Find the indicated term 31st term: 12, 7, 2, -3

To find the equation of the sphere tangent to the plane 2x + 3y - 6z = 5 with center C(-2, 3, 7), we need to find the radius of the sphere. The equation of the sphere is (x + 2)^2 + (y - 3)^2 + (z - 7)^2 = 36.

The distance from the center of the sphere to the plane is equal to the radius. We can use the formula for the distance between a point (x, y, z) and a plane Ax + By + Cz + D = 0:

Distance = |Ax + By + Cz + D| / sqrt(A^2 + B^2 + C^2)

In this case, the plane equation is 2x + 3y - 6z - 5 = 0. Plugging in the coordinates of the center C(-2, 3, 7) into the formula, we have:

Distance = |2(-2) + 3(3) - 6(7) - 5| / sqrt(2^2 + 3^2 + (-6)^2)

= |-4 + 9 - 42 - 5| / sqrt(4 + 9 + 36)

= |-42| / sqrt(49)

= 42 / 7

= 6

So, the radius of the sphere is 6.

The equation of a sphere with center C(-2, 3, 7) and radius 6 is:

(x + 2)^2 + (y - 3)^2 + (z - 7)^2 = 6^2

Simplifying further, we have:

(x + 2)^2 + (y - 3)^2 + (z - 7)^2 = 36

Therefore, the equation of the sphere is (x + 2)^2 + (y - 3)^2 + (z - 7)^2 = 36.

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the magnitude of the cross product of two unit vectors u and v is 1/2, the possible angles between them are π/6 (or 30°) and 5π/6 (or 150°).

The magnitude of the cross product of two vectors u and v is equal to the area of the parallelogram formed by the two vectors. Therefore, the magnitude of the cross product of two unit vectors u and v is equal to the area of the parallelogram formed by the two vectors, which is also equal to the sine of the angle between them. In other words:

∥u × v∥ = sin(θ)

where θ is the angle between the two vectors. If ∥u × v∥ = 1/2, then we have:

1/2 = sin(θ)

This equation has two solutions: θ = π/6 and θ = 5π/6. Therefore, the possible angles between two unit vectors u and v are π/6 and 5π/6, or approximately 30° and 150°.

To see why there are two solutions, we can look at the unit circle, where sine is positive in the first and second quadrants. The angle θ between u and v can lie in either the first or second quadrant, giving us two possible values for sine. These correspond to the angles π/6 and 5π/6, respectively.

In summary, if the magnitude of the cross product of two unit vectors u and v is 1/2, the possible angles between them are π/6 (or 30°) and 5π/6 (or 150°).

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The simplified expression of the expression \((x - y)^p(x^2 + y^2 + q - y)\)while done the simplification through binomial theorm.

\((x - y)^p(x^2 + y^2 + q - y)\)

Expanding the first term using the binomial theorem, we get:

\((x - y)^p = \sum_{k=0}^{p} {p \choose k} x^{p-k} (-y)^k\)

where [ p choose k ] is the binomial coefficient, given by p! / (k! × (p-k)!).

Substituting this expansion into the original expression, we get:

\(\sum_{k=0}^{p} {p \choose k} x^{p-k} (-y)^{k} (x^2 + y^2 + q + y)\)

Expanding the last term, we get:

\(\sum_{k=0}^{p} {p \choose k} x^{p-k} (-y)^{k} (x^2 + y^2) + \sum_{k=0}^{p} {p \choose k} x^{p-k} (-y)^{k} (q+y)\)

The first term can be simplified by distributing the x² and y² terms:

\(\begin{aligned} &\sum_{k=0}^{p} {p \choose k} x^{p-k} (-y)^{k} x^{2} + \sum_{k=0}^{p} {p \choose k} x^{p-k} (-y)^{k} y^{2} \\&= x^{2} \sum_{k=0}^{p} {p \choose k} x^{p-k} (-y)^{k} + y^{2} \sum_{k=0}^{p} {p \choose k} x^{p-k} (-y)^{k} \\&= x^{2}(x-y)^{p} + y^{2}(x-y)^{p} \\&= (x^{2}+y^{2})(x-y)^{p}\end{aligned}\)

The second term can be simplified by distributing the x and y terms:

\(\begin{aligned} &\sum_{k=0}^{p} {p \choose k} x^{p-k} (-y)^{k} q + \sum_{k=0}^{p} {p \choose k} x^{p-k} (-y)^{k} y \\&= q \sum_{k=0}^{p} {p \choose k} x^{p-k} (-y)^{k} - y \sum_{k=0}^{p} {p \choose k} x^{p-k} (-y)^{k} \\&= q (x-y)^{p} - y (x-y)^{p} \\&= (q-y) (x-y)^{p}\end{aligned}\)

Putting these simplified terms together, we get:

\(\begin{aligned}(x-y)^p \cdot (x^2 + y^2 + q - y) &= (x-y)^p \cdot [(x^2 + y^2) + (q - y)] \\&= (x-y)^p \cdot (x^2 + y^2) + (x-y)^p \cdot (q - y) \\&= (x^2 + y^2) \cdot (x-y)^p + (q - y) \cdot (x-y)^p \\&= (x^2 + y^2 + q - y) \cdot (x-y)^p\end{aligned}\)

Therefore, the simplified expression is \((x-y)^p \cdot (x^2 + y^2 + q - y)\)

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Let's start by calculating the value for f(-1).

For this we go to the graph and count the number of squares to the left of the axis to be at -1. In this case f(-1) would be 5.

Now let's calculate f(4)

For this case the value of f(4) would be 0.

Similarly we calculate f(-3). As you can see in the graph it would be 6.

The last scenario would be for when f(x) = 9, we look for that value on the curve and realize that it corresponds to x = 1.

The sample of the speeds of 20 cars and at a 0.05 confidence level, we

have;

(a) There is not enough convincing evidence to suggest that the true mean speed of all cars travelling on Elm Street is more than 25 mph

(b) Type II error

(a) Based on the given data, using MS Excel, we have;

The sample size, n = 20

The mean, \(\overline x\) = 25.86

The standard deviation of the sample, s = 2.284

The null hypothesis, H₀; \(\overline x\) = 25 mph

Alternative hypothesis, Hₐ; \(\overline x\) > 25 mph

Given that the sample size is less than 30, the t-test is used, and the t-score is found as follows;

\(t= \mathbf{\dfrac{\bar{x}-\mu }{\dfrac{s}{\sqrt{n}}}}\)

Which gives;

\(t=\dfrac{25.86-25 }{\dfrac{2.284}{\sqrt{20}}} \approx \mathbf{1.684}\)

Using a graphing calculator, we have;

The probability ≈  0.054

The critical-t ≈ 1.729

Given that the probability is larger than the significance level, we fail to

reject the null hypothesis.

There is not enough convincing statistical evidence to conclude that the

true mean of all cars travelling on Elm Street is more than 25 mph.

(b) The type of error is Type II error, given that there is a possibility that

the null hypothesis is failed to be rejected when it is actually false.

Given that the sample size is less than 30, it may be insufficient for the

Central Limit Theorem to be in effect.

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Answer:

No

Step-by-step explanation:

When \(x=2\), \(y=-6(2)=-12\), not \(4\).

Answer:

C'nD' = {5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33}

Step-by-step explanation:

C' is the complement of C, which contains all the numbers not in C. Since C contains all the even numbers between 4 and 34, C' contains all the odd numbers between 4 and 34 and 4 and 36.

C' = {4, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 36}

Similarly, D' contains all the numbers that are not in D, which are the even numbers between 5 and 33, as well as 4 and 34.

D' = {5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33}

The intersection of C' and D' contains only the numbers in both C' and D'. Therefore,

C' n D' = {5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33}

So, the elements in C' n D' are all odd numbers between 5 and 33.

Answer:

x=-5 i think

Step-by-step explanation:

Answer: 4.8484847e+15

Step-by-step explanation:

Answer:

6666666 × 727272772​ = 4,848,484,661,818,152

The volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis is π cubic units.

To find the volume V of the solid obtained by rotating the region bounded by the curves y = 2 - (1/2)x, y = 0, x = 1, and x = 2 about the x-axis, the method of cylindrical shells.

The volume V can be calculated using the following formula:

V = ∫(2πx × h) dx

where h represents the height of the cylindrical shell at each value of x.

The height h can be determined as the difference between the y-values of the curves y = 2 - (1/2)x and y = 0.

The integral,

V = ∫(2πx × h) dx

= ∫(2πx ×(2 - (1/2)x)) dx

= 2π ∫(2x - (1/2)x²) dx

= 2π [(x²) - (1/6)(x³)] evaluated from 1 to 2

Evaluating the definite integral,

V = 2π [(2²) - (1/6)(2³)] - 2π [(1²) - (1/6)(1³)]

= 2π [4 - (1/6)(8)] - 2π [1 - (1/6)(1)]

= 2π [4 - (4/6)] - 2π [1 - (1/6)]

= 2π [4 - (2/3)] - 2π [1 - (1/6)]

= 2π [4/3] - 2π [5/6]

= (8π/3) - (5π/3)

= (3π/3)

= π

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The solutions to the equation are x = -1, x = -8, and x = 1/2.

The equation 4x³ + 32x² + 42x - 16 = 0 can be divided throuh by 2 as follows:

2x³ + 16x² + 21x - 8 = 0

We can test each of these possible roots by substituting them into the equation and seeing if we get 0. When we substitute -1, we get 0, so -1 is a root of the equation. We can then factor out (x + 1) from the equation to get:

(x + 1)(2x² + 15x - 8) = 0

We can then factor the quadratic 2x² + 15x - 8 by grouping to get:

(x + 8)(2x - 1) = 0

This gives us two more roots, x = -8 and x = 1/2.

Therefore, the solutions to the equation are x = -1, x = -8, and x = 1/2.

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It took Paul a total of 12 hours and 40 minutes to complete all three parts of the Ironman triathlon.

Paul’s time for swimming in the Ironman triathlon was 1 hour 25 minutes. Her time for biking was 5 hours longer than her swimming time, so her biking time was (1 hour + 5 hours) = 6 hours 25 minutes. She ran for 4 hours 50 minutes. The total time it took her to complete all three parts of the race is (1 hour 25 minutes) + (6 hours 25 minutes) + (4 hours 50 minutes) = 12 hours 40 minutes.

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The current supply of dog food will last for 5 days.

Emilio's dog eats 1/2 of a can of food each day. Emilio currently has 2 1/2 cans of dog food. To determine how many days the current supply will last, we need to find the total number of days it takes to consume the available dog food.

If the dog eats 1/2 can of food each day, we can calculate the number of days the current supply will last by dividing the total amount of dog food by the amount consumed each day.

2 1/2 cans of dog food can be written as 2 + 1/2 = 2.5 cans.

The number of days the current supply will last is given by:

(2.5 cans) / (1/2 can per day)

To divide by a fraction, we multiply by its reciprocal:

(2.5 cans) * (2/1 can per day)

The cans cancel out, leaving us with:

2.5 * 2 = 5

Therefore, the current supply of dog food will last for 5 days.

The current supply of dog food, which is 2 1/2 cans, will last for 5 days based on the dog's consumption rate of 1/2 can per day.

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The greatest number of dabcoins the customer can purchase, rounded to the nearest whole number is 22 dabcoins.

Given the official exchange rate, in U.S. dollars per dabcoin, is $34 and a customer can spend a maximum of $750 on dabcoins and is charged the percentage. Let's find the greatest number of dabcoins the customer can purchase, rounded to the nearest whole number.

As per the given problem,

Official exchange rate, in U.S. dollars per dabcoin, is $34.

So, cost of 1 dabcoin = $34

The customer can spend a maximum of $750 on dabcoins.

So, $750 is the amount available to purchase dabcoin.

And, charged the percentage is not given so, let's consider it x%

Now, the amount spent on dabcoin can be represented by the product of the cost of 1 dabcoin and the number of dabcoins purchased plus the percentage charged.

Amount spent = (Cost of 1 dabcoin × Number of dabcoins purchased) + (Percentage charged)

According to the problem, the amount spent on dabcoin is limited to $750.

Therefore, (Cost of 1 dabcoin × Number of dabcoins purchased) + (Percentage charged) = $750... (1)

Also, cost of 1 dabcoin = $34.

Now substitute the value of cost of 1 dabcoin in equation (1).

$34 × Number of dabcoins purchased + (Percentage charged) = $750

Now solve for Number of dabcoins purchased.

Number of dabcoins purchased = $750/$34 - Percentage charged / $34

Number of dabcoins purchased = $22.0588235294 - Percentage charged / $34

To find the greatest number of dabcoins the customer can purchase, rounded to the nearest whole number, we need to round off Number of dabcoins purchased.

So, Number of dabcoins purchased = 22 - Percentage charged / $34

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Answer:

the sum adds up to 180

hope this helped!!.. <3

Step-by-step explanation:

Answer:

B

Step-by-step explanation:

(8,12)=(x1, y1)

(1,3)=(X2, y2)

distance=√(x2-x1)^2+(y2-y1)^2

=√(1-8)^2+(3-12)^2

=√(49+81)

√130

=11.40

Answer:

i think your answer is 8.

Answer:

Step-by-step explanation:

question 13:

the triangle with x is a 30-60-90 triangle which means

x= 17√3

then the traingle with y and z is a 45-45 triangle

z=34

y= 34√2

question 14:

x= 18√3

z=9

y= 18

To dilate a point at a point, you start by selecting a center of dilation and a scale factor.

Then, for each point that you want to dilate, you do the following:

For example, if you wanted to dilate point P by a scale factor of 2, with the center of dilation at O, you would draw a line segment from O to P, double its length, and then draw a new line segment from O to the new point, using the new length that you calculated. The new point would be the point that you arrive at when you complete this line segment.

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Answer:

B

Step-by-step explanation:

I say B because 2 divided by 6 is .333333333 and so on and so forth.

Hope this helped!!

ok

1.- Calcualte the total price for 19 sticks

price = 19 x 26

= $494

2.- Conclusion

there is not enough money to buy the sticks the total cost is

$494 and he only has $480.

3.- He needs 494 - 480 = $14 more money

Answer:

x=5 and x=-8

Step-by-step explanation:

take f(x)=g(x)

2x+30 = x2 +5x-10

you get above values of x

Answer:

43.49

Step-by-step explanation:

let his normal hourly rate be x

16x+7x/2=848

39x/2=848

x=848 ÷ 39/2

=43.48717

In the given problem, random samples of four different models of cars were selected and the gas mileage of each car was measured. The results are shown below:21 226 22 725 21
Given that,The null hypothesis H0: All the population means are equal. The alternative hypothesis H1: At least one population mean is different from the others .

To find the hypothesis test, we will use the one-way ANOVA test. We calculate the grand mean (X-bar) and the sum of squares between and within to obtain the F-test statistic. Let's find out the sample size (n), the total number of samples (N), the degree of freedom within (dfw), and the degree of freedom between (dfb).
Sample size (n) = 4 Number of samples (N) = n × 4 = 16 Degree of freedom between (dfb) = n - 1 = 4 - 1 = 3 Degree of freedom within (dfw) = N - n = 16 - 4 = 12 Total sum of squares (SST) = ∑(X - X-bar)2
From the given data, we have X-bar = (21 + 22 + 26 + 25) / 4 = 23.5
So, SST = (21 - 23.5)2 + (22 - 23.5)2 + (26 - 23.5)2 + (25 - 23.5)2 = 31.5 + 2.5 + 4.5 + 1.5 = 40.0The sum of squares between (SSB) is calculated as:SSB = n ∑(X-bar - X)2
For the given data,SSB = 4[(23.5 - 21)2 + (23.5 - 22)2 + (23.5 - 26)2 + (23.5 - 25)2] = 4[5.25 + 2.25 + 7.25 + 3.25] = 72.0 The sum of squares within (SSW) is calculated as:SSW = SST - SSB = 40.0 - 72.0 = -32.0
The mean square between (MSB) and mean square within (MSW) are calculated as:MSB = SSB / dfb = 72 / 3 = 24.0MSW = SSW / dfw = -32 / 12 = -2.6667
The F-statistic is then calculated as:F = MSB / MSW = 24 / (-2.6667) = -9.0
Since we are testing whether at least one population mean is different, we will use the F-test statistic to test the null hypothesis. If the p-value is less than the significance level, we will reject the null hypothesis. However, the calculated F-statistic is negative, and we only consider the positive F-values. Therefore, we take the absolute value of the F-statistic as:F = |-9.0| = 9.0The p-value corresponding to the F-statistic is less than 0.01. Since it is less than the significance level (α = 0.05), we reject the null hypothesis. Therefore, we can conclude that at least one of the population means is different from the others.

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Ƥ (A) is the set {⊘, {1}, {2}, {{1}}, {1, 2} {1, {1}}, {2, {1}}, {1, 2, {1}}}

To determine Ƥ(A), we need to find the power set of set A, which is the set of all possible subsets of A, including the empty set.

The power set is a set which includes all the subsets including the empty set and the original set itself. It is usually denoted by P. Power set is a type of sets, whose cardinality depends on the number of subsets formed for a given set

Set A = {1, 2, {1}}

The elements of the power set of A, denoted as Ƥ (A), are:

{⊘, {1}, {2}, {{1}}, {1, 2} {1, {1}}, {2, {1}}, {1, 2, {1}}}

Therefore, the correct option is

b. {⊘, {1}, {2}, {{1}}, {1, 2} {1, {1}}, {2, {1}}, {1, 2, {1}}}

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Answer:

Me and my massive brain, among other massive things of mine, believe it to be -1

Step-by-step explanation:

The value of the function F(g(x) will be equal to (7 / 2).

A function is defined as the expression that set up the relationship between the dependent variable and independent variable.

Given functions are:-

The function F(g(x) is written as :-

F(g(x) = 3 [ x + ( 1 / ( x-1 )) -7 ]

F(g(x) = 3 [ (3 + (1/2) - 7 ]

F(g(x) = 3 (7 / 2 ) -7

F(g(x) = ( 21/2) - 7

F(g(x) = ( 21-14 ) ( 2 )

F(g(x) = ( 7/2 )

Therefore the value of the function F(g(x) will be equal to (7 / 2).

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Answer:

84.85

Step-by-step explanation:

\( {a}^{2} + {b}^{2} = {c}^{2} \)

\( {60}^{2} + {60}^{2} = {c}^{2} \\ 3600 + 3600 = {c}^{2} \\ 7200 = {c}^{2} \)

now we should find the square root of 7200 to find c

\( \sqrt[]{7200} = 85.85.....\)

An ordered pair is a solution of an equation in two variables if replacing the variables by the coordinates of the ordered pair results in a true statement.

In mathematics, an equation in two variables represents a relationship between two quantities. An ordered pair consists of two values, typically denoted as (x, y), that represent the coordinates of a point in a two-dimensional plane.

When these values are substituted into the equation, if the equation holds true, then the ordered pair is considered a solution or a solution set to the equation. This means that the relationship described by the equation is satisfied by the values of the ordered pair. In other words, the equation is true when evaluated with the values of the ordered pair.

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The value for the hotspots of the similar triangles ∆ABC and ∆PWR are:

(1). angle B = 68°

(2). PQ = 5cm

(3). BC = 19.5cm

(4). area of ∆PQR = 30cm²

Similar triangles are two triangles that have the same shape, but not necessarily the same size. This means that corresponding angles of the two triangles are equal, and corresponding sides are in proportion.

(1). angle B = 180 - (22 + 90) {sum of interior angles of a triangle}

angle B = 68°

Given that the triangle ∆ABC is similar to the triangle ∆PQR.

(2). PQ/7.5cm = 12cm/18cm

PQ = (12cm × 7.5cm)/18cm {cross multiplication}

PQ = 5cm

(3). 13cm/BC = 12cm/18cm

BC = (13cm × 18cm)/12cm {cross multiplication}

BC = 19.5cm

(4). area of ∆PQR = 1/2 × 12cm × 5cm

area of ∆PQR = 6cm × 5cm

area of ∆PQR = 30cm²

Therefore, the value for the hotspots of the similar triangles ∆ABC and ∆PWR are:

(1). angle B = 68°

(2). PQ = 5cm

(3). BC = 19.5cm

(4). area of ∆PQR = 30cm²

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