The vectors are linearly independent if and only if?

Based on the height of the ball suspended from a spring, the height of the ball at equilibrium is K feet.

The model for the height of the ball when it is suspended from a spring is shown as:

= aSin(b ( t - h)) + k

Given this, we can infer the value of h by looking at the general form of the Sin function which is:

y = Asin(Bx) + c

In the above:

A = amplitude

B = (2 x pi) / Period

C = midline

Comparing the function and the general Sin function, we notice that:

C = K

If C = K then it means that K is the height of the ball at its equilibrium because C as the midline represents the equilibrium height.

In conclusion, the height of the ball at equilibrium is K feet.

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Answer:

  (C)  0

Step-by-step explanation:

As always, the first step is to read and understand the question. You are asked to find the smallest number in the set {M₁, M₂, M₃}. Each of these values is defined to be the largest number in the corresponding sets S₁, S₂, S₃.

The numbers in each set are ordered smallest to largest, so the largest number in each set is the one on the right.

The largest number in S₁ is 8, so M₁ = 8.

The largest number in S₂ is 6, so M₂ = 6.

The largest number in S₃ is 0, so M₃ = 0.

This means the set M is ...

  M = {M₁, M₂, M₃} = {8, 6, 0}

The question asks for the smallest number in set M. Clearly, it is 0.

The smallest number in the set is 0.

The volume of the composite figure shown in the picture is equal to 375 cubic meters.

The volume is the space occupied by a solid, the volume of the composite figure is sum of the volumes of the two elements, the one is a square prism and the other, a triangular prism. The volume of the entire figure is:

V = (5 m)² · (13 m) + (1/2) · (5 m)² · (4 m)

V = 375 m³

The volume of the composite figure shown in the picture is equal to 375 cubic meters.

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The probability that 5 boys would be selected would be 5/6720

From the information given, the probability can be calculated using permutation

The formula for permutation is given as;

Permutation = [tex]\frac{n!}{n - r!}[/tex]

Where;

Substitute the values into the formula

Permutation = [tex]\frac{8!}{8 - 5!}[/tex] = [tex]\frac{8!}{3!}[/tex] = [tex]\frac{40, 320}{6}[/tex] = 6720

The probability that 5 boys would be selected is 5/6720

Thus, the probability that 5 boys would be selected would be 5/6720

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Using hypothesis test, we conclude that there is evidence to believe that the average household income has increased.

According to the question,

The average household income for a recent year (x₁⁻) = $30,000

The average household income for a recent year (x₂⁻) = $24,500

sample sizes n₁ = n₂ = 34

standard deviation of both samples were $24,500. Thus, s₁ = s₂ = $5928.

Null hypothesis:  There is no evidence to believe that the average household income has increased.

H₀ : μ₁ ≤ μ₂

Alternative hypothesis:  There is evidence to believe that the average household income has increased.

H₁: μ₁ > μ₂  

To check hypothesis we have the formula

Standard error=  [tex]\sqrt{\frac{s_{1} ^{2} }{n_{1} } + \frac{s^{2} _{2} }{n_{2} } }[/tex]

= [tex]\sqrt{2067128.4706}[/tex]

= 1437.7512

Thus, standard error is $1437.7512.

Formula for test statistic = [(x₁⁻ -  x₂⁻) - (μ₁ - μ₂)]/standard error

= [(30000 -  24500) - 0]/1437.7512

= 3.8254

The critical value of z at 5% level of significance is 1.6449.

Here, the calculated value is greater than the critical value so we reject H₀.

Hence using hypothesis test, we conclude that there is evidence to believe that the average household income has increased.

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The set of numbers is included in the solution set of the compound inequality is {-7, 5, 18, 24, 32}

Inequalities are expressions not separated by an equal sign.

From the given number line, the equivalent inequality expression are

x > 22 (22 s excluded since it is an open circle) and x≤ 18 (18 is included since the circle is shaded)

Hence the set of numbers is included in the solution set of the compound inequality is {-7, 5, 18, 24, 32}

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The set of numbers that is been  included in the solution set of the compound inequality is [tex]{-7, 5, 18, 24, 32}[/tex]

The inequality can be referred to as the relationship between two values which are not equal and cannot be separated by an equal sign.

From the open circle, Checking the number set, [tex]x > 22[/tex],  22 cannot be involved, from the shaded circle, x≤ 18 , 18 can be included.

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The sum of the pulses allows us to find that the answer for the sum at a point x is

If the waves travel in the same direction, their sum gives resulting waves that can be maximum or minimum depending on the phase between them, un stationary patterns.

If the directions are coincident directions, they add place to the interference processes, in this case, stationary patterns are formed on a screen.

If the waves travel in the opposite direction the sum of the two waves gives rise to a wave that maximum is the sum of the amplitudes of the two pulses

In the case of a pulse that corresponds to a single wavelength, the result is an instantaneous maximum,  which after the pulse passes, returns to zero.

Consequently, there is no stationary pattern, so there is no interference between the two pulses.

In conclusion with the sum of the pulses, we can find that the answer is No interference pattern is produced as the amplitude does not remain constant.

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The measure of angle E is (b) 50°

The given parameters are:

The similarity of the triangles implies that:

From the figure, we have:

angle A = 50

Substitute angle A = 50 in angle A = angle E

angle E = 50

Hence, the measure of angle E is (b) 50°

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The population mean is symbolized as μ. The sample mean is symbolized M.

A population in statistics, is a the total number of people of a group that shares similar characteristics that are of interest to a researcher. Example of population a researcher can understudy include:

A sample is a subset of a population, which is drawn using any sampling technique. An example of a sample is 100 randomly selected persons who patronize a shopping mall.

The population mean can be defined as the average data value of a particular group characteristics that is measured by the researcher. The symbol for population mean is: μ.

The sample mean can be defined as the average data value of the sub-group of an entire population that having characteristics that are of interest to a researcher. The symbol for sample mean is: M.

In summary, the population mean is symbolized as μ. The sample mean is symbolized M.

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The correct order to express cos3x in terms of cosx is; As shown in steps 1 to 9 below.

The steps to simplify the trigonometric identities cos 3x in terms of cos x are as follows;

Step 1; cos x = cos(2x + x)

Step 2; cos(2x) cosx - sin(2x) sinx

Step 3; 1 - 2sin^2x(cosx) -(2sinxcosx)sinx

Step 4; cosx - 2sin²x cosx - 2sin²x cosx

Step 5; cosx - 4sin²x cosx

Step 6; cosx( 1 - 4sin²x)

Step 7; cosx{1 - 4(1 - cos²x)}

Step 8; cosx{-3 + 4cos²x}

Step 9; 4cos³x - 3cosx

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The equation would find the distance between the two points is

[tex]\sqrt{(x_{2}-x_{1}) ^{2}+(y_{2}-y_{1}) ^{2} }[/tex]

According to the question,

The equation would find the distance between the two points is

[tex]\sqrt{(x_{2}-x_{1}) ^{2}+(y_{2}-y_{1}) ^{2} }[/tex]

If (x₁, y₁) and (x₂, y₂) be any point in the plane then the distance between the two points equation is [tex]\sqrt{(x_{2}-x_{1}) ^{2}+(y_{2}-y_{1}) ^{2} }[/tex]

Hence, the equation would find the distance between the two points is

[tex]\sqrt{(x_{2}-x_{1}) ^{2}+(y_{2}-y_{1}) ^{2} }[/tex]

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The value of  [tex]k = 2P/x^2[/tex]

We are aware that an object's location might cause it to store energy. When a bow and arrow are used, the energy that is stored when the bow is pulled is what gives the arrow its kinetic energy when it is released.

when a spring is moved away from its equilibrium position, it gains some energy, which we can notice by the tension our hands experience as we stretch it. Potential energy is a type of energy that arises from a change in its position or condition, according to our definition

We know that P=1/2kx^2

Now solving for k

2P = kx^2

Therefore

k = 2P/x^2

Hence the value of k is 2P/kx^2

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Answer:

[tex]\sqrt{\frac{15}{2} }[/tex] or 2.738

Step-by-step explanation:

Let’s just look at the triangle on the top with the [tex]\sqrt{10}[/tex] on the top and x on the bottom. (Basically the top half to the equilateral triangle)

There is a small square in the bottom right corner, which indicates that this triangle is a right triangle. This means that we can use the Pythagorean Theorem: [tex]a^{2} +b^{2} =c^{2}[/tex]

We know that \sqrt{10} is our hypotenuse, and therefore our c in our equation. Let’s say that x=a in our equation. Therefore we are left to find b. However, b is half the length of the side of the original equilateral triangle. An equilateral triangle means that all three sides are the same length. Therefore our side would also be \sqrt{10} units long. However we know that b is half of that value, so b=[tex]\frac{1}{2}(\sqrt{10})[/tex] or [tex]\frac{\sqrt{10} }{2}[/tex]

Plugging these values into the equation:

x^2+ (\frac{\sqrt{10} }{2})^{2}=\sqrt{10} ^{2}

[tex]x^{2}=\sqrt{10} ^{2}-\frac{\sqrt{10} }{2} ^{2}[/tex]

[tex]x^{2} =10-\frac{10}{4}[/tex]

[tex]x^{2} =\frac{15}{2}[/tex]

[tex]x=\sqrt{\frac{15}{2} }[/tex]

This approximately equals 2.738

Answer:

[tex]x=\sqrt{\frac{15}{2} }[/tex]. Another form of this answer is [tex]x=\frac{\sqrt{15} }{\sqrt{2} }[/tex](it's the same thing)

Step-by-step explanation:

Because the triangle is equilateral, all other sides of the triangle are also [tex]\sqrt{10}[/tex]. Also, x is the altitude of this triangle as it forms a right angle with the base and is therefore perpendicular. We know that this triangle is equilateral and in any triangle that has at least two sides that are equal(in other words isosceles), the altitude is also the median and angle bisector. A median would cut the side it reaches into half. So, x would cut the base into two equal parts. Since the base is [tex]\sqrt{10}[/tex], divide it by two and both halves are [tex]\frac{\sqrt{10}}{2}[/tex]. What we are left with is two right triangles, both with legs [tex]\frac{\sqrt{10} }{2}[/tex] and x. Using the Pythagorean Theorem, we know that [tex]a^{2}+b^{2}=c^{2}[/tex], with [tex]a[/tex] and [tex]b[/tex] being the legs and [tex]c[/tex] being the hypotenuse. So, we plug in the values and get [tex]\frac{\sqrt{10} }{2} ^{2}[/tex][tex]+x^{2}=\sqrt{10}^{2}[/tex]. Squaring, we get [tex]\frac{10}{4}+x^{2}=10[/tex]. This becomes [tex]x^{2}=10-\frac{10}{4}[/tex] which equals [tex]x^{2}=\frac{40}{4}-\frac{10}{4}[/tex]. Finally, [tex]x^{2}=\frac{30}{4}[/tex]. This is [tex]x^{2}=\frac{15}{2}[/tex]. Square rooting this we get [tex]x=\sqrt{\frac{15}{2} }[/tex]. Another form of this answer is [tex]x=\frac{\sqrt{15} }{\sqrt{2} }[/tex]

The value of the standard deviation is σ = 2.20. Using probability distribution, the required standard deviation is calculated.

The formula for the standard deviation of the given probability distribution is

σ = √∑([tex]x_i^2[/tex] × [tex]P(X_i)[/tex]) - μₓ²

Where the mean μₓ = ∑[[tex]x_i[/tex] × [tex]P(X_i)[/tex]]

It is given that,

x: $2, $3, $5, $10

P(X=x): 0.55, 0.26, 0.11, 0.08

Step 1: Calculating the mean:

we have μₓ = ∑[[tex]x_i[/tex] × [tex]P(X_i)[/tex]]

⇒ μₓ = 2 × 0.55 + 3 × 0.26 + 5 × 0.11 + 10 × 0.08

∴ μₓ = 3.23

Step 2: Calculating the standard deviation:

x: 2, 3, 5, 10

x²: 4, 9, 25, 100

P(X=x): 0.55, 0.26, 0.11, 0.08

([tex]x_i^2[/tex]) × [tex]P(X_i)[/tex]: 4 × 0.55 = 2.2; 9 × 0.26 = 2.34; 25 × 0.11 = 2.75; 100 × 0.08 =8

∑[([tex]x_i^2[/tex]) × [tex]P(X_i)[/tex]]: 2.2 + 2.34 + 2.75 + 8 = 15.29

Therefore,

The standard deviation, σ = √∑([tex]x_i^2[/tex] × [tex]P(X_i)[/tex]) - μₓ²

⇒ σ = [tex]\sqrt{15.29-(3.23)^2}[/tex]

      = [tex]\sqrt{15.29-10.43}[/tex]

∴ σ = 2.20

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The complete table of values for the function rule y = -2x +9 is

x -4 -2   0  2  4

y 17    13  9 5  1

Linear equations are equations that have constant average rates of change, slope or gradient

From the question, the function rule is given as:

y = -2x +9

The table of values is given as:

x -4 -2 0 2 4

y 17

Next, we substitute the other values of x in the equation y = -2x +9

y = -2(-2) +9 = 13

y = -2(0) +9 = 9

y = -2(2) +9 = 5

y = -2(4) +9 = 1

So, the complete table of values is

-4     -2   0  2  4

y 17    13  9 5  1

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Answer:

137

Step-by-step explanation:

VYXW is a parallelogram.

The opposite angles of a parallelogram are congruent, meaning their angle measurements are equal.

Following this information we can write an equation to find the value of x:

10x - 27 = 2x + 29 transfer like terms to the same side of the equation

10x - 2x = 29 + 27 add/subtract

8x = 56 divide both sides by 8

x = 7 replace x with 7 to find the given angle measures

now, the consecutive angles in a parallelogram are supplementary meaning their sum is equal to 180

m<V + m<W = 180

m<V + 2*7 + 29 = 180

m<V + 43 = 180 subtract 43 from both sides

m<V = 137

The solution to the system of inequalities is (-3.375, 1.75)

The system of inequalities is given as:

3y > 2x+12

2x+y ≤ -5

Next, we plot the graph of the system using a graphing tool

From the graph, both inequalities intersect at

(-3.375, 1.75)

Hence, the solution to the system of inequalities is (-3.375, 1.75)

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a. Surface area = 172 cm²

Volume = 112 cm²

b. The surface area would be needed.

c. The volume would be needed.

Surface area = 2(wl+hl+hw).

Volume = (length)(width)(height)

a. l = 7 cm

w = 2 cm

h = 8 cm

Surface area =2(wl+hl+hw) = 2·(2·7+8·7+8·2) = 172 cm²

Volume = (7)(2)(8) = 112 cm²

b. The surface area would be needed.

c. The volume would be needed to find the amount of juice the box will hold.

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Answer:

49°

Step-by-step explanation:

Right angled triangle:

This is a type of triangle with one of the angle as 90°

Sum of angles in a Right angled triangle is 90°

Therefore,

y + 41° = 90°

collecting like terms

y = 90 - 41 = 49°

Therefore,

y = 49°

Answer:

see explanation

Step-by-step explanation:

to find the intersection equate g(x) and h(x) , that is

(x + 7)(x - 5) = x - 5 ← expand left side using FOIL

x² + 2x - 35 = x - 5 ( subtract x - 5 from both sides )

x² + x - 30 = 0 ← in standard quadratic form

(x + 6)(x - 5) = 0 ← in factored form

equate each factor to zero and solve for x

x + 6 = 0 ⇒ x = - 6

x - 5 = 0 ⇒ x = 5

substitute these values into either g(x) or h(x) for corresponding values of y

substituting into h(x)

h(- 6) = - 6 - 5 = - 11 ⇒ (- 6, - 11 )

h(5) = 5 - 5 = 0 ⇒ (5, 0 )

Answer:

98, 27

Step-by-step explanation:

In the first figure, the two angles are supplementary, meaning that their sum would equal 180 degrees, giving the equation: b+82 = 180 --> b=98

In the second figure, the two angles are complementary, meaning that they add up to 90 degrees, giving the equation: b+63 = 90 --> b=27

So lets try to prove it,

So let's consider the function f(x) = x^2.

Since f(x) is a polynomial, then it is continuous on the interval (- infinity, + infinity).

Using the Intermediate Value Theorem,

it would be enough to show that at some point a f(x) is less than 2 and at some point b f(x) is greater than 2. For example, let a = 0 and b = 3.

Therefore, f(0) = 0, which is less than 2, and f(3) = 9, which is greater than 2. Applying IVT to f(x) = x^2 on the interval [0,3}.

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The surface area of the cone is: A. 184π units².

Surface area of a cone = πr(l + r), where r is the radius and l is the slant height.

Given the following:

Surface area of a cone = π8(15 + 8)

Surface area of a cone = 184π units²

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Answer:

x = 2

Step-by-step explanation:

                5x - 7 - 5(7x - 12) + 7 = 0

Distribute (-5) to each term of (7x - 12)

   5x - 7 + 7x*(-5) + (-12)*(-5) + 7 = 0

                5x - 7 - 35x + 60 + 7 = 0

                 5x - 35x - 7 + 60 +7 = 0

Combine like terms.

                                 -30x + 60 = 0

Subtract 60 from both sides

                                        -30x   = -60

Divide both sides by (-30)

                                             x   = (-60)/(-30)

                                               x = 2

Answer:

2

Step-by-step explanation:

Simplifying the Algebraic Equation.

( 5x - 7 ) -5 (7x - 12 ) + 7 = 0

Removing brackets.

5x - 7 - 35x + 60 + 7 = 0

Combining like terms.

5x - 35x + 60 + 7 - 7 = 0

- 30x + 60 = 0

- 30x = - 60

dividing bothsides by - 30

[tex] \frac{ - 30x}{ - 30} = \frac{ - 60}{ - 30} \\ \\ x \: = 2[/tex]

The Set C is the subset of A and D.

According to the statement

we have given that the three sets which are A = {2, 4, 6}, B = {2, 6}, C = {4, 6}, and D = {4, 6, 8}.

And we have to find that the subsets of these sets.

So,

For a set S to be a subset of a set T, all of the elements of the set S need to be contained in the set T. Firstly, all of the sets are their own subsets, that is, A⊆A,B⊆B,C⊆C,D⊆D.

It follows that B⊂A because of 2∈A and 6∈A. Since 2 is not belongs to C and D. that's why B is not a subset of C and B is not a subset of D.

Taking into account that 4∈A and 6∈A, we conclude that C⊂A. By analogy, since 4∈D and 6∈D, we conclude that C⊂D.

Since |A|>|B|∣A∣>∣B∣ and |A|>|C|,∣A∣>∣C∣, we conclude that A is not a subset of B and A is not a subset of C.

Taking into account that |D|>|B|∣D∣>∣B∣ and |D|>|C|,∣D∣>∣C∣, we conclude that D is not a subset of B and D is not a subset of C. Since,2 not belongs to D that's why A is not a subset of D.

Since 8 not belongs to A, D is not a subset of A.

Overall, The Set C is the subset of A and D.

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Question:

Suppose that A = {2, 4, 6}, B = {2, 6}, C = {4, 6}, and D = {4, 6, 8}. Determine which of these sets are subsets of which other of these sets.

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Considering right-angled triangle XYW, an approximate ratio WY/WX is equal to: A. 0.34.

Considering right-angled triangle XYW, we would apply the law of cosine to approximate the ratio WY/WX. Mathematically, the law of cosine is given by:

cos(θ) = Adj/Hyp

Where:

Substituting the given parameters into the formula, we have;

cos(θ) = WY/WX

cos(70) = 3.4/10

0.34 = 0.34.

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Complete Question:

Right triangles 1-2 and 3 are given with all their angle measures and approximate side lengths.

Use one of the triangles to approximate the ratio WY/WX

A. 0.34

B. 0.94

C. 1.06

D. 2.76​

Answer: 2x + 6y + 22 units

Step-by-step explanation:

The perimeter of a figure is the total length of all the sides of a figure.

Therefore, the perimeter can be found by adding up all the lengths of the figure.

Suppose that P is the perimeter of this figure, then

[tex]P = x+4+x+4+3y+7+7+3y\\[/tex]

Combining like terms, we get

[tex]P = 2x + 6y + 22[/tex]

Since we do not have any information on what the variables are, this is the perimeter of the figure.