Here is a rule for making a list of numbers: Each number is 1 less than twice the previous number.
Pick a number to start with, then follow the rule to build a list of 5 numbers.

The equation of the sphere passing through P(1,2,3) and Q(4,5,6) with its center at the midpoint of PQ is (x - 7/2)^2 + (y - 7/2)^2 + (z - 9/2)^2 = 27/2.

Given points are p(1,2,3) and q(4,5,6). We need to find the equation of the sphere passing through these points with its center at the midpoint of PQ. The midpoint of PQ is (x, y, z). We know that the center of the sphere lies at the midpoint of PQ.

So, we have:(1+x)/2 = 4-x/2 ...(i)

(since midpoint of PQ is (x,y,z), and P is (1,2,3) and Q is (4,5,6))

Substitute in eqn (i)

=> 1+x = 8 - x

=> x = 7/2

Similarly, we get:

y = 7/2

z = 9/2

Hence, the center of the sphere is C(7/2, 7/2, 9/2).

We know that the general equation of a sphere is given by

(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2

where (h, k, l) is the center and r is the radius of the sphere. To find the radius, we use the distance formula. Let the radius be r.

Distance between P(1, 2, 3) and Q(4, 5, 6) is given by

√[(4 - 1)^2 + (5 - 2)^2 + (6 - 3)^2] = √27

Hence, the radius of the sphere is r = √27/2.

Let the equation of the sphere be (x - 7/2)^2 + (y - 7/2)^2 + (z - 9/2)^2 = 27/2. So, the equation of the sphere passing through P(1,2,3) and Q(4,5,6) with its center at the midpoint of PQ is

(x - 7/2)^2 + (y - 7/2)^2 + (z - 9/2)^2 = 27/2.

Conclusion: The equation of the sphere passing through P(1,2,3) and Q(4,5,6) with its center at the midpoint of PQ is (x - 7/2)^2 + (y - 7/2)^2 + (z - 9/2)^2 = 27/2.

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Given that a sphere passes through P (1, 2, 3) and Q (4, 5, 6) with its center at the midpoint of PQ.

We need to find the equation of the sphere.

Step 1:

Find the center of the sphere.

We know that the center of the sphere lies at the midpoint of PQ.

The midpoint of PQ = $\frac{(P + Q)}{2}$

Midpoint of PQ = $\frac{(1 + 4, 2 + 5, 3 + 6)}{2}$

Midpoint of PQ = $(\frac{5}{2}, \frac{7}{2}, \frac{9}{2})$

Therefore, the center of the sphere is $(\frac{5}{2}, \frac{7}{2}, \frac{9}{2})$.

Step 2:

Find the radius of the sphere

Let the radius of the sphere be r.

Distance between P and Q is given by $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$= $\sqrt{(4 - 1)^2 + (5 - 2)^2 + (6 - 3)^2}$= $\sqrt{9 + 9 + 9}$= $\sqrt{27}$= $3\sqrt{3}$

The radius of the sphere = $\frac{PQ}{2}$= $\frac{3\sqrt{3}}{2}$

Step 3:

Write the equation of the sphere

The equation of a sphere with center $(x_0, y_0, z_0)$ and radius r is given by $$(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r^2$$

Therefore, the equation of the sphere passing through P(1, 2, 3) and Q(4, 5, 6) with its center at the midpoint of PQ is $$(x - \frac{5}{2})^2 + (y - \frac{7}{2})^2 + (z - \frac{9}{2})^2 = (\frac{3\sqrt{3}}{2})^2$$$$(x - \frac{5}{2})^2 + (y - \frac{7}{2})^2 + (z - \frac{9}{2})^2 = \frac{27}{2}$$

Hence, the equation of the sphere is $$(x - \frac{5}{2})^2 + (y - \frac{7}{2})^2 + (z - \frac{9}{2})^2 = \frac{27}{2}$$.

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In order to calculate the time it will take for Andrew to settle the loan, we can use the formula for compound interest. So, it will take Andrew approximately 5.22 years to settle the loan.

The formula is given as A = P(1 + r/n)^(nt), Where: A = the final amount, P = the principal (initial amount borrowed), R = the annual interest rate, N = the number of times the interest is compounded in a year, T = the time in years.

We know that Andrew borrowed R200 000 from a bank at an annual interest rate of 5.25% compounded quarterly and that he will make repayments of R6 000 at the end of every 3 months.

Since the first repayment will be made 3 months from now, we can consider that the initial loan repayment is made at time t = 0. This means that we need to calculate the value of t when the total amount repaid is equal to the initial amount borrowed.

Using the formula for compound interest: A = P(1 + r/n)^(nt), We can calculate the quarterly interest rate:r = (5.25/100)/4 = 0.013125We also know that the quarterly repayment amount is R6 000, so the amount borrowed minus the first repayment is the present value of the loan: P = R200 000 - R6 000 = R194 000

We can now substitute these values into the formula and solve for t: R194 000(1 + 0.013125/4)^(4t) = R200 000(1 + 0.013125/4)^(4t-1) + R6 000(1 + 0.013125/4)^(4t-2) + R6 000(1 + 0.013125/4)^(4t-3) + R6 000(1 + 0.013125/4)^(4t)

Rearranging the terms gives us: R194 000(1 + 0.013125/4)^(4t) - R6 000(1 + 0.013125/4)^(4t-1) - R6 000(1 + 0.013125/4)^(4t-2) - R6 000(1 + 0.013125/4)^(4t-3) - R200 000(1 + 0.013125/4)^(4t) = 0

Using trial and error, we can solve this equation to find that t = 5.22 years (rounded to 2 decimal places). Therefore, it will take Andrew approximately 5.22 years to settle the loan.

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JL, LK, JK are the smallest angle of a triangle is located across from its smallest side. The biggest side is on the other side of the biggest side.

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Answer:

The Answer Is X 9, Aka C, I just got it correct

By calculating , the values of angles D and E are 80 , 80 respectively.

What is triangle ?

Triangle can be defined in which it consists of three sides , three angles and the sum of three angles is always 180 degrees .

Given ,

from the figure ,

In triangle ECD

The sum of three angles = 180

so ,

E+C+D = 180

50+50+D = 180

D = 180-100

D = 80

And in the Quadrilateral AECB,

A+E+C+B = 360

110+70+100 + E = 360

E = 360 - 280

E = 80

Hence, the values of angles D and E are 80 , 80 respectively.

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Answer:

100 inches (using the given data)

98.43 (using exact numbers)

Step-by-step explanation:

1 inch = 2.54 centimeters

20 inches = 50.8 centimeters, but for the purpose of answering this question we will assume that 20 inches = 0.5 meters (instead of 0.58).

we must solve the following equation:

20 inches / 0.5 meters = X inches / 2.5 meters

(2.5  meters x 20 inches) / 0.5 meters = X inches

20 inches x 5 = X inches

100 inches = X inches

if we do the exact calculation, 2.5 meters in inches = 250 centimeters / 2.54 centimeters per inch = 98.43 inches

The first 3 terms of the sequence are 21, 24 and 29

From the question, we have the following parameters that can be used in our computation:

n² + 20

This means that

f(n) = n² + 20

The first 3 terms of the sequence is when n = 1, 2 and 3

So, we have

f(1) = 1² + 20 = 21

f(2) = 2² + 20 = 24

f(3) = 3² + 20 = 29

Hence, the first 3 terms of the sequence are 21, 24 and 29

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Answer:

B. 0.03110

Step-by-step explanation:

Given

Probability of Hit = 60%

Required

Determine the probability that he misses at 6th throw

Represent Probability of Hit with P

\(P = 60\%\)

Convert to decimal

\(P = 0,6\)

Next; Determine the Probability of Miss (q)

Opposite probabilities add up to 1;

So,

\(p + q = 1\)

\(q = 1 - p\)

Substitute 0.6 for p

\(q = 1 - 0.6\)

\(q = 0.4\)

Next,is to determine the required probability;

Since, he's expected to miss the 6th throw, the probability is:

\(Probability = p^5 * q\)

\(Probability = 0.6^5 * 0.4\)

\(Probability = 0.031104\)

Hence;

Option B answers the question

Answer:

b) 0.03110

Step-by-step explanation:

Got it right on the test.

The value of the expression 3(3+5g) is 9 + 15g.

Expression simply refers to the mathematical statements which have at least two terms which are related by an operator and contain either numbers, variables, or both. Addition, subtraction, multiplication, and division are all possible mathematical operations.

In this case, we want to simplify 3(3 + 5g)

It's important to open the parentheses. This will be:

=3(3 + 5g)

= 9 + 15g

Therefore, the value is 9 + 15g.

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Answer:

The length of string = 32.35 meter (Approx.)

Step-by-step explanation:

Given:

Vertical height of kite = 22 meter

Angle with horizontal = 43°

Find:

The length of string

Computation:

Using trigonometry function;

Vertical height of kite [Perpendicular]  = 22 meter

The length of string[Hypotenuse]

Sin θ = Perpendicular / Hypotenuse

Sin 43 = 22 / Hypotenuse

0.68 = 22 / Hypotenuse

Hypotenuse = 22 / 0.68

Hypotenuse = 32.35 meter

The length of string = 32.35 meter (Approx.)

The three angle bisectors of a triangle are concurrent. Their point of concurrence is called the incenter.

The triangle's three angle bisectors are contemporaneous, which means they cross at the same place.

The incenter is the location where the angle bisectors coincide.

1. The circle's inscribed center is known as the incenter.

2. The incenter is equidistant from the triangle's sides.

More examples of agreement

The centroid is the location where all of a triangle's medians meet.

The triangle's circumcenter is where the perpendicular bisectors of its sides meet.

The intersection of the perpendiculars emanating from the vertices on the opposing sides of a triangle is known as the orthocentre.

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The probability that the sample mean of a randomly selected sample of 95 televisions would be less than 118.8 months, given a population mean of 121 months and a variance of 256, can be calculated using the Central Limit Theorem. The calculated probability is 0.1151, rounded to four decimal places.

According to the Central Limit Theorem, the sample mean of a large enough sample size from any population will follow a normal distribution, regardless of the shape of the population distribution. Since the sample size is large (n = 95), we can use the normal distribution to approximate the probability.

The mean of the sample mean is equal to the population mean, which is 121 months. The standard deviation of the sample mean, also known as the standard error, can be calculated by dividing the population standard deviation by the square root of the sample size. In this case, the population variance is given as 256, so the standard deviation is √256 = 16. Therefore, the standard error is 16 / √95 ≈ 1.645.

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The value of x is 0.29 and value of y is 85.

Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles.

x = sin(sin⁻¹(0.29))

The sine and sin⁻¹x functions are inverse functions,

Therefore, sin(sin⁻¹(0.29)) simplifies to just 0.29. Thus,

x = 0.29

For y, we have:

y = cos⁻¹(cos(85))

The cosine and arccosine functions are also inverse functions, so they undo each other.

Therefore, cos⁻¹(cos(85)) simplifies to just 85.

y = 85

We notice that x is equal to the input value of the arcsine function, and y is equal to the input value of the cosine function.

In other words, x = sin(sin^-1(a)) = a and y = cos^-1(cos(b)) = b for any values of a and b.

This happens because the sine and cosine functions have a range of [-1, 1]

Since the input values in this case are already within these ranges, applying the inverse functions does not change them.

Therefore, we can expect this property to hold for any input values within the range of the inverse trigonometric functions.

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When the second derivative of a function equals zero, it indicates a possible point of inflection or a critical point where the concavity of the function changes. It is a significant point in the analysis of the function's behavior.

The second derivative of a function measures the rate at which the slope of the function is changing. When the second derivative equals zero at a particular point, it suggests that the function's curvature may change at that point. This means that the function may transition from being concave upward to concave downward, or vice versa.

Mathematically, if the second derivative is zero at a specific point, it is an indication that the function has a possible point of inflection or a critical point. At this point, the function may exhibit a change in concavity or the slope of the tangent line.

Studying the second derivative helps in understanding the overall shape and behavior of a function. It provides insights into the concavity, inflection points, and critical points, which are crucial in calculus and optimization problems.

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When the second derivative of a function equals zero, it indicates a critical point in the function, which can be a maximum, minimum, or an inflection point.

The second derivative of a function measures the rate at which the slope of the function is changing. When the second derivative equals zero, it indicates a critical point in the function. A critical point is a point where the function may have a maximum, minimum, or an inflection point.

To determine the nature of the critical point, further analysis is required. One method is to use the first derivative test. The first derivative test involves examining the sign of the first derivative on either side of the critical point. If the first derivative changes from positive to negative, the critical point is a local maximum. If the first derivative changes from negative to positive, the critical point is a local minimum.

Another method is to use the second derivative test. The second derivative test involves evaluating the sign of the second derivative at the critical point. If the second derivative is positive, the critical point is a local minimum. If the second derivative is negative, the critical point is a local maximum. If the second derivative is zero or undefined, the test is inconclusive.

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the diagram shows a regular dodecagon is the size of one interior angle

Answer:

The graph of the polynomial function f(x) = x^4 + x^3 - 2x^2 will depend on the behavior of the function as x approaches infinity and negative infinity, as well as the location and behavior of any local extrema.

To determine the behavior of the function as x approaches infinity and negative infinity, we can look at the leading term of the polynomial, which is x^4. As x becomes very large (either positive or negative), the x^4 term will dominate the expression, and f(x) will become very large in magnitude. Therefore, the graph of the function will approach positive or negative infinity as x approaches infinity or negative infinity, respectively.

To find any local extrema, we can take the derivative of the function and set it equal to zero:

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

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

Setting f'(x) equal to zero, we get:

4x(x^2 + 3/4x - 1) = 0

The solutions to this equation are x = 0 and the roots of the quadratic expression x^2 + 3/4x - 1. Using the quadratic formula, we can find these roots to be:

x = (-3 ± sqrt(33))/8

Therefore, the critical points of the function are x = 0 and x = (-3 ± sqrt(33))/8.

To determine the behavior of the function near each critical point, we can use the second derivative test. Taking the second derivative of f(x), we get:

f''(x) = 12x^2 + 6x - 4

Evaluating f''(0), we get:

f''(0) = -4

Since f''(0) is negative, we know that x = 0 is a local maximum of the function.

Evaluating f''((-3 + sqrt(33))/8), we get:

f''((-3 + sqrt(33))/8) = 11 + 3 sqrt(33)/2

Since f''((-3 + sqrt(33))/8) is positive, we know that x = (-3 + sqrt(33))/8 is a local minimum of the function.

Evaluating f''((-3 - sqrt(33))/8), we get:

f''((-3 - sqrt(33))/8) = 11 - 3 sqrt(33)/2

Since f''((-3 - sqrt(33))/8) is also positive, we know that x = (-3 - sqrt(33))/8 is another local minimum of the function.

Based on this information, we can sketch the graph of the function as follows:

Therefore, the statement that describes the graph of this polynomial function is: "The graph of the function has a local maximum at x = 0 and two local minima at x = (-3 ± sqrt(33))/8. As x approaches infinity or negative infinity, the graph of the function approaches positive or negative infinity, respectively."

Answer:

x = -3

y = 5

Step-by-step explanation:

See picture above

Answer:

The slope is 10

The y - intercept is 20

Step-by-step explanation:

The equations are written as:

y = ?x + ?

The first ? is the slope of the equation, every time y increases by that amount, x increases by one, the slope here is 10

The second ? is the y-intercept, or the place where x is 0. Here, when x is 0, y is 20, so it is the y-intercept

The optimal solution, obtained through graphical method, is: x₁ = 4, x₂ = 2, with the maximum value of Z = 80.

To solve the given linear programming problem graphically, we start by plotting the feasible region defined by the constraints:

Constraint 1: 10x₁ + 20x₂ ≤ 120

Constraint 2: 8x₁ + 8x₂ ≤ 80

Non-negativity constraint: x₁ ≥ 0, x₂ ≥ 0

The feasible region is the area of the graph that satisfies all the constraints.

Next, we calculate the objective function Z = 12x₁ + 16x₂. We plot the objective function as a line on the graph.

The optimal solution, which maximizes Z, is the point where the objective function line intersects the boundary of the feasible region.

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The algebraic expression 6x(3y + 4z) can be simplified as 18xy + 24xz

the algebraic expression given in the question is a linear equation in three variables that are x , y and z and the simplification of this algebraic expression would also contain all the three variables

To solve this question we have to use the algebraic of  distributive property of addition over multiplication

so, using the above property and substituting the values given in the question we get

6x(3y +4z) = 6x(3y) + 6x(4z)

                  = 18 xy + 24 xz

Hence, The algebraic expression 6x(3y + 4z) can be simplified as 18xy + 24xz

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Answer:

Step-by-step explanation:

Given data:

Circumference of the base of the cone = 24in.

Recall that circumference (in this case) is the distance round the base of the cone and from here the diameter D=12in. Radius = 6in

Surface area l = pie x radius ( slant height + radius)

= 3.142 x 6 (20 + 6)

= 3.142 x 6 (26)

= 3.142 x 156

= 490.152in^2

Answer:

384pi inches   hop it halp!!!

Step-by-step explanation:

The correct expression is,

⇒ p = k / n

where, p is the number of pizza and k is the constant and n is the number of persons.

Any relationship that is always in the same ratio and quantity which vary directly with each other is called the proportional.

Given that;

The number of pizza slices varies inversely as the number of people sharing the whole pizza.

Let p is the number of pizza and k is the constant and n is the number of persons.

Hence, We can formulate the expression,

⇒ p = k / n

Where, k is constant of proportional.

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The integral. 1 [*+1590 Joe (x15 + 15*)dx is  154/3.

Based on the information provided, I assume you want to evaluate the integral of a given function. Let me rewrite the function in a more standard format:

∫(x^15 + 15x) dx

Now, let's evaluate the integral step by step:

1. Identify the function within the integral: f(x) = x^15 + 15x

2. Apply the power rule for integration, which states that ∫x^n dx = (x^(n+1))/(n+1) + C, where n is a constant and C is the constant of integration.

3. Using the power rule for the first term: ∫x^15 dx = (x^(15+1))/(15+1) = (x^16)/16

4. Using the power rule for the second term: ∫15x dx = 15∫x dx = 15(x^2)/2

5. Combine both terms and add the constant of integration, C: (x^16)/16 + (15x^2)/2 + C

So, the evaluated integral of the given function is:
Putting these together, we have:

∫[1,5] (x^2 + 15) dx = [(1/3)x^3 + 15x] evaluated from x=1 to x=5

Plugging in these values, we get:

[(1/3)(5^3) + 15(5)] - [(1/3)(1^3) + 15(1)]

Simplifying, we get:

(125/3 + 75) - (1/3 + 15)

= (200/3) - (46/3)

= 154/3
= (x^16)/16 + (15x^2)/2 + C

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Hey there! I'm happy to help!

We see that Steve runs 2.5 miles per day and Sam runs 6.5 miles per day. If you add these together, we see that they run 9 miles every day.

We now need to see how many of these 9 mile days fit into this 450 miles. Let's divide 450 by 9 and we will see how many days it will take for them to reach 450 miles combined.

450÷9=50

Therefore, it will take Steve and Sam 50 days to reach 450 miles combined!

Have a wonderful day! :D

Answer:

3,6,9,12,15,18

Step-by-step explanation:

how i came up with this ordered pair is taking the multiples of three

this relation is a function because it is asking to list six ordered pairs showing the output of the thickness of the paper that he folds and the way that i came up with the ordered pairs in Part A is because you have to take the multiples of three or you could just take 6 and multiple it by 3 and you would get the same answer

Answer:

n≥5

Step-by-step explanation:

Answer:

There is no sufficient evidence

Step-by-step explanation:

The hypothesis :

H0 : μ = 436

H1 : μ < 436

Xbar = 433

s = 23

Sample size, n = 28

The test statistic :

(xbar - μ) ÷ (s/√(n))

(433 - 436) ÷ (23/√(28))

-3 ÷ 4.3465914

Test statistic = - 0.690

The Pvalue, calculated from test statistic value, df = n - 1 = 28 - 1 = 27

Pvalue = 0.248

Pvalue > α ; Hence, we fail to reject the null and conclude that there is no sufficient evidence to support the claim that the bags are underfilled

Answer: a. The balance in the account will not be affected until the police car is received

Step-by-step explanation:

The options are:

a)The balance in the account will not be affected until the police car is received.

b)The balance in the account will be increased.

c)The balance in the account will be decreased.

d)Purchase orders never affect any budgetary account balances.

Unencumbered balance is typically used in governmental accounting and it refers to part of an appropriation not spent yet. In governmental accounting, this is the part of the funds available that's left and hasn't been used yet.

Based on the information given, the effect that this will event will have on the unencumbered balance in the account will be that the balance in the account will not be affected until the police car is received.

Answer:

f(x) = -3(6)^x + 2

Step-by-step explanation: