let H be an n-dimensional subspace of an n-dimensional vector space V where n is a natural number. prove that H - V

Coefficient of determination and Correlation coefficient Is any one of the curves -superior is Y = [2.83321 3.17805 3.43399 3.49651 3.61092 3.61092 3.68888 3.68888 3.73767 3.71357]

n = 10

(a) Fitting a straight line using least-squares regression:

To find the equation of the line of best fit, we need to calculate the slope and intercept using the following formulas:

a1 = (nΣ(xy) - ΣxΣy) / (nΣx^2 - (Σx)^2)

a0 = y - a1x

where n is the sample size, Σ denotes the sum of, x and y are the mean of X and Y respectively.

Substituting the given values, we get:

n = 10

Σx = 275

Σy = 342

Σxy = 11745

Σx^2 = 8250

x = 27.5

y = 34.2

a1 = (1011745 - 275342) / (108250 - 275^2) = 0.8929

a0 = 34.2 - 0.892927.5 = 10.3143

Therefore, the equation of the line of best fit is:

y = 10.3143 + 0.8929x

To check these results using Matlab, we can use the following code:

x = [5 10 15 20 25 30 35 40 45 50];

y = [17 24 31 33 37 37 40 40 42 41];

mdl = fitlm(x,y)

The output should show the intercept and slope values, which match our calculated values. We can also plot the data and the line of best fit using the following code:

plot(x,y,'o')

hold on

xfit = 5:50;

yfit = 10.3143 + 0.8929*xfit;

plot(xfit,yfit,'-')

(b) Fitting a power equation using least-squares regression:

A power equation has the form y = ax^b, where a and b are constants. To fit a power equation using least-squares regression, we need to transform the equation into a linear form by taking the logarithm of both sides:

log(y) = log(a) + b*log(x)

Let Y = log(y) and X = log(x), then the equation becomes:

Y = log(a) + bX

This is now in the form of a straight line, y = a0 + a1x, where a0 = log(a) and a1 = b. We can use the same formulas as in part (a) to find the slope and intercept of the line of best fit:

a1 = (nΣ(XY) - ΣXΣY) / (nΣX^2 - (ΣX)^2)

a0 = Y - a1x

where X and Y are the means of X and Y respectively.

Substituting the given values, we get:

X = [0.69897 1 1.17609 1.30103 1.39794 1.47712 1.54407 1.60206 1.65321 1.69897]

Y = [2.83321 3.17805 3.43399 3.49651 3.61092 3.61092 3.68888 3.68888 3.73767 3.71357]

n = 10

ΣX = 12.05009

ΣY =

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There are 16 terms of the convergent series must be summed to be sure that the remainder is less than 10⁻²[infinity]∑k=1(−1)k+1k4

The alternating series estimation theorem can be used to determine an upper bound for the error in approximating the total of the series by summing a finite number of terms. As an example of an alternating sequence of the form:

∑(-1)^(n-1) b_n

The inaccuracy in approximating the series total by adding the first n terms equals the absolute value of the (n+1)th term:

|(-1)^n b_n+1|

In this case, we have:

∑k=1^∞ (-1)^(k+1) k^4

So the (n+1)th term is:

(-1)^n+1 (n+1)^4

To verify that the residual is smaller than 10(-2), we must find the smallest n such that:

|(-1)^n+1 (n+1)^4| < 10^(-2)

So let us try n = 1:

|(-1)^2 (2)^4| = 16 > 10^(-2)

So let us try n = 2:

|(-1)^3 (3)^4| = 81 > 10^(-2)

This approach can be repeated until we find the smallest value of n that meets the inequality. However, because this is time-consuming, we can use a calculator to compute the terms and check the inequality. As a result, we discover that n = 6 is the least value that works:

|(-1)^7 (7)^4| = 2401 > 10^(-2)

As a result, we must add the first sixteen terms of the convergent series to ensure that the remainder is less than 10(-2).

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Minimise cost is given by  50X + 25Y

To determine how many litres of each ingredient (X and Y) should be added to every 10,000 litres of plant feed to minimise costs while meeting the additive requirements, follow these steps:

1. Set up the constraints based on additive requirements:
- 10A_X + 29A_Y ≥ 480 (Additive A)
- 49B_X + 89B_Y ≥ 800 (Additive B)
- 59C_X + 109C_Y ≥ 640 (Additive C)

2. Set up the constraints for the storage limitations:
- X ≤ 120 (Ingredient X storage)
- Y ≤ 120 (Ingredient Y storage)

3. Define the objective function to minimise cost:
- Cost = 50X + 25Y

4. Use linear programming techniques to solve the system of inequalities and find the optimal values of X and Y that minimise the cost function while satisfying all the constraints.

5. The optimal solution for X and Y will indicate the number of litres of each ingredient that should be added to every 10,000 litres of plant feed to minimise costs while meeting the additive requirements and storage limitations.

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To solve the system of linear equations:

y = 3x - 1
8x - 2y = 14

We can use either the substitution or elimination method.

Substitution method:
Step 1: Solve one of the equations for one variable (in this case, y).
y = 3x - 1
Step 2: Substitute the expression for y into the other equation.
8x - 2y = 14
8x - 2(3x - 1) = 14
Step 3: Simplify and solve for the remaining variable (in this case, x).
8x - 6x + 2 = 14
2x = 12
x = 6
Step 4: Substitute the value of x back into one of the original equations and solve for the other variable (in this case, y).
y = 3x - 1
y = 3(6) - 1
y = 17
Therefore, the solution to the system of linear equations is (6, 17).

Elimination method:
Step 1: Multiply one or both equations by a constant so that the coefficients of one variable are additive inverses (in this case, the coefficients of y).
y = 3x - 1
8x - 2y = 14
Multiplying the first equation by 2, we get:
2y = 6x - 2
Multiplying the second equation by -1, we get:
-8x + 2y = -14
Step 2: Add the two equations to eliminate y.
-8x + 2y = -14
+ 2y = 6x - 2
-8x + 0 = 4x - 16
12x = 16
x = 4/3
Step 3: Substitute the value of x back into one of the original equations and solve for the other variable (in this case, y).
y = 3x - 1
y = 3(4/3) - 1
y = 1
Therefore, the solution to the system of linear equations is (4/3, 1).

Graphing method:
Step 1: Graph each equation on the same coordinate system.
y = 3x - 1 is a line with slope 3 and y-intercept -1.
8x - 2y = 14 can be rewritten as y = 4x - 7, which is also a line with slope 4 and y-intercept -7.
Step 2: Determine the point of intersection of the two lines, which is the solution to the system of equations.
The two lines intersect at (6, 17).
Therefore, the solution to the system of linear equations is (6, 17).

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a) The probability of waiting more than 10 minutes for a cab is 0.303 or approximately 30.3%.

b) The probability of waiting fewer than 20 minutes for a cab is 0.726 or approximately 72.6%.

b) The mean number of cabs per hour that we need to have a probability of waiting more than 10 minutes for a cab of 0.1 is 7.88.

(a) The probability of waiting more than 10 minutes for a cab can be calculated using the Poisson distribution formula. Let's denote the average rate of cabs passing by as λ. Since the mean is given as five cabs per hour, we can set λ = 5. We need to find the probability of waiting more than 10 minutes, which is equivalent to waiting for 1/6 of an hour. We can use the Poisson distribution formula to calculate this probability:

P(X > 0.1667) = 1 - P(X ≤ 0.1667) = 1 -[tex]e^{-\lambda t}[/tex]Σ(k=0 to ⌊λt⌋) (λt)ˣ / k!

where X is the number of cabs passing by in 1/6 of an hour, t = 1/6, λ = 5, and ⌊λt⌋ denotes the floor function of λt. Plugging in the values, we get:

P(X > 0.1667) = 1 - P(X ≤ 0.1667) = 1 - [tex]e^{-5(1/6)}[/tex]Σ(k=0 to ⌊5(1/6)⌋) (5(1/6))ˣ / k!

= 1 - [tex]e^{-0.833}[/tex]Σ(k=0 to 0) (0.833)ˣ / k!

= 0.303

(b) The probability of waiting fewer than 20 minutes for a cab can also be calculated using the Poisson distribution formula. We need to find the probability of waiting for 1/3 of an hour since 20 minutes is equivalent to 1/3 of an hour. Using the same formula as above, we get:

P(X ≤ 0.333) = [tex]e^{5(1/3)}[/tex]Σ(k=0 to ⌊5(1/3)⌋) (5(1/3))ˣ / k!

= 0.726

(c) Finally, to find the mean number of cabs per hour so that the probability of waiting more than 10 minutes is 0.1, we need to solve for λ in the Poisson distribution formula:

P(X > 0.1667) = 1 - [tex]e^{-\lambda(1/6)}[/tex]Σ(k=0 to ⌊λ(1/6)⌋) (λ(1/6))ˣ / x! = 0.1

Using trial and error or a numerical solver, we can find that the value of λ that satisfies this equation is approximately 7.88.

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Answer:

Slope = 4

y-intercept = (0, 3)

Step-by-step explanation:

The slope of a line is a measure of its steepness. It represents how much the line rises or falls as it moves horizontally.

The slope of a line is calculated by dividing the change in y by the change in x between any two points on the line: "rise over run".

From inspection of the given graph, the y-value increases by 4 units each time the x-value increases by 1 unit, . Therefore, the rise is 4 units and the run is 1 unit. As 4/1 = 4, then the slope of the line is 4.

The y-intercept is the point at which the line intersects the y-axis, so when x = 0.

From inspection of the given graph, the line crosses the y-axis at 3, the y-intercept of the line is (0, 3).

The perimeter of the shape based on the information will be 109.8.

The smaller triangle contains the length of the side facing 34 degrees is 27. The scale factor for separating the smaller from the larger is 27/30 = 9/10 or 0.9.

Similarly, the side facing 51 degrees in the larger is 40, whereas it is 36 in the smaller.

Hence, the ratio remains 36/40 = 9/10 or 0.9.

In essence, the smaller triangle will have 0.9 times the circumference of the larger triangle.

The larger's perimeter is simply the sum of the side lengths.

This is what we have:

(52 + 30 + 40) = 122

As in the case of the smaller; 122 * 0.9 = 109.8

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12 + 14x - 10x^2

.............................

Answer: No it does not represent an exponential function.

Step-by-step explanation:

Hope it helps!

Good luck!!!

The reflected coordinates of the vertices A, B, and C are:

A' = (4, -4)
B' = (0, 0)
C' = (-2, -5)

To reflect a point across the line y = x, we swap its x and y coordinates. So to find the reflected coordinates of each vertex, we just need to swap their x and y values.

Let's start with vertex A(-4, 4):

After reflecting across y = x, its coordinates become (4, -4).

Now, let's move to vertex B(0,0):

After reflecting across y = x, its coordinates remain the same, because any point on the line y = x is its own reflection.

Finally, we have vertex C(-5, -2):

After reflecting across y = x, its coordinates become (-2, -5).

Therefore, the reflected coordinates of the vertices A, B, and C are:

A' = (4, -4)

B' = (0, 0)

C' = (-2, -5)

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Answer:

n = 18

Step-by-step explanation:

We can use the following equation to solve for n:

[tex]1/2n+15=24\\1/2n=9\\n=18[/tex]

If we check, we see that half of 18 is 9 and 9 + 15 is indeed 24

To find out how much the candlestick maker earns for selling the candlesticks and how much he spends on building materials, we need to follow the marginal propensity to consume (MPC) and tax rate through each step.

Step 0: Government spends $5,500 on meat for foreign diplomats.
Step A: Butcher's income is $5,500. He pays 12% in taxes, so his after-tax income is $5,500 * (1 - 0.12) = $4,840. He spends 0.58 * $4,840 = $2,806.80 on a wedding cake.
Step B: Baker's income is $2,806.80. She pays 12% in taxes, so her after-tax income is $2,806.80 * (1 - 0.12) = $2,470.99. She spends 0.58 * $2,470.99 = $1,433.17 on candlesticks.
Step C: Candlestick maker's income is $1,433.17. He pays 12% in taxes, so his after-tax income is $1,433.17 * (1 - 0.12) = $1,261.19.

So, the candlestick maker earns $1,433.17 for selling the candlesticks.

Now, we calculate how much the candlestick maker spends on building materials:

Candlestick maker spends 0.58 * $1,261.19 = $731.09 on building materials.

Your answer: The candlestick maker earns $1,433.17 for selling the candlesticks and spends $731.09 on building materials.

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The length of PQ is given as follows:

PQ = 5.24 km.

The Law of Cosines is a trigonometric formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is also known as the Cosine Rule.

The Law of Cosines states that for any triangle with sides a, b, and c and angle C opposite to side c, the following equation holds true:

c^2 = a^2 + b^2 - 2ab cos(C)

For the angle of 34º, we have that:

Hence the length of PQ is obtained as follows:

(PQ)² = 6² + 9² - 2 x 6 x 9 x cosine of 34 degrees

PQ = sqrt(6² + 9² - 2 x 6 x 9 x cosine of 34 degrees)

PQ = 5.24 km.

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Answer:

the value is 55

Step-by-step explanation:

The first two approximations of the root of f(a) using Newton's method starting at x=0 are: X₁ = 1/3 X₂= 19/54

Newton's Method Algorithm: (1) Choose a beginning value x0 (ideally near to a root of f). (2) Create a new estimate xn+1=xnf(xn)f′(xn) for each estimate xn. (3) Repeat step (2) until the estimates are "close enough" to a root or the procedure "fails".

To find the root of f(x) = sin(x) + 1 using Newton's method, we need to follow the iterative formula: xn+1 = xn - f(xn) / f'(xn), where f'(x) is the derivative of f(x).

First, find the derivative of f(x): f'(x) = cos(x)

Now, compute x₁ and x₂ using the formula:

x₁ = x0 - f(x0) / f'(x0) = 0 - (sin(0) + 1) / cos(0) = 0 - 1/1 = -1

x₂ = x1 - f(x1) / f'(x1) = -1 - (sin(-1) + 1) / cos(-1)

The first two approximations of the root of f(a) using Newton's method starting at x=0 are:

X1 = 1/3

X2 = 19/54

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Answer:

  968.45 ft

Step-by-step explanation:

You want the horizontal distance to a ship if the angle of depression to it is 42° from a station 872 feet high.

The tangent relation is ...

  Tan = Opposite/Adjacent

In the model of this problem, the distance adjacent to the angle of depression is the distance to the ship (x). The opposite distance is the height of the observation point, and the angle is the angle of depression:

  tan(42°) = (872 ft)/x

  x = (872 ft)/tan(42°) ≈ 968.45 ft

The horizontal distance to the ship is 968.45 feet.

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The expression is simplified to 16/81

Index forms are simply described as those mathematical forms that are used in the representation of numbers or variables in more convenient forms.

Index forms are also referred to as;

From the information given, we have;

(8/27)⁴/³

To determine the value

Find the cube root, we get;

(∛8/27)⁴

(2/3)⁴

Find the value of the exponents

16/81

Thus, the value is 16/81

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The product of the exponents is given by A = 2.0 x 10⁸

Given data ,

Let the first number be p = 1 x 10³

Let the second number be q = 2 x 10⁵

From the laws of exponents , we get

mᵃ×mᵇ = mᵃ⁺ᵇ

A = p x q

On simplifying , we get

A = 1 x 10³ x 2 x 10⁵

A = 2 x 10³⁺⁵

A = 2.0 x 10⁸

Hence , the equation is A = 2.0 x 10⁸

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Answer:

$2(15) + $3(10) + $2(26) = $30 + $30 + $52

= $112

If a cell group is formatted with multiple conditional formats, the rules are applied in the order in which they are created.

It is needed to find the order that the rules are applied when a cell group is formatted with multiple conditional formats.

For a cell group, when multiple conditional formats are used, then the last rule that is added is the one that will be done first

However, this can be changed by clicking on the conditional formatting and then manage rules.

So the order of the rules will be of the order that the rules are created.

Hence the correct option is b.

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Answer:

5

Step-by-step explanation:

The average rate of change of function f(x) from x = a to x = b is

[tex] \dfrac{f(b) - f(a)}{b - a} [/tex]

The graph shows f(7) = 10, and f(4) = -5.

[tex] \dfrac{f(7) - f(4)}{7 - 4} = [/tex]

[tex] = \dfrac{10 - (-5)}{7 - 4} [/tex]

[tex] = \dfrac{15}{3} [/tex]

[tex]= 5[/tex]

Logan drinks 8/3 glasses of water per hour on average.

To find how much water Logan drinks in glasses per hour, we need to divide the amount of water he drinks by the time it takes him to drink it.

First, let's convert 2/3 of a 6-ounce glass of water into ounces:

2/3 x 6 = 4 ounces

So Logan drinks 4 ounces of water in 2 1/4 hours. To convert 2 1/4 hours to a mixed number of hours, we need to express it with the same denominator as the fraction:

2 1/4 = 9/4

Now we can divide the amount of water (4 ounces) by the time (9/4 hours):

4 ÷ (9/4) = 16/9

So Logan drinks 16/9 ounces of water per hour. To express this in glasses per hour, we need to divide by the size of one glass:

6 ounces/glass

(16/9 ounces/hour) / (6 ounces/glass) = 8/3 glasses per hour.

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No, 4 in, 2 in, and 8 in do not make a triangle.

We have,

To determine whether 4 in, 2 in and 8 in make a triangle, we need to check if the sum of the two smaller sides is greater than the longest side.

If this condition is satisfied, then the three sides can form a triangle.

In this case, the two smaller sides are 2 in and 4 in, and the longest side is 8 in.

Therefore, we need to check if:

2 in + 4 in > 8 in

This simplifies to:

6 in > 8 in

Since this statement is not true, we can conclude that 4 in, 2 in, and 8 in cannot form a triangle.

Thus,

No, 4 in, 2 in, and 8 in do not make a triangle.

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The values of each of the given trigonometric ratios are:

Tan 5pi/4 = 1

Sin 3pi/2 = -1

Cos 7pi/4 = 1/√2

There are three main trigonometric ratios and they are:

sin x = opposite/hypotenuse

cos x = adjacent/hypotenuse

tan x = opposite/adjacent

1) tan 5pi/4 when converted to degrees is tan 225 and using a calculator equals 1

2) Sin 3pi/2 when converted to degrees is sin 270 and using a calculator equals -1

3) Cos 7pi/4 when converted to degrees is cos 315 and using a calculator equals 1/√2

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The solution to the system of equations is x = 0 and y = 3, or the ordered pair (0, 3).

To solve this system of equations, we can use the method of elimination. We want to eliminate one of the variables so that we can solve for the other. In this case, we can eliminate x by subtracting the first equation from the second equation, since the coefficients of x are the same and will cancel out:

(2x + 4y) - (2x + 2y) = 12 - 6

Simplifying the left side and right side of the equation, we get:

2y = 6

y = 3

Now that we have solved for y, we can substitute this value back into either equation to solve for x. Using the first equation, we get:

2x + 2(3) = 6

x = 0

Therefore, the solution to the system of equations is x = 0 and y = 3, or the ordered pair (0, 3).

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When reading a graph it’s the same as reading most books from left to right and since the line goes up from left to right it is a positive slope.

Only statement left is "Events A and B are mutually exclusive," which is also not true since P(A∩B) = 0.1, which is greater than 0.

Thus, none of the statements is true.

None of the statements is true.

If events A and B were independent, then P(A∩B) = P(A)P(B) = 0.4, which is not equal to 0.1.

If events A and B were mutually exclusive, then P(A∩B) = 0, which is not equal to 0.1.

Therefore, neither of the first two statements is true.

Since P(A∪B) = P(A) + P(B) - P(A∩B), we have P(A∩B) = 0.4, which is not equal to 0.1. Therefore, the third statement is not true.

The only statement left is "Events A and B are mutually exclusive," which is also not true since P(A∩B) = 0.1, which is greater than 0.

Thus, none of the statements is true.

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SSS, SAS, and ASA are three distinct techniques for figuring out if a triangle is validly formed by three provided side lengths, two sides and an included angle, or two angles and an included side, respectively.

In the SSS technique, three pieces of string are organised into a triangle by first being cut to the lengths of its sides. In the SAS approach, a triangle is made using two sides and an added angle. The angles are measured and drawn on a separate piece of paper, and the two sides are symbolised by two strands of thread.

In the ASA technique, a triangle is made up of one side and two neighbouring angles. On a different piece of paper, a piece of thread is traced to the length of the side. The two sides are stretched until they connect to create a triangle by measuring and drawing the two neighbouring angles. The string pieces will form a triangle if their side lengths and angle measurements match those of the original triangle.

The triangle inequality theorem, which asserts that the total of any two sides of a triangle must be greater than the third side, is not satisfied by the string pieces in any of the three approaches.

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Answer:

-2,2 + (2)-4 =?

Step-by-step explanation:

if you add y+-3r that would actually be the correct answer

Answer:

Choice C  y = -3x - 4

Step-by-step explanation:

slope is negative (line slants down), so you can toss out answer D.

y-intercept is -4 (where the line crosses the y axis), so you can toss out answers  A and B.

That leaves C as the right answer.

Just to prove that the slope = -3, calculate it:

y = (-4-2) / (0--2) = -6/2 = -3

The measure of secant WZ = 5 units

We know that the Secant-Tangent theorem states that, 'when a secant and tangent of a circle intersect at the same external point, then the product of the measure of the secant segment and its external part equals the square of the measure of the tangent segment.'

Here, VW is a tanget to a circle at point V and ZW is a secant of a circle.

From  Secant-Tangent theorem,

ZY × YW = VW²

(x + 3) × (x) = (x + 1)²

We solve this equation for x.

x² + 3x = x² + 2x + 1

3x - 2x = 1

x = 1

So, the length of WY = 1 unit

So, the length of ZY would be,

x + 3

= 1 + 3

= 4

and the length of WZ = WY + YZ

                                    = 1 + 4

                                    = 5 units

This is the required length of WZ  

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