Rectangle ABCD and DEFG are congruent
AB=7cm
AD=17cm
Work out the length of CE

Given the volume of the initial solid, V1 = 36 cubic units. Let's assume the dilated scale factor is K and the volume of the dilated solid is V2 = 4 cubic units.

We need to find the value of K using the given data. Relation between volumes of two similar solids: Let the scale factor between the corresponding sides of the two similar solids be k, then the ratio of their volumes is given [tex]by:$$\frac{Volume \ of \ Dilated \ Solid}{Volume \ of \ Initial \ Solid} = k^3$$Let's apply this formula to solve this problem. Substitute V1 = 36 cubic units, and V2 = 4 cubic units.$$k^3 = \frac{V2}{V1}$$On substituting the given values, we get;$$k^3 = \frac{4}{36}$$$$k^3 = \frac{1}{9}$$$$\sqrt[3]{k^3} = \sqrt[3]{\frac{1}{9}}$$$$k = \frac{1}{3}$$Therefore, the value of K is 1/3.[/tex]

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The series converges.

To determine the convergence or divergence of the given series, we can use the limit comparison test.

Let's consider the series:

∑(11n^2 - 4n^4)/(3n)

We can simplify the series by dividing both numerator and denominator by n^3, which gives:

∑(11/n - 4/n^3)

Now we can use the limit comparison test by comparing this series to the series ∑(1/n^2).

We have:

lim n→∞ (11/n - 4/n^3)/(1/n^2)

= lim n→∞ (11n^2 - 4)/(n^2)

= 11

Since the limit is finite and positive, and the series ∑(1/n^2) is a known convergent p-series with p=2, by the limit comparison test, the given series also converges.

Therefore, the main answer is that the series converges.

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The profit from selling garlands is higher than the profit from manufacturing wreaths. FN should focus on selling garlands instead of wreaths.

To compare the profitability of manufacturing wreaths and garlands, we need to calculate the profit for each product.

Profit from manufacturing wreaths:

Revenue from selling 1,200 wreaths = 1,200 x $600 = $720,000

Total cost of making 1,200 wreaths = 1,200 x $465 = $558,000

Profit from selling 1,200 wreaths = $720,000 - $558,000 = $162,000

Profit from selling garlands:

Revenue from selling 1,400 garlands = 1,400 x $440 = $616,000

Total cost of making 1,400 garlands = $315 + (1,400 x $315) = $441,315

Profit from selling 1,400 garlands = $616,000 - $441,315 = $174,685

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The meterstick must be moving at a velocity of approximately 0.836 times the speed of light (or about 251,547,246 m/s) for its length to be measured as 0.737 m.

According to this theory, the length of an object moving relative to an observer appears to be shorter than its rest length. The amount of length contraction depends on the relative velocity between the observer and the object, as well as the direction of motion.

The formula for length contraction is given by:

[tex]L' = L \times \sqrt{(1 - v^2/c^2)}[/tex]

where L is the rest length of the object, L' is its length as measured by the observer, v is the relative velocity between the observer and the object, and c is the speed of light.

In this case, we are given that the measured length of the meterstick is 0.737 m. We can assume that the rest length of the meterstick is the standard length of a meterstick, which is 1.0 m. We want to find the velocity v at which this length contraction occurs.

So, we can rearrange the formula above to solve for v:

[tex]v = c \times \sqrt{(1 - (L'/L)^2)}[/tex]

Plugging in the values given, we get:

[tex]v = c \times \sqrt{(1 - (0.737/1.0)^2)} \\= c \times \sqrt{(1 - 0.542^2)} \\= c \times \sqrt{v} \\= 0.836c[/tex]

where c is the speed of light (299,792,458 m/s).

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Answer:

Step-by-step explanation:

Answer: 1) linear pairs are supplementary

2) subtraction property

3) transitive property

Step-by-step explanation:

transitive property is also vertical angles showing that angle 4 and angle 2 are equal

both angles 1 and 2 lay on the same line causing them to be supplementary angles.

The acceleration at t=10 seconds is approximately 0.2222 m/s^2.

Using Equation 23.9 of the book, we can calculate the acceleration at t=4 seconds and t=10 seconds as follows:

At t=4 seconds:

The first-order divided difference for velocity between t=2.5 and t=6.0 is:

f[t_2, t_1] = (V(t_2) - V(t_1))/(t_2 - t_1) = (18 - 15)/(6.0 - 2.5) = 1.7143 m/s^2

The first-order divided difference for velocity between t=1.0 and t=2.5 is:

f[t_1, t_0] = (V(t_1) - V(t_0))/(t_1 - t_0) = (15 - 10)/(2.5 - 1.0) = 10 m/s^2

The second-order divided difference for velocity between t=2.5, t=6.0, and t=1.0 is:

f[t_2, t_1, t_0] = (f[t_2, t_1] - f[t_1, t_0])/(t_2 - t_0) = (1.7143 - 10)/(6.0 - 1.0) = -1.6571 m/s^2

Therefore, the acceleration at t=4 seconds is approximately -1.6571 m/s^2.

At t=10 seconds:

The first-order divided difference for velocity between t=9.0 and t=12.0 is:

f[t_2, t_1] = (V(t_2) - V(t_1))/(t_2 - t_1) = (30 - 22)/(12.0 - 9.0) = 2.6667 m/s^2

The first-order divided difference for velocity between t=6.0 and t=9.0 is:

f[t_1, t_0] = (V(t_1) - V(t_0))/(t_1 - t_0) = (22 - 18)/(9.0 - 6.0) = 1.3333 m/s^2

The second-order divided difference for velocity between t=9.0, t=12.0, and t=6.0 is:

f[t_2, t_1, t_0] = (f[t_2, t_1] - f[t_1, t_0])/(t_2 - t_0) = (2.6667 - 1.3333)/(12.0 - 6.0) = 0.2222 m/s^2

Therefore, the acceleration at t=10 seconds is approximately 0.2222 m/s^2.

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Answer:

if you want to know which polynomial is exactly divisible by (x+2) then where is the equation ?

Both Tom's and Saul's methods will work to find the correct answer for the subtraction problem of 357 - 216. Adding a constant value to each number before subtracting does not change the relative difference between the numbers, ensuring the same result.

In the given problem, Tom adds 4 to each number (357 + 4 = 361, 216 + 4 = 220) and then subtracts the adjusted numbers (361 - 220 = 141). Similarly, Saul adds 3 to each number (357 + 3 = 360, 216 + 3 = 219) and then subtracts the adjusted numbers (360 - 219 = 141).
Both methods yield the same result of 141. This is because adding a constant value to each number before subtracting does not affect the relative difference between the numbers. The difference between the original numbers (357 - 216) remains the same when the same constant is added to both numbers.
Therefore, both Tom's and Saul's methods will work to find the correct answer. Adding a constant to each number before subtracting does not alter the result as long as the same constant is added to both numbers consistently.

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Answer:

Angle 1 is 113, Angle 2 is 34

Step-by-step explanation:

Check the attachment:

The focus of the parabola y = -132x² is located at (0, -1/528) and the equation of the directrix is y = 1/528.

In the general equation of a parabola, y = ax², the focus is located at (0, 1/(4a)), and the directrix is given by the equation y = -1/(4a). In this case, the coefficient of x² is -132, so we substitute this value into the formulas.

To find the coordinates of the focus, we set a = -132 in the focus formula: (0, 1/(4(-132))) = (0, -1/528). Therefore, the focus of the parabola is located at (0, -1/528).

For the equation of the directrix, we substitute a = -132 into the directrix formula: y = -1/(4(-132)) = -1/528. Hence, the equation of the directrix is y = 1/528.

conclusion, the focus of the parabola y = -132x² is located at (0, -1/528), and the equation of the directrix is y = 1/528.

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The approximate percent of the price increase is 8.5%.

A subwoofer box for sound costs $260.40 after a price increase. The cost before the price increase was $240.00. What was the approximate percent of the price increase?To calculate the percent increase, you can use the formula:percent increase = (new value - old value) / old value * 100In this case, the old value is $240.00 and the new value is $260.40. Therefore,percent increase = (260.40 - 240.00) / 240.00 * 100 ≈ 8.5%So, the approximate percent of the price increase is approximately 8.5%.Explanation:This is a problem involving percent increase.

The formula to calculate percent increase is:percent increase = (new value - old value) / old value * 100Let's plug in the given values. The old value is $240.00 and the new value is $260.40.percent increase = (260.40 - 240.00) / 240.00 * 100percent increase = 20.40 / 240.00 * 100percent increase ≈ 0.0850 or 8.5%Therefore, the approximate percent of the price increase is 8.5%.

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Answer:

16.12

Step-by-step explanation:

The exact distance between Gary and the school is 16.12 blocks.

The integral does not have a closed-form solution, but it can be expressed using the exponential integral function,where u = 3x and c is the constant of integration.

To evaluate the integral ∫2e^(3xe^(3x)) dx, we can use the substitution method. Let u = 3x, then du = 3dx.

Although this integral does not have a closed-form solution in terms of elementary functions, it can be expressed using special functions such as the exponential integral.

Thus, the integral evaluates to (2/3)Ei(uˣ e^u) + c, where Ei(x) is the exponential integral function and c is the constant of integration.

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The graph of the Quadratic function y = f(x) = x^2 + 3x - 4 has a minimum value. The vertex of the graph is located at (-1.5, -6.25). The domain of the function is (-∞, ∞), and the range is (-∞, -6.25].

The graph of the quadratic function y = f(x) = x^2 + 3x - 4 has a maximum or minimum value, we can examine its leading coefficient. In this case, the coefficient of the x^2 term is positive (1), indicating that the graph opens upward and therefore has a minimum value.

To find the vertex of the quadratic function, we can use the formula x = -b/(2a), where a is the coefficient of the x^2 term and b is the coefficient of the x term. In our function, a = 1 and b = 3.

x = -3/(2*1) = -3/2 = -1.5

Substituting this x-value back into the function, we can find the corresponding y-value:

y = f(-1.5) = (-1.5)^2 + 3(-1.5) - 4 = 2.25 - 4.5 - 4 = -6.25

Therefore, the vertex of the graph is (-1.5, -6.25).

The domain of the function represents all the possible x-values for which the function is defined. In this case, since the function is a quadratic polynomial, it is defined for all real numbers. Hence, the domain is (-∞, ∞), indicating that there are no restrictions on the x-values.

The range of the function represents all the possible y-values that the function can take. Since the graph opens upward and has a minimum value, the y-values increase indefinitely as x approaches positive or negative infinity. Thus, the range is (-∞, f(-1.5)], where f(-1.5) represents the minimum value of the function.

In summary, the graph of the quadratic function y = f(x) = x^2 + 3x - 4 has a minimum value. The vertex of the graph is located at (-1.5, -6.25). The domain of the function is (-∞, ∞), and the range is (-∞, -6.25].

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For each consecutive term we need to subtract 9 to the previous one, using that rule, we can see that the six row will have 27 rocks.

Here we have an arithmetic sequence, such that the first 3 terms are:

a₁ = 72

a₂ = 63

a₃ = 54

We can see that in each consecutive term, we subtract 9 from the previous value:

72 - 9 = 63

63 - 9 = 54

And so on.

Then the fourth term is:

a₄ = 54 - 9 = 45

The fifth term is:

a₅ = 45 - 9 = 36

And the sixth term is:

a₆ = 36 - 9= 27

That is the number of rocks.

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Complete question:

"Retha is building a rock display for her science project. She put 72 rocks in the first row, 63 rocks in the second row, and 54 rocks in the third row.

If the pattern continues, how many rocks will be on the sixth row?"

The question does not provide enough information to answer this question. Please provide the relevant part c of the question to be able to determine the sample size and make a judgment on whether it was large enough for inference.

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Answer choice C ($1000) is the most plausible option, as it corresponds to a relatively high value of R.

We can find the maximum profit by finding the value of s that maximizes the profit function P(s).

To do this, we first take the derivative of P(s) with respect to s and set it equal to zero to find any critical points:

P'(s) = 20 - sR = 0

Solving for s, we get:

s = 20/R

To confirm that this is a maximum and not a minimum or inflection point, we can take the second derivative of P(s) with respect to s:

P''(s) = -R

Since P''(s) is negative for any value of s, we know that s = 20/R is a maximum.

Therefore, to find the amount of advertising that gives the maximum profit, we need to substitute this value of s back into the profit function:

P = 230 + 20s - 1/2 s^2 R

P = 230 + 20(20/R) - 1/2 (20/R)^2 R

P = 230 + 400/R - 200/R

P = 230 + 200/R

Since R is not given, we cannot find the exact value of the maximum profit or the corresponding value of s. However, we can see that the larger the value of R (i.e. the more revenue generated for each unit of advertising spent), the smaller the value of s that maximizes profit.

So, answer choice C ($1000) is the most plausible option, as it corresponds to a relatively high value of R.

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The value of a rare coin in dollars can be approximated by the model Y = 0.25(1.06)^t, where t represents the number of years since the coin was minted. The model indicates that the value of the coin increases over time.

The given model Y = 0.25(1.06)^t represents an exponential growth model. In this model, the value of the coin is determined by multiplying an initial value of 0.25 dollars by the growth factor (1.06) raised to the power of the number of years since the coin was minted (t).

The growth factor of 1.06 indicates that the value of the coin increases by 6% per year. Each year, the value of the coin is multiplied by 1.06, resulting in continuous growth over time.

The initial value of 0.25 dollars represents the starting value of the coin when it was minted. As time passes, the value of the coin increases exponentially according to the model.

Therefore, the given model provides an approximation of the value of the rare coin in dollars based on the number of years since it was minted, with a growth rate of 6% per year.

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The values of the three slack variables at the optimal solution are x = 4, y = 0, and z = 20.

a. To write the problem in standard form, we need to introduce slack variables. Let x, y, and z be the slack variables for the first, second, and third constraints, respectively. Then the problem becomes:

Maximize: 3A + 4B
Subject to:
-lA + 2B + x = 8
lA + 2B + y = 12
24 + B + z = 16A
B, x, y, z >= 0

b. To solve the problem using the graphical solution procedure, we first graph the three constraint lines: -lA + 2B = 8, lA + 2B = 12, and 24 + B = 16A.

We then identify the feasible region, which is the region that satisfies all three constraints and is bounded by the x-axis, y-axis, and the lines -lA + 2B = 8 and lA + 2B = 12. Finally, we evaluate the objective function at the vertices of the feasible region to find the optimal solution.

After graphing the lines and identifying the feasible region, we find that the vertices are (0, 4), (4, 4), and (6, 3). Evaluating the objective function at each vertex, we find that the optimal solution is at (4, 4), with a maximum value of 3(4) + 4(4) = 24.

c. To find the values of the slack variables at the optimal solution, we substitute the values of A and B from the optimal solution into the constraints and solve for the slack variables. We get:

-l(4) + 2(4) + x = 8
l(4) + 2(4) + y = 12
24 + (4) + z = 16(4)

Simplifying each equation, we get:

x = 4
y = 0
z = 20

Therefore, the values of the three slack variables at the optimal solution are x = 4, y = 0, and z = 20.

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The correct statements are:

1: The expression has a term containing the variable p.

Statement 3: The expression has four terms.

The expression M + (5n)(9 - p) - 6 - r^2 is given, and you have to determine which of the statements are correct.

Statement 1: The expression has a term containing the variable p. - True.

Statement 2: The expression has a term containing the variable q. - False.

Statement 3: The expression has four terms. - True.

Statement 4: The expression has a term with a coefficient of 5. - True.

Statement 5: The expression has a term with a coefficient of -6. - True.

Statement 6: The expression has a term with a coefficient of r. - False.

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The probability that in 8 progeny plants, four or more plants will express the gene is approximately 0.892.

To find the probability that four or more plants will express the gene, we sum up the probabilities of these individual outcomes:P(X ≥ 4) = P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8). Calculating these probabilities and summing them up will give you the final result.

To calculate the probability that in 8 progeny plants, four or more plants will express the gene, we can use the binomial probability formula.

The binomial probability formula is given by:

[tex]P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)[/tex]

Where:

P(X = k) is the probability of getting exactly k successes

n is the total number of trials

k is the number of successful outcomes

p is the probability of success in a single trial

C(n, k) is the number of combinations of n items taken k at a time (given by n! / (k! * (n - k)!)

In this case, we want to find the probability of getting four or more plants expressing the gene in 8 progeny plants. Let's calculate it step by step:

[tex]P(X = 4) = C(8, 4) * 0.20^4 * (1 - 0.20)^(8 - 4)\\P(X = 5) = C(8, 5) * 0.20^5 * (1 - 0.20)^(8 - 5)\\P(X = 6) = C(8, 6) * 0.20^6 * (1 - 0.20)^(8 - 6)\\P(X = 7) = C(8, 7) * 0.20^7 * (1 - 0.20)^(8 - 7)\\P(X = 8) = C(8, 8) * 0.20^8 * (1 - 0.20)^(8 - 8)[/tex]

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The given sequence diverges.

The nth term of the sequence is given by an = 3n + 7. As n approaches infinity, the term 3n dominates over the constant term 7, and the sequence increases without bound. Mathematically, we can prove this by contradiction. Assume that the sequence converges to a finite limit L.

Then, for any positive number ε, there exists an integer N such that for all n>N, |an-L|<ε. However, if we choose ε=1, then for any N, we can find an integer n>N such that an > L+1, contradicting the assumption that the sequence converges to L. Therefore, the sequence diverges.

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The explicit solution to the given initial value problem

y′=(1−y)5/4 with y(0)=0 is

y(x) = [tex]1 - (1 - e^x)^4/5[/tex]

The given first-order differential equation is separable, which means that we can separate the variables and write the equation in the form

[tex]dy/(1-y)^(5/4) = dx.[/tex]

Integrating both sides, we get [tex](1-y)^(-1/4)[/tex] = 5/4 * x + C, where C is the constant of integration. Solving for y, we get y(x) = 1 -[tex](1 - e^x)^4/5[/tex].

Using the initial condition y(0) = 0, we can solve for C and get C = 1. Therefore, the explicit solution to the initial value problem is

[tex]y(x) = 1 - (1 - e^x)^4/5.[/tex]

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d^966 / dx^939 (cos x) = d^2/dx^2 (cos x) = -cos x.

To find the derivative of d^966 / dx^939 (cos x), we can examine the pattern of derivatives and look for a recurring pattern.

Let's start by calculating the first few derivatives of cos x:

d/dx (cos x) = -sin x

d^2/dx^2 (cos x) = -cos x

d^3/dx^3 (cos x) = sin x

d^4/dx^4 (cos x) = cos x

We can observe that the derivatives of cos x repeat with a period of 4. Specifically, the derivatives repeat in the pattern: {-sin x, -cos x, sin x, cos x}.

Since d^966 / dx^939 is much larger than the period of the pattern (4), we can divide 966 by 4 to determine the remainder:

966 divided by 4 gives a remainder of 2.

This means that the derivative at the 966th derivative position will correspond to the second derivative in the pattern.

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Answer:

9 seconds

Step-by-step explanation:

The height of the rocket is given by the function h(t) = -16t² + 144t, where t represents the time in seconds after launch.

The rocket will hit the ground when its height is zero, so when h(t) = 0.

Set the function h(t) to zero:

[tex]-16t^2+144t=0[/tex]

Factor out the common term -16t:

[tex]-16t(t-9)=0[/tex]

Apply the Zero Product Property by setting each factor equal to zero and solving for t:

[tex]\implies -16t=0 \implies t=0[/tex]

[tex]\implies t-9=0 \implies t=9[/tex]

When t = 0, the rocket is launched.

Therefore, the rocket hits the ground at 9 seconds.

The critical points are:

At c1 ≈ -2.108, f is Local Min.

At c2 ≈ -0.416, f is Neither.

At c3 ≈ 1.524, f is  Local Min.

To find the critical points, we need to find where the derivative of the function is equal to zero or undefined. Let's calculate the derivative:

[tex]f'(x) = 7x^3 + 14x^2 - 7x - 14[/tex]

To find the critical points, we set f'(x) equal to zero and solve for x:

[tex]7x^3 + 14x^2 - 7x - 14 = 0[/tex]

We can simplify this equation by factoring out a common factor of 7:

[tex]7(x^3 + 2x^2 - x - 2) = 0[/tex]

Now, we have a cubic equation. Unfortunately, the roots of this equation cannot be found easily by factoring or simple methods. We can approximate the roots using numerical methods or calculators.

Using numerical methods or a calculator, we find the approximate values of the three critical points:

c1 ≈ -2.108

c2 ≈ -0.416

c3 ≈ 1.524

To determine the nature of each critical point, we apply the First Derivative Test. We evaluate the sign of the derivative on either side of each critical point:

For c1 ≈ -2.108:

Evaluate f'(-3): f'(-3) ≈ -77.364 < 0

Evaluate f'(-2): f'(-2) ≈ 4.000 > 0

Since the sign changes from negative to positive, c1 ≈ -2.108 corresponds to a local minimum.

For c2 ≈ -0.416:

Evaluate f'(-1): f'(-1) ≈ -20.083 < 0

Evaluate f'(0): f'(0) ≈ -14.000 < 0

Since the sign does not change, c2 ≈ -0.416 does not correspond to a local maximum or minimum (neither).

For c3 ≈ 1.524:

Evaluate f'(1): f'(1) ≈ -11.083 < 0

Evaluate f'(2): f'(2) ≈ 42.000 > 0

Since the sign changes from negative to positive, c3 ≈ 1.524 corresponds to a local minimum.

Therefore, the answers are:

At c1 ≈ -2.108, f is B) Local Min.

At c2 ≈ -0.416, f is C) Neither.

At c3 ≈ 1.524, f is B) Local Min.

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The length of YXN is √34 cm, and YN is 5 cm, using the Pythagoras theorem and the midpoint theorem. The triangle LMN is right-angled at L, LM, and LN are the legs of the triangle, and MN is its hypotenuse.

We know that X and Y are the midpoints of MN and LN, respectively. Therefore, from the midpoint theorem, we know that.

MY=LY = LN/2 (as Y is the midpoint of LN) and

MX=NX= MN/2 (as X is the midpoint of MN).

We have given LM=8cm and MN=6cm. Now we will use the Pythagoras theorem in ΔLMN.

Using Pythagoras' theorem, we have,

     LN2=LM2+MN2

        LN = 82+62=100

       =>LN=10 cm

As Y is the midpoint of LN, YN=5 cm

MX = NX = MN/2 = 6/2 = 3 cm

Therefore, ΔNYX is a right-angled triangle whose hypotenuse is YN = 5 cm. MX = 3 cm

From Pythagoras' theorem, NY2= YX2+ NX2

= 52+32= 34

=>NY= √34 cm

Therefore, YXN is √34 cm, and YN is 5 cm.

Thus, we can conclude that the length of YXN is √34 cm, and YN is 5 cm, using the Pythagoras theorem and the midpoint theorem.

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These are the symmetric equations for the line passing through the origin and the point (5, 9, -1).

To find the parametric equations for the line passing through the origin (0, 0, 0) and the point (5, 9, -1), we can use the parameter t.

Let's assume the parametric equations are:

x(t) = at

y(t) = bt

z(t) = c*t

where a, b, and c are constants to be determined.

We can set up equations based on the given points:

When t = 0:

x(0) = a0 = 0

y(0) = b0 = 0

z(0) = c*0 = 0

This satisfies the condition for passing through the origin.

When t = 1:

x(1) = a1 = 5

y(1) = b1 = 9

z(1) = c*1 = -1

From these equations, we can determine the values of a, b, and c:

a = 5

b = 9

c = -1

Therefore, the parametric equations for the line passing through the origin and the point (5, 9, -1) are:

x(t) = 5t

y(t) = 9t

z(t) = -t

To find the symmetric equations, we can eliminate the parameter t by equating the ratios of the variables:

x(t)/5 = y(t)/9 = z(t)/(-1)

Simplifying, we have:

x/5 = y/9 = z/(-1)

Multiplying through by the common denominator, we get:

9x = 5y = -z

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By applying the Intermediate Value Theorem for continuous functions, we can conclude that if the integral of a continuous function f over the interval [a, b] is equal to zero, then there exists at least one point c in the interval [a, b] where f(c) is also equal to zero.

To prove that there exists a point c in the interval [a, b] where f(c) is equal to zero, we will make use of the Intermediate Value Theorem.

The Intermediate Value Theorem states that if a function f is continuous on a closed interval [a, b] and f(a) and f(b) have opposite signs (i.e., f(a) < 0 and f(b) > 0, or f(a) > 0 and f(b) < 0), then there exists at least one point c in the interval (a, b) where f(c) is equal to zero.

In our case, we are given that the integral of f over the interval [a, b] is equal to zero, i.e., ∫[a,b] f(x) dx = 0. Since the integral represents the signed area under the curve of f(x), the fact that the integral is zero indicates that the positive and negative areas cancel each other out.

Now, let's assume, for the sake of contradiction, that there does not exist any point c in the interval [a, b] where f(c) is equal to zero. This would mean that f(x) maintains a constant sign (either positive or negative) throughout the interval [a, b].

If f(x) is always positive or always negative, then the integral of f over [a, b] cannot be zero, as it would represent a nonzero positive or negative area under the curve. This contradicts the given condition that the integral is equal to zero.

Therefore, by contradiction, we can conclude that there must exist at least one point c in the interval [a, b] where f(c) is equal to zero. This completes the proof.

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We need to evaluate the double integral:

∬R 5x sin(x+y) dA, R=(0, π/6)×(0, π/3)

Using iterated integrals, we have:

∬R 5x sin(x+y) dA = ∫[0, π/6] ∫[0, π/3] 5x sin(x+y) dy dx

= ∫[0, π/6] [(-5/2)x cos(x+y)]|[0,π/3] dx

= ∫[0, π/6] [(-5/2)x cos(x+π/3) + (5/2)x cos(x)] dx

= (-5/2) ∫[0, π/6] x cos(x+π/3) dx + (5/2) ∫[0, π/6] x cos(x) dx

Let's evaluate each integral separately:

∫[0, π/6] x cos(x+π/3) dx

= ∫[π/3, 2π/3] (u-π/3) cos(u) du   (where u = x+π/3)

= ∫[π/3, 2π/3] u cos(u) du - (π/3) ∫[π/3, 2π/3] cos(u) du

= sin(2π/3) - sin(π/3) - (π/3)(sin(2π/3) - sin(π/3))

= -π/3√3

Similarly,

∫[0, π/6] x cos(x) dx = sin(π/6)/2 = 1/4

Therefore,

∬R 5x sin(x+y) dA = (-5/2) (-π/3√3) + (5/2)(1/4) = (5π)/(6√3)

Hence, the value of the double integral is (5π)/(6√3).

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