What is the common ratio?

n f(n)

1 300

2 375

3 468.75

4 585.9375

Write an explicit rule for the geometric sequence

What is f(12)?

The eigenvectors v1 and v2 are such that when multiplied by matrix A, they result in scalar multiples of themselves (scaled by their respective eigenvalues).

An eigenvector is a non-zero vector that, when multiplied by a matrix, results in a scalar multiple of itself. In other words, if A is a square matrix and v is an eigenvector of A, then Av = λv, where λ is the eigenvalue associated with v.

Given the eigenvalues and eigenvectors for matrix A as λ1 = 2, v1 = [-5 1]ᵀ and λ2 = 6, v2 = [5 4]ᵀ, we can say that:

- Av1 = λ1v1 = 2[-5 1]ᵀ = [-10 2]ᵀ
- Av2 = λ2v2 = 6[5 4]ᵀ = [30 24]ᵀ

Therefore, the eigenvectors v1 and v2 are such that when multiplied by matrix A, they result in scalar multiples of themselves (scaled by their respective eigenvalues).

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Answer:

x^2+y^2+12x+36

Step-by-step explanation:

Expand the square

(x+6)^2+y^2

(x+6)(x+6)+y^2

Distribute

(x+6)(x+6)+y^2

x(x+6)+6(x+6)+y^2

Distribute#2

x(x+6)+6(x+6)+y^2

x^2+6x+6(x+6)+y^2

Distribute

x^2+6x+6(x+6)+y^2

x^2+6x+6x+36+y^2

Combine line terms

x^2+6x+6x+36+y^2

x^2+12x+36+y^2

Rearrange Terms

x^2+12x+36+y^2

x^2+y^2+12x+36

Suppose that foreign citizens decide to purchase more U.S. pharmaceuticals and U.S. citizens decide to buy stock in foreign corporations. Other things being the same, these actions would lead to an increase in demand for U.S. pharmaceuticals and a decrease in demand for U.S. stocks, leading to changes in the relative prices of these assets.

An increase in demand for U.S. pharmaceuticals would lead to higher prices for these products, assuming that the supply of pharmaceuticals remains constant. This increase in demand could benefit U.S. pharmaceutical companies, as they would be able to charge higher prices for their products, resulting in increased revenues and profits. However, higher prices could also result in reduced affordability and access to these products for U.S. consumers.

On the other hand, a decrease in demand for U.S. stocks could lead to lower prices for these securities, assuming that the supply of foreign stocks remains constant. This decrease in demand could negatively impact U.S. investors who hold these stocks, as the value of their holdings would decline. However, it could also lead to increased investment in foreign corporations, potentially benefiting these companies and the economies in which they operate.

Overall, the actions of foreign and U.S. citizens can significantly affect the prices of different assets and the performance of various industries and economies. Understanding these relationships and dynamics is important for investors and policymakers in making informed decisions about investment and trade.

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The amount invested was approximately P239,534.52.

To solve this problem, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

where A is the amount after t years, P is the principal amount, r is the annual interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the time in years.

In this problem, we know that the amount grew to P250,500 after some time. We also know that the interest rate is 5 3/4%, or 0.0575, and that the investment was made on July 12, 2020, which is about 8 months before March 11, 2021.

Let's first convert the interest rate to a monthly rate, since we need to compound the interest monthly:

r = 0.0575/12 = 0.004792

Next, let's calculate the number of months between July 12, 2020 and March 11, 2021:

8 months + 31 days/365 days = 8.0849 months

Now we can use the formula to solve for P:

250500 = P(1 + 0.004792/12)^(12*8.0849)

250500/P = 1.045305

P = 250500/1.045305

P = 239534.52

Therefore, the amount invested was approximately P239,534.52.

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The value of the angle x formed by the two intercepted arcs is: 29.5

The central angle is the angle that a circle's arc occupies in its center. The arms of the central angle are formed by the radius vectors.

Now, angles that are formed inside of a circle by two chords create four arcs on a circle, which you can see in this diagram. The measure of the angle is equal to half the sum of the intercepted arcs.

We are given that:

mAB = 34

mCD = 25

Thus:

x = (34 + 25)/2

x = 59/2

x = 29.5

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If a particular bowling ball has a mass of 12 kilograms, and has a volume of 4 liters, the volume of a ball with a mass of 9 kilograms is 3 liters.

If the mass of a bowling ball varies directly with its volume, then we can use the formula:

mass = k * volume

where k is the constant of proportionality.

To find the value of k, we can use the given information:

12 = k * 4

k = 3

Now, we can use the value of k to find the volume of a ball with a mass of 9 kilograms:

9 = 3 * volume

volume = 3

This result makes sense, as we would expect the volume to be smaller if the mass is smaller, assuming the density of the ball remains constant.

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Answer: 87

Step-by-step explanation:

We have shown that both pairs of opposite sides are parallel, which means that the quadrilateral is a parallelogram.

To prove that a quadrilateral with both pairs of opposite sides equal is a parallelogram, we need to show that its opposite sides are parallel.

Let ABCD be the given quadrilateral with AB = CD and BC = DA. We need to show that AB || CD and BC || DA

Since opposite sides are equal, we have AB = CD and BC = DA. Adding these two equations, we get:

AB + BC = CD + DA

By the triangle inequality, we know that AB + BC > AC and CD + DA > AC. Therefore, we can write:

AB + BC > AC > CD + DA

Subtracting BC and CD from both sides, we get:

AB > AC - BC > DA

Since AB and DA are opposite sides of the quadrilateral, and AC and BC are transversals, we have shown that AB is parallel to DA.

Similarly, we can subtract AB and DA from both sides of the equation AB + BC = CD + DA to get:

BC > CD - DA > AB

Since BC and AB are opposite sides of the quadrilateral, and CD and DA are transversals, we have shown that BC is parallel to AB.

Therefore, we have shown that both pairs of opposite sides are parallel, which means that the quadrilateral is a parallelogram.

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Answer: 32 km

Step-by-step explanation:

If he travelled 4/5 of the trip by bike, then he travelled 1/5 on foot.

so he travelled 160/5 = 32 km on foot. Phew! thats a long walk.

The after-tax present worth according to the given values is $732,140.56 and the internal rate of return is 22.65%.

(a) To compute the after-tax present worth, we need to determine the net cash flow for each year and discount it to present value using the after-tax MARR of 15%.

Year 0: Initial cost of equipment = -$18,600
Years 1-10:
Revenue from bags = (100.3 bags/carton) x ($0.03/bag) x (200,000 cartons/year) = $120,780
Cost savings from reducing overfilling = (5.5%) x ($0.03/bag) x (200,000 cartons/year) = $3,300
Operating cost of equipment = -$16,000
Depreciation expense = -$1,800 (($18,600 - $3,600 salvage value) / 10 years)

Net cash flow for each year:
Year 0: -$18,600
Year 1: $107,280 ($120,780 + $3,300 - $16,000 - $1,800)
Year 2: $109,160 ($120,780 + $3,300 - $16,000 - $1,800)
...
Year 10: $113,640 ($120,780 + $3,300 - $16,000 - $1,800)

Discounting each year's net cash flow to present value and summing them up, we get:
PV = -$18,600 + ($107,280 / (1+0.15)^1) + ($109,160 / (1+0.15)^2) + ... + ($113,640 / (1+0.15)^10)
PV = -$18,600 + $750,740.56
PV = $732,140.56

Therefore, the after-tax present worth is $732,140.56.

(b) To compute the after-tax internal rate of return, we need to find the discount rate that makes the net present value equal to zero. We can use trial and error or a financial calculator to solve this.

Using trial and error, we find that a discount rate of approximately 22.65% makes the net present value equal to zero. Therefore, the after-tax internal rate of return is approximately 22.65%.

(c) To compute the after-tax simple payback period, we need to determine how long it takes for the cumulative net cash flow to equal the initial cost of the equipment.

Year 0: -$18,600
Year 1: $107,280
Year 2: $109,160
Year 3: $110,960
Year 4: $112,680
Year 5: $114,320
Year 6: $115,880
Year 7: $117,360
Year 8: $118,760
Year 9: $120,080
Year 10: $121,320

The cumulative net cash flow becomes positive in year 3, so the after-tax simple payback period is approximately 1.9 years (between year 2 and year 3).

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(a) First step of implicit differentiation:
-0.82dP/dt = dq/dt
The equation that results from differentiating each side of the equation with respect to P is:
-0.82 - 0.82P(d^2q/dP^2) = (dQ/dP)(dP/dt)/(dq/dt)
To find the elasticity of demand, we need to use the formula:
E = (dQ/Q)/(dP/P)
We can rewrite this as:
E = (dQ/dP) * (P/Q)
We know that dQ/dP = -0.82P(d^2q/dP^2), so we substitute that into the formula:
E = (-0.82P(d^2q/dP^2)) * (P/q)
Simplifying this expression, we get:
E = -0.82P^2(d^2q/dP^2)/q

(b)  We can solve the original equation for q by dividing both sides by -0.82:
q = (-1/0.82)P + D
Taking the derivative of q with respect to P, we get:
dq/dP = -1/0.82
We can use this result to calculate the elasticity of demand using the formula:
E = (dQ/dP) * (P/Q)
Substituting the values we found, we get:
E = (-1/0.82) * (P/((-1/0.82)P + D))
Simplifying this expression, we get:
E = -1/(P/((-1/0.82)P + D))
E = -1/((D/P) - 1.22)

(a) Interpretation: The elasticity of demand is a measure of how much the quantity demanded changes in response to a change in price. If E > 1, demand is considered elastic, meaning that a small change in price will result in a large change in quantity demanded. If E < 1, demand is considered inelastic, meaning that a change in price will result in a small change in quantity demanded. If E = 1, demand is unit elastic, meaning that a change in price will result in an equal proportional change in quantity demanded.
(b) Interpretation: The elasticity of demand in this case depends on the values of D and P. If D is relatively small compared to P, then the elasticity of demand will be close to -1.22, which is the upper limit of the elasticity. If D is relatively large compared to P, then the elasticity of demand will be close to zero, which means that demand is very inelastic.

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The correct graph of function f(x) =-3ˣ - 2 is shown in option 4.

We have to given that;

Function is,

⇒ f(x) = -3ˣ - 2

Now, In option 4;

Take a point (0, - 3) into function as;

Plug x = 0, y = - 3;

⇒ f(x) = -3ˣ - 2

⇒ - 3 = - 3⁰ - 2

⇒ - 3 = - 1 - 2

⇒ - 3 = - 3

Thus, The correct graph of function f(x) =-3ˣ - 2 is shown in option 4.

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A call that lasts for a length of 4 minutes would cost the same at either hotel.

For Hotel A: Cost = 2(4) + 5 = 13

For Hotel B: Cost = 4(4) - 3 = 13 minutes

We can determine the length of phone call that would cost the same at either hotel by the following equations.

Given:

Hotel A, cost of the call:

Cost for Hotel A = 2x + 5

For Hotel B, the cost of the call:

Cost for Hotel B = 4x - 3

For the length of the call that would cost the same at either hotel, we shall set these two equations equal to each other and solve for x:

2x + 5 = 4x - 3

Subtracting 2x from both sides, we get:

5 = 2x - 3

Adding 3 to both sides, we get:

8 = 2x

Dividing both sides by 2, we get:

x = 4

So, a call that lasts 4 minutes would cost the same at either hotel.

We can verify this, let's plug x = 4 into both equations:

For Hotel A: Cost = 2(4) + 5 = 13

For Hotel B: Cost = 4(4) - 3 = 13

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In this scenario, we are trying to estimate a positive random variable, which has a known mean and variance. We are given a measurement X, which is equal to the product of the random variable and another variable W. We assume that W is independent of O, has a zero mean, unit variance, and known fourth moment.

In order to estimate the random variable, we need to find the linear LMS estimator of it based on X. The LMS estimator is a method of finding an estimator that minimizes the mean squared error of the estimation. In this case, we are looking for a linear function of X that gives us the best estimate of the random variable.

Using the linear LMS estimator, we can find the estimator of the random variable based on X = 1. The estimator is given by E[Y|X] = y + (1/O) Cov[Y,X] = y + (1/O) Var[W] = y.

In part (b) of the question, we are asked to find the linear LMS estimator of the random variable based on Y = y. In this case, the estimator is given by E[X|Y] = y + (Cov[X,Y]/Var[Y]) (Y - y) = y + (Cov[X,Y]/o?) (Y - y).

In summary, we can estimate a positive random variable based on a measurement X using the linear LMS estimator. The estimator can be found by finding a linear function of X that minimizes the mean squared error of the estimation. In the given scenario, we can find the estimator of the random variable based on X = 1 and Y = y.

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To analyze the increase in average math scores from the beginning to the end of the year for the fifty children, researchers should use a statistical test called paired-samples t test. So fourth option is the correct answer.

The paired-samples t-test is used when comparing the means of two related groups or when analyzing data with repeated measures on the same group. In this case, the scores of the same group of children are measured at the beginning and end of the year, making it a paired design.

The test would determine whether there is a significant difference between the mean math scores at the beginning and end of the year, indicating an increase or decrease in scores over time for the same group of children.

So the correct answer is fourth option paired sample t test.

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Here is the code for the condition given:

```python
age = 65  # Replace this with the given age

if age >= 60:
   print("Senior Citizen")
else:
   print("Not Senior Citizen")

Let us explain the answer in detail:
1. Store the given age in a variable called `age`.
2. Use an if statement to check if the age is greater than or equal to 60.
3. If the condition is true, print "Senior Citizen".
4. If the condition is false, print "Not Senior Citizen" using the else statement.

Here's a code example:

```python
age = 65  # Replace this with the given age

if age >= 60:
   print("Senior Citizen")
else:
   print("Not Senior Citizen")
```

Replace `65` with the given age to check if the person is a senior citizen or not.

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The evaluated lateral surface area is 261.8 square meters, under the condition that the base length is 20 m and height is 13 m.

The lateral surface area of a cylinder is given by the formula 2πrh
Here,
r = radius of the base
h = height of the cylinder.
For the given case, the base length is 20 m and height is 13 m. Then the base length is stated  instead of the radius, we have  to evaluate the radius first.
The radius of a cylinder can be found applying the formula r = l/2π
Here,
l = base length.
So, staging l = 20 m
, we get
r = 20/(2π)
≈ 3.18 m
Now that we have received the radius and height, we can evaluate the lateral surface area applying the formula mentioned above.
Staging
r = 3.18 m
h = 13 m,
we get:
Lateral surface area
= 2πrh
≈ 261.8 m²
Then, the lateral surface area of the given cylinder is approximately 261.8 square meters.
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The time the ball will hit the ground after the free throw has been shot is 6.7 seconds

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

f(t) = -0.8t² + 4t + 9

The graph is added as an attachment

From the graph, we have

Maximum height = 14 ft

From the graph, we have

Time to reach maximum height = 2.5 seconds

From the graph, we have

Time to hit the ground = 6.7 seconds

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The expression is factorized to give j = -10  and j = 4

From the information given, we have the quadratic equation as;

j²+ 6j - 40

Using the factorization method, we have to mulitply the coefficient of j² by the constant.

After this, find the pair factors of the product that adds up to give 6

Substitute the values

Then, we have;

j² + 10j - 4j - 40

group the expression in pairs

(j² + 10j) - (4j- 40)

factor the common terms

j(j + 10) - 4(j + 10)

We have;

(j + 10) (j - 4)

j + 10 = 0

collect the terms

j = -10

j = 4

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The distance between the two points plotted include the following: A. 12 units.

In Mathematics and Geometry, the distance between two (2) points that are on a coordinate plane can be calculated by using the following mathematical equation:

Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]

Where:

x and y represent the data points (coordinates) on a cartesian coordinate.

By substituting the given points into the distance formula, we have the following;

Distance = √[(-4 + 4)² + (-6 - 6)²]

Distance = √[(0)² + (-12)²]

Distance = √[0 + 144]

Distance = √144

Distance = 12 units.

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The formula that can be used to calculate the density of books in the box is a. Density = the number of books/volume of the box.

This formula relates the number of books in the box to the volume of the box, allowing Bennie to determine how densely packed the books are in the given space. The term "density" refers to the amount of mass (in this case, the number of books) per unit volume, and this formula helps to quantify that relationship.

The volume of the box is the area of ​​the box or cube. A box is a three-dimensional object in the shape of a cube, a three-dimensional object with a square face. The cube, also known as a regular hexahedron, is one of the five Platonic bodies.

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Answer:

I believe the correct answer here would be C.

Step-by-step explanation:

The height of the cone is h = 3V / πr². Then the correct option is B.

Given that:

Volume of the cone, V = πr²h / 3

Simplify the equation for the value of h, then we have

V = πr²h / 3

3V = πr²h

h = 3V / πr²

The height of the cone is h = 3V / πr². Then the correct option is B.

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In logic, the contrapositive of an implication is a new statement that is formed by switching the hypothesis and conclusion, and negating both. In other words, the contrapositive of "if p then q" is "if not q, then not p."

Please find the contrapositives of the given statements below:

(a) Original statement: If m or n is even, then mn is even.
Contrapositive: If mn is not even, then neither m nor n is even.

(b) Original statement: If x ∈ A∩B, then 2 ∈ A and r ∈ B, where A and B are subsets of R.
Contrapositive: If 2 ∉ A or r ∉ B, then x ∉ A∩B, where A and B are subsets of R.

(c) Original statement: Let S be a subset of R. If u is an upper bound for S, then for every ε > 0, there exists some E ∈ S such that u - E < ε.
Contrapositive: Let S be a subset of R. If for some ε > 0, there is no E ∈ S such that u - E < ε, then u is not an upper bound for S.

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There are approximately 3.72 x 10⁴¹ ways that Anna can hand out 1 treat to each of the 36 children who come to her door.

Anna has 2 dozen Rollo bars, which is equivalent to 2 x 12 = 24 Rollo bars.

She also has 1 dozen apples, which is equivalent to 1 x 12 = 12 apples.

So, Anna has a total of 24 + 12 = 36 treats to hand out.

Now, she needs to hand out 1 treat to each of the 36 children who come to her door.

This can be thought of as selecting 1 treat from the total of 36 treats for each child, without replacement, as each child can only receive 1 treat.

The number of ways to do this is given by the concept of permutations, denoted by "nPr", which is calculated as;

nPr = n! / (n - r)!

where n is the total number of items (treats) to choose from, and r is the number of items (treats) to choose.

In this case, n = 36 (total number of treats) and r = 36 (number of children).

Plugging in the values, we get;

36P36 = 36! / (36 - 36)! = 36! / 0! = 36!

Since 0! (0 factorial) is equal to 1, we can simplify further:

36! / 1 = 36!

36! ≈ 3.72 x 10⁴¹

So, there are 3.72 x 10⁴¹ ways that Anna can hand out 1 treat to each of the 36 children who come to her door.

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Answer: 24√6 in simplest radical form.

Explanation: We can simplify this expression by first combining the coefficients (the numbers in front of the square roots) and then multiplying the square roots together.

3√2 * 2√8 * √3 * √6

= (3 * 2 * √2 * √8) * (√3 * √6)

= (6 * √16) * √18

= (6 * 4) * √2 * √3 * √3

= 24√6

Therefore, the answer is 24√6 in simplest radical form.

the answer is  -2 (6 x + 7)

The volume of the solid obtained by cylindrical shells method/ rotating the region bounded by xy = 5, x = 0, y = 5, y = 7 about the x-axis is 8π cubic units and by y = x^3, y = 27, x = 0 about the x-axis is 57π/5 cubic units.

To use the method of cylindrical shells, we need to consider an infinitesimal vertical strip at a distance x from the y-axis with width dx. This strip will have height y, which we can find using the equation of the curve.

The circumference of the shell will be 2πx, and the volume of the shell will be its height times its circumference times its thickness, which is dx.

We want to rotate this region about the x-axis, so the height of the shell will be y - 5, and its circumference will be 2πx. The volume of the shell will be (y - 5) * 2πx * dx.

Integrating this expression from x = 1 to x = 5 (since y = 5/x intersects xy = 5 at x = 1 and y = 7/x intersects y = 7 at x = 5), we get:

V = ∫(y=5/x to y=7/x) (y - 5) * 2πx dx
 = 2π ∫(x=1 to x=5) (7/x - 5/x) * x dx
 = 2π ∫(x=1 to x=5) (7 - 5) dx
 = 2π * 4
 = 8π

Therefore, the volume of the solid obtained by rotating the region bounded by xy = 5, x = 0, y = 5, y = 7 about the x-axis is 8π cubic units.

For the second problem, the region bounded by y = x^3, y = 27, x = 0

We want to rotate this region about the x-axis, so the height of the shell will be 27 - y, and its circumference will be 2πx. The volume of the shell will be (27 - y) * 2πx * dx.

Integrating this expression from x = 0 to x = 3 (since y = 27 intersects y = x^3 at x = 3), we get:

V = ∫(y=x^3 to y=27) (27 - y) * 2πx dx
 = 2π ∫(x=0 to x=3) (27 - x^3) * x dx
 = 2π (∫(x=0 to x=3) 27x dx - ∫(x=0 to x=3) x^4 dx)
 = 2π (81/2 - 243/5)
 = 2π * 57/10
 = 57π/5
Therefore,the volume of the solid obtained by rotating the region bounded by y = x^3, y = 27, x = 0 about the x-axis is 57π/5 cubic units.

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Answer:

[tex]f(x) = 1000 {(0.80)}^{x - 1} [/tex]

B is the correct function.

Step-by-step explanation:

10 to the power of 6 is 1,000,000

10 x 10 x 10 x 10 x 10 x 10 = 1,000,000

10 to the power of 7 is 10,000,000

10 x 10 x 10 x 10 x 10 x 10 x 10 = 10,000,000

=(5 x 1,000,000)(5 x 10,000,000)

=5,000,000 x 50,000,000

= 250,000,000,000,000