Find the dot product for the pair of vectors

The value of the Expression is 610, 919.

We have,

Expression : [tex]\sum_{n=0}^{61[/tex] 700 (1.07[tex])^{n+1[/tex]

The expression is a summation formula, representing the sum of a series of values.

Here r= 1.07 and n is number of terms.

So, [tex]\sum_{n=0}^{61[/tex] 700 (1.07[tex])^{n+1[/tex]

= 700 (1) +  [tex]\sum_{n=0}^{61[/tex] 700 (1.07[tex])^{n[/tex]

= 700 + 700 ([tex]r^{61[/tex] - 1)/ (r-1)

= 700 + 700 ([tex](1.07)^{61[/tex]-1)/ (0.07)

= 700+ 700 (871.7428)

= 610, 919

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The value of f(1) is 3.

An estimate of the value of f(-1) is -0.2.

The values of x for which f(x) = 1 are: x = (0, 3).

The value of x such that f(x) = 0 is x = -0.5.

The domain of f is {-2, 4}.

The range of f is {-1, 3}.

The interval over which f is increasing is {-2, 1}.

In Mathematics and Geometry, a domain is the set of all real numbers for which a particular function is defined.

Furthermore, the vertical extent of any graph of a function represents all range values and they are always read and written from smaller to larger numerical values, and from the bottom of the graph to the top.

By critically observing the graph of the function shown in the image attached above, we can reasonably and logically deduce the following domain and range:

Domain = {-2, 4}.

Range = {-1, 3}.

In conclusion, we can logically deduce that this function is increasing over the [-2, 1] and decreasing over the interval [1, 4].

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The solution is, the simplification of the equation is :

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

Here, we have,

given that,

the expression is:

4x^2 - 4

now, we have to simplify this.

we know that ,

The a^2 - b^2 formula is also known as "the difference of squares formula". The a square minus b square is used to find the difference between the two squares without actually calculating the squares.

It is one of the algebraic identities.

It is used to factorize the binomials of squares.

The a^2 - b^2 formula is given as:

a^2 - b^2 = (a - b) (a + b).

so, we have,

4x^2 - 4

= (2x)^2 - 2^2

= (2x -2 ) (2x+2)

Hence, The solution is, the simplification of the equation is :

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

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a) The polar coordinates of point A are (√(18), π/4).

b) The curve is a circle centered at the origin with radius 6.

c) The directed distance is the value of r, which is 6 √(3).

To find the polar coordinates of point A on the curve, we need to convert the point from Cartesian to polar coordinates. The conversion formula is:

r = √(x² + y²)

θ = arctan(y/x)

Using the values of point A, we have:

r = √((-3)² + (-3)²) = √(18)

θ = arctan((-3)/(-3)) = arctan(1) = π/4

To find the polar form of the equation x² + y² + 6y = 0, we need to convert it from Cartesian to polar coordinates. The conversion formulae are:

x = r cos(θ)

y = r sin(θ)

Using these formulae, we can rewrite the equation as:

r² cos²(θ) + r² sin²(θ) + 6r sin(θ) = 0

Simplifying this equation, we get:

r = -6 sin(θ) / (1 - cos²(θ))

To find the directed distance when θ equals 4 π over 3, we need to substitute this value of θ into the polar equation we found in Part B. Doing so, we get:

r = -6 sin(4 π/3) / (1 - cos²(4 π/3))

r = -6(-√(3)/2) / (1 - (-1/2)²)

r = 6 √(3)

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Answer:

That statement describes an action taken by Keng to add a 3-inch-wide frame around all sides of his canvas, likely for artistic or aesthetic purposes. This would result in the canvas being extended by 3 inches in each direction, effectively increasing its overall dimensions and adding a border or frame around the original artwork. The purpose and effect of adding a frame may vary depending on the artistic intention of the artist, but it is a common practice in art and design to enhance the presentation and visual appeal of the artwork.

Step-by-step explanation:

The binomial whose greatest common factor is 7a³ is 14a⁴ + 35a³.

Given, the a³ is not common in A and C.

Now, in 14a⁴ + 35a³

14a⁴ = 2 x 7 x a³ x a

35a³ = 7 x 5 x a³

So, the common factors of 14 and 35 is 7.

Then, 14a⁴ + 35a³

= ( 2 x 7 x a³ x a) + (  5 x 7 x a³)

= 7a³ (2a + 5)

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Answer:

Well, if Alice buys 12 shirts which each cost $18.99 the equation would be 18.99 * 12 which = 227.88

Alice spent $227.88 on 12 shirts

Step-by-step explanation:

Answer:

Well, if Alice buys 12 shirts which each cost $18.99 the equation would be 18.99 * 12 which = 227.88

Step-by-step explanation:

The solution is , price of jeans = RD$ 1140 and, price of  handbag =

RD$ 2110.

Here, we have,

given that,

Margarita went to the store and bought a bag and some jeans for a total of RD$3,250.

Knowing that the handbag exceeds the jeans by RD$970,

now, we have to find that, how many do she pay for each article.

let, price of jeans = RD$ x

so, price of  handbag = RD$ (x +970)

ATQ, we get,

RD$ x + RD$ (x +970) = RD$3,250

or, RD$ 2x + 970 = RD$3,250

or, RD$ 2x = RD$ 2280

or, x = RD$ 1140

Hence, price of jeans = RD$ 1140 and, price of  handbag = RD$ 2110.

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The area of the parallelogram is 8.5 square miles. This is found by multiplying the length of one of the parallel sides, 2 miles, by the height, which is given as 4 1/4 miles.

To find the area of a parallelogram, we can multiply the length of one of its parallel sides by the length of its perpendicular height. Therefore, to find the area of this parallelogram, we need to determine its height.

We are given that one of the parallel sides has a length of 4 1/4 mi and the other has a length of 2 mi. We are also given that the length of the perpendicular on one of the parallel sides is 4 1/4 mi, which means that this is the height of the parallelogram.

So, the area of the parallelogram is

Area = base x height

Area = 2 mi x 4 1/4 mi

Area = 8 1/2 mi²

Therefore, the area of the parallelogram is 8 1/2 square miles or 8.5 square miles.

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Answer: 12 people

Step-by-step explanation:

0.30 x 40 = 12

A belt runs a pulley at 80 revolutions per minute.

(a) To find the angular speed of the pulley in radians per second, we can use the formula

ω = 2πf

Where

ω = angular speed in radians per second

f = frequency in Hertz

We know that the pulley rotates at a rate of 80 revolutions per minute. To convert this to a frequency in Hertz, we can divide by 60 seconds per minute

f = 80 rev/min ÷ 60 min/s = 4/3 Hz

Now we can plug in this value for f into the formula

ω = 2πf = 2π(4/3) = 8π/3 rad/sec

Therefore, the angular speed of the pulley is 8π/3 radians per second.

(b) To find the linear speed of the belt in centimeters per second, we can use the formula

v = rω

Where

v = linear speed in centimeters per second

r = radius of the pulley in centimeters

ω = angular speed in radians per second

We know the radius of the pulley is given in centimeters. Let's assume it is r cm. We just found the angular speed to be 8π/3 radians per second. Now we can plug in these values into the formula

v = rω = r(8π/3) = (8π/3)r cm/s

Therefore, the linear speed of the belt is (8π/3)r centimeters per second.

The given question is incomplete and the complete question is ''A belt runs a pulley at 80 revolutions per minute. Find the angular velocity of the pulley in radians per second. (b)Find the linear speed of the belt in centimeters per second''.

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Answer:

The formula looks like this:

where SA = surface area, l = length, w = width, and h = height. In the rectangular prism net above, l = 8 inches, w = 5 inches, and h = 3 inches. Simply put these numbers into the formula and solve for surface area.

The rectangular coordinates from the polar coordinates are: (-4√2, -4√2)

The steps to  to convert polar coordinates to rectangular coordinates are:

Step 1: Find the x -coordinate for the rectangular coordinate form of the point by using the equation x = r cos(θ)

Step 2: Find the y -coordinate for the rectangular coordinate form of the point by using the equation y = rsin(θ)

Step 3: Write the rectangular coordinates as (x,y) using the results from steps 1 and 2.

We are given the polar coordinates as (8, 225°)

Thus:

x-coordinate of rectangular coordinate = 8 cos 225 = -4√2

y-coordinate of rectangular coordinate = 8 sin 225 = -4√2

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The only equation that represents a direct variation is: y - 3x = 0

Direct Variation is defined as the relationship that exists between two variables in which one is a constant multiple of the other. For example, when one variable changes the other, then they are said to be in proportion. If b is directly proportional to a the equation is of the form b = ka (where k is a constant).

Making y the subject in each of the options gives:

A) y = 4x + 8

B) y = 7x - 2

C) y = 3x

D) y = 5x - 2

Looking at them, the only one where there is a relationship that exists between two variables in which one is a constant multiple of the other is option C

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You can use the notation P(A), read “the probability of event B, given event A” to write a conditional probability. The correct answer is C.

Conditional probability refers to the probability of one event occurring given that another event has already occurred. In this case, we are interested in the probability of event B occurring given that event A has already occurred, and we can represent this using the notation P(B|A), where '|' means 'given'.

For example, let's say we are interested in the probability of getting a head on a coin toss (event B), given that the coin was flipped and landed on heads (event A). We could represent this using the notation P(B|A). The value of P(B|A) would be 1, because if the coin already landed on heads, then the probability of getting a head on the next flip is certain.

Conditional probability is an important concept in probability theory and is often used in real-world applications, such as predicting the likelihood of a disease given certain symptoms, or the probability of an event occurring given certain conditions.

The correct answer is C.

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Answer:

(i) To compute the value of the fund based on the withdrawals required, we can use the formula for the future value of an annuity due:

FV = P * ((1 + r)^n - 1) / r) * (1 + r)

where FV is the future value of the annuity, P is the annual payment, r is the interest rate per period, n is the total number of periods, and the extra (1 + r) factor is because the payments are made at the beginning of each period.

In this case, P = €25,000, r = 0.065, n = 20. We want to find the future value at the end of the 20-year period:

FV = 25000 * ((1 + 0.065)^20 - 1) / 0.065) * (1 + 0.065)

FV ≈ €743,704.96

Therefore, the value of the fund based on the withdrawals required is approximately €743,704.96.

(ii) To compute the amount of each deposit needed in order to maintain the fund, we can use the formula for the present value of an ordinary annuity:

PV = P * ((1 - (1 + r)^(-n)) / r)

where PV is the present value of the annuity, P is the annual payment, r is the interest rate per period, and n is the total number of periods.

In this case, PV = €743,704.96, r = 0.065, n = 20. We want to find the annual payment:

PV = P * ((1 - (1 + 0.065)^(-20)) / 0.065)

P ≈ €22,630.53

Therefore, the amount of each deposit needed in order to maintain the fund is approximately €22,630.53.

(iii) To compute the total interest earned over the entire 50 years, we can subtract the total deposits from the total withdrawals, and then subtract the initial balance. The total deposits are the annual deposit amount times the number of years (30), and the total withdrawals are the annual withdrawal amount times the number of years (20). The initial balance is the present value of the annuity that we calculated in part (ii).

Total deposits = €22,630.53 * 30 = €678,915.90

Total withdrawals = €25,000 * 20 = €500,000

Initial balance = €743,704.96

Total interest earned = Total withdrawals - Total deposits - Initial balance

Total interest earned = €500,000 - €678,915.90 - €743,704.96

Total interest earned ≈ -€922,620.86

Note that the negative sign indicates that the insurance company actually earned interest on this annuity, rather than the couple earning interest on their investment. This is because the withdrawals are greater than the deposits, and the interest rate earned by the insurance company is greater than the interest rate paid to the couple.

Step-by-step explanation:

The probability that the first student selects Pennsylvania and the second student selects Virginia to the nearest hundredth of a percent is 0.64%.

The probability that the first student selects Pennsylvania is 1/13. Once Pennsylvania has been drawn, there are only 12 colonies left in the bag, so the probability that the second student selects Virginia is 1/12.

To find the probability that both events occur, we multiply the probabilities:

(1/13) x (1/12) = 1/156

To express this probability as a percentage, we multiply by 100:

(1/156) x 100 ≈ 0.64%

Therefore, the probability that the first student selects Pennsylvania and the second student selects Virginia is approximately 0.64%.

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The value of angle between the two vectors is 86⁰.

The value of angle between the two vectors is calculated as follows;

tan θ = vy/vx

where;

( -8, 9), (-9, -6)

vy = (-6 - 9) = -15

vx = (-9 + 8) = -1

tan θ = ( -15 ) / ( -1 )

tan θ = 15

The value of θ is calculated  by taking arc tan of the fraction,;

θ = tan ⁻¹ ( 15 )

θ =  86⁰

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17. 15 degrees

18. ABC

19. 5.88

20. 11.59

let me know if you'd like an explanation

The domain of the quadratic function in this problem is given as follows:

All real values.

The domain of a function is the set of all the possible input values that can be assumed by the function.

On the graph, the domain of the function is given by the values of x of the function.

A quadratic function has no restrictions on the domain, hence it is defined by all the real values.

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Answer: z = 8

Step-by-step explanation:

The diagram shows us that 8z + 3z + 2 = 90

so we can say that: 11z = 88

therefore z = 8.

Note: diagrams can be misleading! this diagram technically shows us that 64 < 26!

The dimensions of the island on the blueprint are 14 inches by 3.5 inches.

We have,

The actual dimensions of the island are 3 1/2 feet by 1 3/4 feet.

We need to find the dimensions of the island on the blueprint, given that the scale of the blueprint is 1 inch = 2 feet.

To convert the actual dimensions to the dimensions on the blueprint, we need to use the scale factor of 1 inch = 2 feet.

We can set up a proportion to relate the actual dimensions to the dimensions on the blueprint:

Actual dimension/blueprint dimension = scale factor

Let x be the length of the island on the blueprint.

Then we can set up the following proportion:

3.5 feet / (1.75 feet)

= x inches / 7 inches

Simplifying,

2 = x / 7

Multiplying both sides by 7, we get:

x = 14 inches

The length of the island on the blueprint is 14 inches.

Similarly, we can find the width of the island on the blueprint:

1.75 feet / 3.5 feet

= y inches / 7 inches

Simplifying, we have:

0.5 = y / 7

Multiplying both sides by 7, we get:

y = 3.5 inches

The width of the island on the blueprint is 3.5 inches.

Thus,

The dimensions of the island on the blueprint are 14 inches by 3.5 inches.

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The difference between the stratified and systematic random sampling is explained below.

Systematic random sampling is a method where every Kth person of the population is chosen to be part of the sample, whereas stratified random sampling is a method where the population is first divided into subgroups, and then drawing a simple random sample from each subgroup.

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Answer:

13: (2,3) 14: A. Yes, 90, 180, 270 B. No C. No

Step-by-step explanation:

13. Right: -2+4, Down: 5-2

14. If it can be rotated and be similar, it has rotational.

The inequality that represents the sentence "The stadium held less than 25,000 people" is given as follows:

c < 25,000.

The four inequality symbols, along with their meaning on the number line and the coordinate plane, are presented as follows:

The amount is less than in this problem, hence the symbol is given as follows:

<.

As the amount is less than 25000, the inequality is given as follows:

c < 25,000.

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After two years, Brooklyn will have more money in her savings account than Matthew.

Brooklyn's savings account grows at a slower rate than Matthew's initially but eventually catches up and surpasses Matthew's account due to the higher interest rate and monthly compounding.

a.

[tex]A = P(1 + r/n)^{nt}[/tex]

where A is the final amount, P is the initial principal (or net pay), r is the interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the number of years.

For Brooklyn saving account,

P = $625, r = 0.025, n = 12 (monthly compounding), and t = 2.

A = $625(1 + 0.025/12)^(12*2) = $686.71

For Matthew saving account,

P = $446, r = 0.04, n = 4 (quarterly compounding), and t = 2.

A = $446(1 + 0.04/4)^(4*2) = $506.84

b.

To graph Brooklyn's equation and Matthew's equation on the same graph, we can use the following equations:

Brooklyn's equation: A = 625(1 + 0.025/12)^(12t)

Matthew's equation: A = 446(1 + 0.04/4)^(4t)

Now,

The resulting graph will show the growth of each saving account over time.

The points of intersection on the graph represent the times when the two savings accounts have the same amount of money.

From the graph, we can see that the two accounts intersect at around 11 months and 23 months.

We can say that Brooklyn's savings account grows at a slower rate than Matthew's initially, but eventually catches up and surpasses Matthew's account due to the higher interest rate and monthly compounding.

Thus,

After two years, Brooklyn will have more money in her savings account than Matthew.

Brooklyn's savings account grows at a slower rate than Matthew's initially but eventually catches up and surpasses Matthew's account due to the higher interest rate and monthly compounding.

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Answer:

first option

Step-by-step explanation:

differentiate using the power rule

[tex]\frac{d}{dx}[/tex] (a[tex]x^{n}[/tex] ) = na[tex]x^{n-1}[/tex]

then

[tex]\frac{dy}{dx}[/tex] = [tex]\frac{dy}{dt}[/tex] × [tex]\frac{dt}{dx}[/tex] = [tex]\frac{\frac{dy}{dt} }{\frac{dx}{dt} }[/tex]

y = t² + 4t

[tex]\frac{dy}{dt}[/tex] = 2t + 4

x = t - 3

[tex]\frac{dx}{dt}[/tex] = 1

then

[tex]\frac{dy}{dx}[/tex] = [tex]\frac{2t+4}{1}[/tex] = 2t + 4

Answer:

2t + 4

Step-by-step explanation:

A parametric equation is one where x and y are defined separately in terms of a third variable (often the parameter t).

To find dy/dx from parametric equations, differentiate each equation with respect to the parameter t, then use the chain rule:

[tex]\boxed{\dfrac{\text{d}y}{\text{d}x}=\dfrac{\text{d}y}{\text{d}t} \times \dfrac{\text{d}t}{\text{d}x}}[/tex]

Differentiate the two parametric equations with respect to t:

[tex]x=t-3 \implies \dfrac{\text{d}x}{\text{d}t}=1[/tex]

[tex]y=t^2+4t \implies \dfrac{\text{d}y}{\text{d}t}=2t+4[/tex]

Use the chain rule to combine them:

[tex]\begin{aligned}\implies \dfrac{\text{d}y}{\text{d}x}&=\dfrac{\text{d}y}{\text{d}t} \times \dfrac{\text{d}t}{\text{d}x}\\\\&=(2t+4) \times \dfrac{1}{1}\\\\&=2t+4\end{aligned}[/tex]

Therefore:

[tex]\boxed{\dfrac{\text{d}y}{\text{d}x}=2t+4}[/tex]

There 1365 ways to choose four people from the group of fifteen.

b. There are 70 ways to choose  four Republicans from the group of eight Republicans.

C. The probability is about  0.0513, or  5.13%.

a.  To know the ways that four people can be selected from this group of fifteen is by:

nCr = n! / (r! x (n-r)!),

Where:

n  = total number of items

r = is the number of items to be selected,

!  = the factorial of a number.

Putting in the values into the the formula:

15C4 = 15! / (4! x (15-4)!)

(15-4)! = 11!

15C4 = 1365

B.  Since:

n = 8

r = 4

Putting in the values into the the formula:

8C4 = 8! / (4!  x (8-4)!)

(8-4)! = 4!

8C4 = 70

c. The  Probability = Number of ways to choose four Republicans / Number of ways to choose four people

Hence  Probability = 70 / 1365

                             =  0.0513

Therefore, the probability that the selected group will consist of all Republicans is about  0.0513, or  5.13%.

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The bearing of C to A is 239⁰.

The bearing of C to A is calculated by finding the value of angle C using cosine rule since we know the value of all the sides of the triangle.

AB² = AC² + CB² - 2(AC)(CB) cosC

12.5² = 16.25² + 8.5² - 2(16.25 x 8.5) x cos C

156.25 = 336.3125 - 276.25cosC

276.25cosC = 180.06

cosC = 180.06/276.25

cos (C) = 0.6518

C = cos⁻(0.6518)

C = 49.3⁰

The value of angle A is calculated as follows;

Sin A/CB = Sin C/AB

sin A/8.5 = sin 49.3/12.5

sin A = 8.5 [sin 49.3/12.5]

sin A = 0.5157

A = sin⁻¹ (0.5157)

A = 31⁰

The bearing of C to A is calculated as;

= 270⁰ - 31⁰

= 239⁰

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