Solve the simultaneous equation x-y=5 and 2x-y=13​

Answer:

43,5311

Step-by-step explanation:

Cos A= 6²+10²-7²/2×6×10

this is the cosin theorem in triangle

Answer:

For n = 1, 2, 3,.....,

[tex] a(n) = \frac{10n(n + 1)}{2} + 1 = 5n(n + 1) + 1 [/tex]

[tex] = 5 {n}^{2} + 5n + 1[/tex]

Harold had originally 100 pencils.

Let X be the number of pencils Harold had originally. After giving 17 to his friend Carl, he had X-17 left. Then, after giving 35 to his teacher, he had X-17-35 left, which is equal to 48.

Therefore, we can set up an equation: X-17-35=48. Solving for X, we get X=100 pencils.

Thus, Harold originally had 100 pencils. To check, we can verify that after giving 17 to Carl and 35 to his teacher, he is left with 48, which matches the information given in the problem.

This problem can also be solved by using algebraic equations, but since there are only two steps involved, we can use simple arithmetic to arrive at the answer.

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The required, D. A study of the popularity of a new sneaker conducted by the sneaker manufacturer is more likely to produce results that are skewed by bias.

This is because the sneaker manufacturer has a vested interest in promoting the new sneaker and may use biased methods to ensure that the study produces favorable results. For example, the manufacturer may selectively sample individuals who are more likely to be interested in the sneaker or use leading questions to bias the responses in favor of the sneaker. In contrast, the other studies listed are less likely to be influenced by bias because they rely on random sampling or objective measurements.

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The expression cos(α+β)/sinβ can be simplified using trigonometric identities to give tanαcotβ+1/cosαcotβ−sinαsinα+cosαtanβcotαcosβ+sinβcosβ−tanαsinβ.

To simplify the expression cos(α+β)/sinβ, we first use the trigonometric identity cos(A+B) = cosAcosB - sinAsinB to obtain:

cos(α+β) = cosαcosβ - sinαsinβ

Substituting this into the numerator of the original expression, we have:

cosαcosβ - sinαsinβ -------------------      sinβ

We then use the identity cotθ = 1/tanθ to simplify the expression further. This gives: tanαcotβ + 1 --------------cosαcotβ - sinαsinβ/sinβ

We can then use the identities tanθ = sinθ/cosθ and cotθ = cosθ/sinθ to obtain:

sinαcosβ + cosαsinβ--------------------cosαcosβsinβ - sinαsinβsinβ

Simplifying this expression gives:

sin(α+β)---------cosαsinβ

Finally, we use the identities sin(A+B) = sinAcosB + cosAsinB and cos(A+B) = cosAcosB - sinAsinB to obtain:

sinαcosβ + cosαsinβ--------------------cosαcosβsinβ - sinαsinβsinβ

= sin(α+β)/(cosαsinβ)

Therefore, we have successfully simplified the expression cos(α+β)/sinβ to tanαcotβ+1/cosαcotβ−sinαsinα+cosαtanβcotαcosβ+sinβcosβ−tanαsinβ.

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Answer:

D. Yes

Step-by-step explanation:

The geometric Transformation shown in the line of music is: Reflection

In mathematics, a geometric transformation is referred to as any bijection of a set to itself (or perhaps to another such set) possessing some salient geometrical underpinning. Furthermore, it is a function whose domain and range are a defined sets of points  such that the function is bijective so that its inverse exists.

Now, there are different transformations such as:

Rotation

Reflection

Translation

From the given image, we can see that the left side is a mirror image of the right side of music notes and as such we can say that the transformation undergone is called Reflection.

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Without that information, I cannot interpret or evaluate the validity of the interval. However, I can provide general guidance on how to interpret a confidence interval.

A 95% confidence interval for a parameter (such as the mean age of the six presidents at their inaugurations) is an interval estimate calculated from sample data that has an associated probability of capturing the true population parameter. Specifically, it means that if we were to repeat the sampling process many times and construct 95% confidence intervals each time, approximately 95% of those intervals would contain the true population parameter.

Therefore, if the confidence interval you provided was calculated correctly and is valid, it means that we are 95% confident that the true mean age of the six presidents at their inaugurations falls within the given interval. However, if there are any issues with the data or the assumptions made in calculating the interval, it may not be appropriate to interpret it in this way.

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Answer: [tex](x+4)^2 + (y+2)^2 = 9[/tex]

Step-by-step explanation:

The general form for the equation of a circle is:

[tex](x-h)^2 + (y-k)^2 = r^2[/tex], where (h,k) is the center, and r is the radius:

We see the center to be at the point (-4,-2). and the radius can also be seen to be 3 units:

with this in mind:

[tex](x-(-4))^2 +(y-(-2))^2 = 3^2[/tex]

[tex](x+4)^2 + (y+2)^2 = 9[/tex]

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The half-life of a radioactive material is the time it takes for half of its atoms to decay. The half-life of the given radioactive material is approximately 4.5 hours.

To calculate the half-life of the given radioactive material, we need to use the formula:
Nt = N0 [tex](1/2)^{(t/T)}[/tex]
Where Nt is the number of radioactive atoms at time t, N0 is the initial number of radioactive atoms, T is the half-life of the material, and t is the time elapsed since the initial measurement.
Using the given data, we can set up two equations:
1450 = N0 [tex](1/2)^{(0/T)}[/tex]
380 = N0 [tex](1/2)^{(8/T)}[/tex]
Dividing the second equation by the first equation, we get:
380/1450 = [tex](1/2)^{(8/T)} / (1/2)^{(0/T)}[/tex]
Simplifying this expression, we get:
380/1450 = [tex](1/2)^{(8/T)}[/tex]
Taking the natural logarithm of both sides, we get:
ln(380/1450) = ln[tex](1/2)^{(8/T)}[/tex]
Simplifying this expression, we get:
T = -8ln(380/1450)/ln(1/2) ≈ 4.5 hours

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Thus, we have proved that for all integers a, b, and c, if a|bc, then a|b or a|c.

To prove that for all integers a, b, and c, if a|bc, then a|b or a|c, we need to use the definition of divisibility.

Assume that a|bc. This means that there exists an integer k such that bc = ak.

We can consider two cases:

Case 1: a and b are coprime.
In this case, a does not share any factors with b. Therefore, a cannot divide b. However, since a|bc and b and c share no factors, a must divide c. Hence, a|c.

Case 2: a and b have a common factor.
In this case, we can write a = dx and b = dy, where d is the greatest common divisor of a and b. Therefore, bc = dxy*c = ak = dxy*k.
Dividing both sides by dxy, we get c/k = a/dy.

Since a and d share no factors, d|c/k. Therefore, there exists an integer m such that c/k = dm. This means that c = dkm.

Since a = dx, we have a|dxk. Since a|bc = dxy*k, we have d|a and d|k. Therefore, we can write k = dh and a = dg, where h and g are integers.

Substituting these expressions into c = dkm, we get c = dg*dh*m. Since d|c and d|a, we have d|b and a|b.

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The volume of the prism in the image attached is 27cm³. This type of prism is called Trapezoidal.

The attached image shows a Trapezoidal prism. To calculate the volume of the prism, we need to multiply the area of the base by the height.

Given dimensions:

Top width = 2 cm

Base width = 4 cm

Length = 6 cm

Height = 1.5 cm

To find the area of the base, we need to consider the average of the top width and base width.

average width = (2 + 4)cm / 2 = 6 cm / 2 = 3 cm.

The area of the base (A) can be calculated as follows:

A = Length × Width = 6 cm × 3 cm = 18 cm²

Now, we can calculate the volume (V) of the prism:

V = A × Height = 18 cm² × 1.5 cm = 27 cm³

Therefore, the volume of the prism is 27 cm³

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The answer to the question is 3v - 7.9w + 9.3.

We are given that;

The expression 3v−(7.9w−9.3)

Now,

we need to simplify the expression by applying the distributive property and combining like terms. Here are the steps:

First, we need to distribute the negative sign to the terms inside the parentheses. This gives us 3v - 7.9w + 9.3.

Next, we need to combine any like terms that have the same variable or are constants.

In this case, there are no like terms, so we cannot simplify further.

3v - 7.9w + 9.3.

Therefore, by the given expression the solution will be 3v - 7.9w + 9.3.

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The value of the autonomous expenditure multiplier would be equal to 1 divided by (1 minus the tax multiplier).

The value of the autonomous expenditure multiplier would depend on the size of the tax multiplier.

The tax multiplier is the inverse of the marginal propensity to consume (MPC) from taxes, which is the portion of each dollar of taxes that households and businesses will spend.

If all taxes are autonomous, this means that taxes are not influenced by changes in income or other economic factors, and will remain constant.

In this case, the tax multiplier would be equal to the MPC from taxes.
The autonomous expenditure multiplier is calculated by dividing 1 by the marginal propensity to save (MPS), which is the portion of each dollar of income that households and businesses will save.

Since the MPC and MPS always add up to 1, we can use the tax multiplier to calculate the MPS from taxes as 1 minus the tax multiplier.

Therefore, the value of the autonomous expenditure multiplier would be equal to 1 divided by (1 minus the tax multiplier).

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The formula for f(x) is f(x) = -10x^4 + 3x^2 + 4.

How can we find the derivative of both sides of the equation?

We can find the derivative of both sides of the equation using the Fundamental Theorem of Calculus:

d/dx [−2x^5 + x^3 + 4x] = d/dx [∫_0^x f(t) dt]

-10x^4 + 3x^2 + 4 = f(x)

Therefore, the formula for f(x) is:

f(x) = -10x^4 + 3x^2 + 4

We can check that this formula satisfies the original equation by taking the definite integral:

∫_0^x f(t) dt = ∫_0^x (-10t^4 + 3t^2 + 4) dt = [-2t^5 + t^3 + 4t]_0^x = -2x^5 + x^3 + 4x

Thus, the formula for f(x) is f(x) = -10x^4 + 3x^2 + 4.

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The proof that AC=2AB when B is the midpoint of segment AC involves the use of the given statement, segment addition postulate, definition of midpoint, and substitution method.

To prove that AC=2AB when B is the midpoint of segment AC, follow these steps:

Thus, by these steps, you can see that when B is the midpoint of AC, the length of segment AC is twice the length of segment AB.

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Answer: A.) π x D

Step-by-step explanation:

The diameter of a circle is 2x the radius

Answer:

A.πx D

Step-by-step explanation: The formula for the circumference of a circle is C=π×d, or it can be written as C=2×π×r.

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Answer: C(2)

Step-by-step explanation: Count by 2's like 4,6,8

Answer:

Step-by-step explanation:

y-intercept is when x=0.

The rule here is y-x=3.

So when x=0, we get y=3.

SOLUTION: (D) 3

The work done by the force field on the particle moving along the given helix is 60π units of work.

To find the work done by the force field F on the particle that moves along the helix r, we use the formula:

W = ∫ F · dr

where · denotes the dot product, and dr is the differential displacement vector along the path of the particle.

First, we need to calculate dr. Since the particle moves along the helix r, we can write:

dr = dx i + dy j + dz k

where dx, dy, and dz are the differentials of x, y, and z with respect to t, respectively. We have:

dx = -2sin(t) dt

dy = 2cos(t) dt

dz = 5 dt

Therefore, we can write:

dr = (-2sin(t) i + 2cos(t) j + 5k) dt

Next, we need to calculate F · dr. We have:

F · dr = (6x i + 6y j + 2k) · (-2sin(t) i + 2cos(t) j + 5k) dt

= -12sin(t) + 12cos(t) + 10 dt

Finally, we can integrate F · dr over the interval 0 ≤ t ≤ 2π to obtain the work done by the force field F on the particle that moves along the helix r:

W = ∫ F · dr = ∫ (-12sin(t) + 12cos(t) + 10) dt

= [-12cos(t) + 12sin(t) + 10t]0[tex]^(2π)[/tex]

= (-12cos(2π) + 12sin(2π) + 10(2π)) - (-12cos(0) + 12sin(0) + 10(0))

= 20π

Therefore, the work done by the force field F on the particle that moves along the helix r is 20π.

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The mulch will cost the homeowner $49.

To find the cost of the mulch, we need to first find the volume of mulch required to cover the garden.

Using the formula for a circle:

Area of a half circle = (1/2) * π * r^2

Plugging in the values, we get:

Area of half circle = (1/2) * π * 6^2 = 56.55 square feet

We need to find the volume of mulch required to cover the garden with a 3 inch layer.

Volume of mulch = Area of half circle * Height of mulch

Volume of mulch = 56.55 * 0.25 = 14.14 cubic feet (rounded to two decimal places)

Finally, we can find the cost of the mulch by multiplying the volume by the cost per cubic foot:

Cost of mulch = Volume of mulch * Cost per cubic foot

Cost of mulch = 14.14 * 3.50 = $49 (rounded to the nearest dollar)

Therefore, the mulch will cost the homeowner $49.

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The value of y is ± (3√6/2) when x =2

To find the value of y when x = 2, we can use the given differential equation and initial condition.

Given: dy/dx = (x³ + 1)/y

Let's solve the differential equation using the separation of variables:

Separating the variables, we have:

y dy = (x³ + 1) dx

Integrating both sides, we get:

∫ y dy = ∫ (x³ + 1) dx

Integrating the left side:

(1/2) y² = (1/4) x⁴ + x + C

Now, using the initial condition y = 2 when x = 1, we can solve for C:

(1/2) (2²) = (1/4) (1)⁴ + 1 + C

2 = 1/4 + 1 + C

2 = 5/4 + C

C = 2 - 5/4

C = 3/4

Substituting the value of C back into the equation:

(1/2) y² = (1/4) x⁴ + x + 3/4

Multiplying both sides by 2 to eliminate the fraction:

y² = (1/2) x⁴ + 2x + 3/2

Now, we can substitute x = 2 into the equation to find y:

y² = (1/2) (2)⁴ + 2(2) + 3/2

y² = 8 + 4 + 3/2

y² = 27/2

y = ± (3√6/2)

Therefore, The value of y is ± (3√6/2) when x =2

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Among the following pairs of triangles, the triangle in option D can be proven through SSS similarity.

Similar triangles are the pair of triangles with corresponding angles of the same magnitude and the corresponding sides are proportional. The following are the criteria for proving similarities:

In the given question,

In option A,

Given:

DF / JK = FG / KL

∠F = ∠K

Therefore, ΔDFG ≈ ΔJKL by SAS criterion

In option B,

Given:

∠G = ∠L

∠F = ∠K

Therefore, ΔDFG ≈ ΔJKL by AA criterion

In option C,

Given:

∠G = ∠L

∠F = ∠K

Therefore, ΔDFG ≈ ΔJKL by AA criterion

In option A,

Given:

DF / JK = FG / KL = GD / LJ

Therefore, ΔDFG ≈ ΔJKL by SSS criterion

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The F-test should only be used to test the joint significance of multiple variables or restrictions, and should not be used to reject all the hypotheses.

When testing joint hypothesis, you should use the F-statistics and reject at least one of the hypothesis if the statistic exceeds the critical value.

This is because the F-test is designed to test the joint significance of multiple variables or restrictions. In other words, it tests whether the combination of the hypotheses is statistically significant.

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The probability of selecting an executive with a type A personality can be calculated by dividing the number of executives with a type A personality by the total number of executives in the group. the probability of selecting an executive with a type A personality from this group is 0.54, or 54%.

P(type A) = number of executives with type A personality / total number of executives
P(type A) = 27 / 50
P(type A) = 0.54 or 54%
Therefore, the probability of selecting an executive with a type A personality from this group is 54%. Probability = (Number of desired outcomes) / (Total number of possible outcomes)
Probability = 27 / 50
Now, we can simplify the fraction:
Probability = 0.54
So, the probability of selecting an executive with a type A personality from this group is 0.54, or 54%.

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The type of sampling described in the scenario is known as systematic sampling.

Systematic sampling involves selecting elements from a population in a systematic and predetermined manner. In this case, the researcher is selecting every 20th name from the student directory to form her sample. This method of sampling is relatively simple to execute and can be less time-consuming compared to other methods such as random sampling. However, it is important to ensure that the selected interval does not coincide with any underlying patterns in the population that may bias the results.

Overall, systematic sampling can be a useful method for obtaining a representative sample from a large population, but it is important to consider the potential limitations and biases associated with this approach.

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Answer: 34

I really hope that this helps

So, approximately 7.71 grams of radium will be present after 100 years.

To calculate the remaining amount of radium after 100 years, we will use the half-life formula:
Remaining Amount = Initial Amount * (1/2)^(time passed / half-life)
Here, the initial amount is 8 grams, the half-life is 1690 years, and the time passed is 100 years.
Step 1: Plug in the values
Remaining Amount = 8 * (1/2)^(100 / 1690)
Step 2: Calculate the exponent
Exponent = 100 / 1690 ≈ 0.05917
Step 3: Calculate the base raised to the exponent
(1/2)^0.05917 ≈ 0.9634
Step 4: Multiply the initial amount by the base raised to the exponent
Remaining Amount = 8 * 0.9634 ≈ 7.7072
So, approximately 7.71 grams of radium will be present after 100 years.
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Answer:

x=-1

y=-10

x=-2

y=0

Step-by-step explanation:

we pit why into the first formula, wimplifying it and using the quadratic formula to get the two values for x, we then put the two values into x into the value for y

To provide a solution for this problem, we can factor the polynomial or find its roots using the rational root theorem.

One possible factorization of the polynomial is:

3x^3 + 13x^2 + 18x - 12 = (x+1)(3x^2 + 10x - 12)

To obtain this factorization, we can start by trying factors of the constant term -12 that might work as roots of the polynomial.

One such factor is -1, so we can use synthetic division or long division to divide the polynomial by x+1. This gives us a quotient of 3x^2 + 10x - 12, which can be factored using the quadratic formula or other methods.

The roots of the polynomial can be found by setting each factor equal to zero and solving for x. This gives us:

x+1 = 0 or 3x^2 + 10x - 12 = 0

The first equation has a single root of x = -1. The second equation can be solved using the quadratic formula or factoring, giving us two more roots:

x = (-10 ± sqrt(100 + 4312)) / (2*3) = (-10 ± 2sqrt(19)) / 3

Therefore, the roots of the polynomial are -1, (-10 + 2sqrt(19))/3, and (-10 - 2sqrt(19))/3.

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