280 feet/mile. For example, if we want to convert 2 miles to feet, we multiply 2 by 5280 to get 10560 feet. Hence, 2 miles is equal to 10560 feet. This conversion factor can be used to convert any number of miles to feet, as long as we use it consistently and accurately.

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Answer:

y = 2x

Step-by-step explanation:

We know

They earn $2 for each cup of lemonade they sell. So, the equation is

y = 2x

A joule is one unit of energy, while a kilojoule is 1000 joules.

KJ x 1,000 = J.

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The actual distance in miles between the campsite and the trail is 9 miles.

Map can be defined as  a symbolic depiction emphasizing relationships between elements of some space such as objects and regions.

To convert from centimeters on the map to miles, we use the scale of 1 centimeter represents 5 miles.

So, 1.8 centimeters on the map represents 1.8 * 5 = 9 miles in real life.

Therefore, The actual distance in miles between the campsite and the trail is 9 miles.

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Answer:

Step-by-step explanation:

Equivalent rates can be used to compare different sets of quantities that have the same value. A rate that compares a quantity to one is called a unit rate. The unit rate has a denominator equal to one when written as a fraction. Unit rates can be used to find larger equivalent rates.In a unit rate, the denominator is always 1. So, to find unit rate, divide the denominator with the numerator in a way that the denominator becomes 1. For example, if 50km is covered in 5.5 hours, the unit rate will be 50km/5.5 hours = 9.09 km/hour

The measure of the indicated angle in each triangle is ∠U=80° and ∠Q =93°

The measure of each triangle's specified angle.

Comprehensive explanation:

Calculate the indicated angle's measurement.

Given: The additional two triangle angles.

Solution:

We are aware that a triangle's three angles add up to 180 degrees.

So,

The measure of the indicated angle in each triangle.​

1st TRIANGLE

∠U+∠V+∠T =180°

∠U+63°37° =180°

∠U= 180°-100°

∠U=80°

2ND TRIANGLE

∠A+∠B+∠C=180°

46°+90°+∠C =180°

∠C=44°

The measure of the indicated angle in each triangle.​

3rd TRIANGLE

∠R+∠P+∠Q= 180°

36°+51°+∠Q =180°

∠Q =93°

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Discrete mathematics is focused on discrete objects and the relationships between them, while calculus is focused on continuous change and motion.

Discrete mathematics and calculus are two branches of mathematics that deal with different types of mathematical objects and concepts. Discrete mathematics studies discrete objects, such as integers, graphs, and algorithms, while calculus deals with continuous and smooth objects and their derivatives and integrals.

Discrete mathematics provides the mathematical foundations for computer science, especially in areas such as algorithm design, data structures, and complexity theory.

In contrast, calculus is a branch of mathematics that focuses on continuous change, and provides a foundation for many other branches of mathematics and science, including physics, engineering, and economics.

In discrete mathematics, concepts are studied in a finite and countable manner, meaning that the objects and structures being studied can be counted and identified.

For example, in graph theory, discrete mathematical structures like vertices and edges are studied and their properties are analyzed. In discrete mathematics, the focus is on finding relationships between discrete objects and how they can be manipulated, such as in combinatorics and number theory.

On the other hand, calculus deals with the study of change and motion, specifically the rate at which objects change and the accumulation of these changes over time.

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The average rate of change for a city population of the city was growing steadily and can be used to predict future population growth.

The average rate of change for a city can be calculated by subtracting the initial population (23,000 in 2010) from the final population (25,000 in 2015) and then dividing that number by the total number of years (five). The equation for this calculation is:

(Final Population - Initial Population) / (Number of Years)

Using this equation, the average rate of change for the city is calculated as follows:

(25,000 - 23,000) / 5 = 2,000 / 5 = 400

Therefore, the average rate of change for the city between 2010 and 2015 was 400 people per year. This means that on average, the population of the city increased by 400 people each year during this time interval. This calculation shows that the population of the city was growing steadily and can be used to predict future population growth.

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The equation which would help determine the cost of 21 glue sticks is 5 / 9.65 = 21 / x, where x is the cost of 21 glue sticks.

Proportions are defined as the concept where two or more ratios are set to be equal to each other.

Suppose we have two ratio p : q and r : s.

If both these ratios are proportional, then we can write it as p: q : : r : s.

This is same as p : q = r : s or p/q = r/s

Given that,

Cost of 5 glue sticks = $9.65

Let x be the cost of 21 glue sticks.

Using proportional concept,

5 / 9.65 = 21 / x

Hence the equation is 5 / 9.65 = 21 / x

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The inverse of the function defined by the ordered pairs are (7, -49), (8, -56), (9,-63), (10, 70)

An inverse is a function that serves to “undo” another function. That is, if f(x) produces y, then putting y into the inverse of the function produces the output x. x. A function that has an inverse is called invertible, and the inverse is denoted by f⁻¹.

Given that, a set of ordered pairs defines a function.

{(-49,7), (-56,8), (-63,9), (-70,10)}

We need to find its inverse function,

Since, we know that,

The inverse of a function means that the input becomes the output of the function and the output becomes input.

When the function is given in the form of ordered pairs, the first element of each ordered pair represents the input and the second element represents the output.

Therefore,

The inverse will be:

{(7, -49), (8,-56), (9,-63), (10, -70)}

Hence, the inverse of the function defined by the ordered pairs are (7, -49), (8, -56), (9,-63), (10, 70)

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6 cm, 8 cm, 10 cm meaurement repreent the ide length of a right triangle.

A right triangle, like the triangle below, is one in which one of the angles measures 90 degrees (a right angle). In most cases, right angles are shown by drawing a square at the angle's vertex.

For a right triangle, where c is the hypotenuse and a and b are the smaller sides, the Pythagorean Theorem gives us a2 + b2 = c2.

The hypotenuse is usually the longest. Any triple would still represent the sides of a right triangle if we multiplied it by a constant. Therefore, the sides of a right triangle would be 6, 8, and 10.

a2 + b2 = c2.

= 6^2 + 8^2  =  10^2

36 + 64 = 100.

so,

6 cm, 8 cm, 10 cm meaurement repreent the ide length of a right triangle.

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The total surface area of the triangular pyramid is 170.4 inches².

The area of a three dimensional object on it's outer surface is called the surface area of the object.

A triangular pyramid has a triangular base and three triangular faces.

Given that, base of a triangular pyramid is an equilateral triangle. Each side of the base measures 12 in. The area of the base is 62.4 in².

Area of the base = 62.4 inches²

Slant height = 6 inches

Length of edge of each of the three triangular faces = 12 inches

Area of a triangular face = [tex]\frac{1}{2}[/tex] × 12 × 6 = 36 inches²

Area of three triangular faces = 3 × 36 inches² = 108 inches²

Total surface area = 108 inches² + 62.4 inches² = 170.4 inches²

Hence 170.4 inches² is the total surface area of the pyramid.

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The probability of failure after 1 year of use will be approximately 109/1,000,000 = 0.000109 or 0.0109%.

The failure rate (fpmh) is the number of failures per million hours. If a device is used for 5 hours per day, every day, it will be used for 365 days * 5 hours/day = 1,825 hours in a year.

The probability of failure after 1 year of use can be calculated as:

Failure rate * total hours used = 0.06 fpmh * 1,825 hours = 0.06 * 1,825 failures = 109.35 failures

So, there will be approximately 109 failures after 1 year of use. Since the failure rate is per million hours, and there were 1,825 hours of use, the probability of failure after 1 year of use will be approximately 109/1,000,000 = 0.000109 or 0.0109%.

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The required answer is she needs to number each plant from 1 to 60 and randomly select  10 plant. The correct option is C

Probability can be defined as a number that reflects the chance or likelihood that a particular event will occur.

As she has 60 seeds, firstly she must number each plant in order or randomly see the height, After that he can peek at random 10 plants to test the fertilizer.

The required answer is she needs to number each plant from 1 to 60 and randomly select 10 plant.

A greenhouse owner wants to test. She has 60 violet seedlings that were grown for 8 weeks. She wants to test the new fertilizer on 10 of the plants and decides to use a random number table to select a simple random sample. Which of the following correctly labels the population of violets Is to be determined.

The correct option is C

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For, -3 |2x + 6| = -12 the value of x = -1 and - 5

Correct option is: x = -1 and x = -5. This can be solved using the concept of equation.

Equations simultaneously, or a system of equations Multiple equations in algebra must be solved simultaneously. There must be an equal number of equations and unknowns for a system to have a singular solution. When two or more equations use the same variables, they are said to be in a system of equations. Graphing, substitution, and elimination are the three techniques used to solve systems of equations.

-3 |2x + 6| = -12

or, 2x + 6 = 4

or, 2x + 6 = ± 4

or, 2x + 6 = 4

or, 2x = -2

or, x = -1

on the other hand,

2x + 6 = -4

or, 2x = - 10

or, x = -5

So, the value of x = -1, and -5

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To achieve a 95% confidence level and ±1.5% precision, you would need a sample size of approximately 3845 people.

To estimate the proportion of people who feel that unemployment compensation should be expanded with 95% confidence to within ±1.5%, you would need to determine the sample size necessary to achieve this level of precision.

The formula for sample size estimation in proportion estimation is given by:

[tex]n = (Z^2 * p * (1 - p)) / E^2[/tex]

Where:

[tex]n = (1.96^2 * 0.5 * 0.5) / 0.015^2 = 3844.83[/tex]

Therefore, to achieve a 95% confidence level and ±1.5% precision, you would need a sample size of approximately 3845 people.

In conclusion, to estimate the proportion of people who feel that unemployment compensation should be expanded with 95% confidence to within ±1.5%, a sample size of approximately 3845 people is necessary. This calculation assumes that the estimated proportion of people in favor of expanding unemployment compensation is 0.5, and that the Z-score for a confidence level of 95% is 1.96.

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