Steve invests $2,100 in an account that earns 6.9% interest, compounded continuously. What is the value of the account after 13 years? Round your answer to the nearest hundredth.

The inverse of the radical equation x = [√(a + 2) + √(a - 2)] / [√(a + 2) - √(a - 2)] is the equation a = x + 1 / x. (Correct choice: A)

In this problem we find the case of a radical equation in explicit form, that is, a variable x only in terms of a variable a, and we need to obtain the inverse of the function, that is, an expression in terms of a variable x. This problem can be solved by algebra properties.

First, write the original equation:

x = [√(a + 2) + √(a - 2)] / [√(a + 2) - √(a - 2)]

Second, rationalize the expression:

x = [[√(a + 2) + √(a - 2)] · [√(a + 2) + √(a - 2)]] / [[√(a + 2) - √(a - 2)] · [√(a + 2) + √(a - 2)]]

x = [√(a + 2) + √(a - 2)]² / [(a + 2) - (a - 2)]

x = [a + 2 - 2 ·√(a² - 4) + a - 2] / 4

x = [a - √(a² - 4)] / 2

2 · x = a - √(a² - 4)

Third, eliminate the square root operator by algebra properties:

√(a² - 4) = a - 2 · x

a² - 4 = (a - 2 · x)²

Fourth, find the solution to the expression:

a² - 4 = a² - 4 · x · a + 4 · x²

- 4 = - 4 · x · a + 4 · x²

1 = x · a - x²

1 + x² = x · a

a = (1 + x²) / x

a = 1 / x + x

a = x + 1 / x

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a) The average rate of change is -0.6333.

b) For $500 after the inflation rate from 1917 to 1920 the worth will be 242.20631771406.

The rate of change (ROC) measures how quickly a variable alters over a predetermined amount of time.

Given:

a) The average rate of change

= (15.9- 17.8)/ (1920 - 1917)

= -1.9 / 3

= -0.6333

So, the equation can be

y = -0.6333x + b

and, 17.8 = -0.63333(1917)+ b

b= 1231.79

So, y= -0.6333x + 1231.79

b) For $500 after the inflation rate from 1917 to 1920 the worth will be 242.20631771406.

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The value of the trigonometric equation sin D = 0.923

Trigonometry is the study of the relationships between the angles and the lengths of the sides of triangles

The six trigonometric functions are sin , cos , tan , cosec , sec and cot

Let the angle be θ , such that

sin θ = opposite / hypotenuse

cos θ = adjacent / hypotenuse

tan θ = opposite / adjacent

tan θ = sin θ / cos θ

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

Given data ,

Let the triangle be represented as ΔBCD

Now , the measure of BC = 24

The measure of side CD = 10

So , the measure of hypotenuse BD = √ ( BC )² + ( CD )²

Substituting the values in the equation , we get

The measure of BD = √ ( 576 ) + 100

The measure of BD = √ ( 676 )

The measure of BD = 26

Now , from the trigonometric relations ,

sin θ = opposite / hypotenuse

Substituting the values in the equation , we get

sin D = 24 / 26

On simplifying the equation , we get

sin D = 0.923

Hence , the equation is sin D = 0.923

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The length of the segment QV is  9.2 units

What is a square?

Regular quadrilaterals, such as a square, have four equal sides and four equal angles (right angles, 90-degree angles, or angles of /2 radian). It is also known as a rectangle with two neighbouring sides of identical length. It is the only regular polygon whose internal angle, central angle, external angle, and diagonal length are all equal (90°).

diagonal length, d=a√2 , where,  d--> diagonal length ,  a= side length

So, d= 13* 1.4142 = 18.3847 units

Diagonals bisect each other

QV = d/2 = 18.3847/2  = 9.1923

Rounding to 1 decimal place

QV = 9.2 units

The length of the segment QV is  9.2 units

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As per the question Sales revenue is $198,000 (from the sale of inventory)

Sales revenue, which is sometimes referred to as revenue from sales or just sales, is the entire sum of money that a business makes from the selling of goods or services to clients. It is one of the main revenue streams for the majority of businesses and is frequently used as a key performance indicator (KPI) to gauge a business's success.

Calculating sales income involves multiplying the quantity sold by the cost per unit. As an illustration, if a business sells 1,000 pieces of a product at $100 each, its sales revenue would be $100,000 ($100 x 1,000).

Sales revenue is a crucial indicator for businesses to monitor since it indicates how much demand there is for a company's goods or services.

According to question:

Cost of goods sold = $110,000 (the cost of the inventory that was sold)

Gross margin = Sales revenue - Cost of goods sold = $198,000 - $110,000 = $88,000

Expenses = Cost of goods sold + Operating expenses = $110,000 + $36,000 = $146,000

Operating income = Gross margin - Operating expenses = $88,000 - $36,000 = $52,000

Non-Operating Items = Gain on sale of land = $80,000 (the difference between the selling price and the original cost of the land)

Net income = Operating income + Non-Operating Items = $52,000 + $80,000 = $132,000

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Using trigonometric functions and Pythagoras' theorem, the x value of different right-angled triangles is found.

What is a right-angled triangle?

A right triangle, also known as a right-angled triangle, is a triangle with one right angle, or two perpendicular sides. The sum of the other two angles is 90 degrees.

In all of these figures, one of the angles is 90°. So we can use Pythagoras' theorem and trigonometric relations, to solve the questions.

1) Opposite side = 5

sin 27° = Opposite side/ hypotenuse

0.45 = 5/x

x = 11.11

2) Hypotenuse = 16

   cos 63 = adjacent side/hypotensue

    0.45 = x / 16

    x = 7.2

3) Hypotenuse = 29

    sin 67 = opposite side/hypotensue

    0.92 = x/29

      x =  26.68

4) Opposite side = 27

  tan 39° = opposite side/ adjacent side = 27/x

0.81 = 27/x

x = 33.33

5) sin Q = opposite side / hypotensue = 14/50 = 7/25

   sin R = opposite side / hypotensue = [tex]\sqrt{50^2 - 14^2}[/tex]/50 = 48/50 = 24/25

  cos Q = PQ/ 50 = 48/50 = 24/25

   cos R = PR/50 = 14/50 = 7/25

tan Q = sin Q/cos Q = 7/24

tan R = sin R/cos R = 24/7

Therefore using trigonometric functions and Pythagoras' theorem, the x value of different right-angled triangles is found.

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[tex](\stackrel{x_1}{-7}~,~\stackrel{y_1}{5})\qquad (\stackrel{x_2}{g}~,~\stackrel{y_2}{-2}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{-2}-\stackrel{y1}{5}}}{\underset{\textit{\large run}} {\underset{x_2}{g}-\underset{x_1}{(-7)}}} ~~ = ~~\stackrel{\stackrel{\textit{\small slope}}{\downarrow }}{ 7 }\implies \cfrac{-7}{g+7}=7\implies -7=7(g+7) \\\\\\ \cfrac{-7}{7}=g+7\implies -1=g+7\implies \boxed{-8=g}[/tex]

The volume of the solid obtained by rotating about the x-axis the region enclosed by the curves is 32π/5 cubic units.

In this problem, we are asked to find the volume of a solid obtained by rotating a region enclosed by curves about the x-axis. We will use integration to calculate the volume.

First, let's sketch the region and the solid we need to find the volume of.

The region is enclosed by the curves

=> y = 16 /(x² + 16), y = 0, x = 0, and x = 4.

When we rotate this region about the x-axis, we get a solid with a hole in the center, shaped like a donut.

To find the volume of this solid, we can use the method of cylindrical shells.

We start by taking a thin strip of the region, parallel to the y-axis, and of width dy. This strip has height y, and we rotate it about the x-axis to get a cylindrical shell of thickness dy and radius x.

The volume of this cylindrical shell is given by the formula V = 2πxy dy, where x is the distance from the y-axis to the edge of the shell.

To express x in terms of y, we can solve for x in the equation

y = 16 /(x² + 16).

y(x² + 16) = 16

x² = 16/y - 16

x = √(16/y - 16)

Now we can substitute x into the formula for V to get:

V = 2πx(16/(x² + 16)) dy

[tex]= 2\pi(16/y - 16)^{1/2}(16/y) dy \\\\= 32\pi(y - y^{3/2}) dy[/tex]

To find the total volume, we integrate this expression from y = 0 to y = 1 (since the maximum value of y is 16/16 = 1):

[tex]\int_0^1 32\pi(y - y{(3/2)}) dy = 32\pi/5[/tex]

Therefore, the volume of the solid obtained by rotating the region about the x-axis is 32π/5 cubic units.

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Answer:

1.) M=9,100(1 + 0.05)^6

Step-by-step explanation:

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

In this formula the "P" represents the principle amount, or the original amount. The "r" represents the interest rate in decimal form, while the "n" represents the number of compounds in the time unit (usually years), and the "t" represents the amount of time that has passed (usually years)

In this case it says annual interest rate of 5% and nothing of compound monthly, etc... so n=1, and r=0.05. We're also given the principle amount of 9,100 and the time passed is just 6 years, so t=6. Plugging all this information into the equation we get:

[tex]A=9,100(1+0.05)^6[/tex]

which makes the first option correct.

[tex]\cfrac{6}{4+\sqrt{2}}\implies \cfrac{6}{4+\sqrt{2}}\cdot \cfrac{4-\sqrt{2}}{4-\sqrt{2}}\implies \cfrac{6(4-\sqrt{2})}{\underset{\textit{difference of squares}}{(4+\sqrt{2})(4-\sqrt{2})}} \implies \cfrac{6(4-\sqrt{2})}{(4)^2 - (\sqrt{2})^2} \\\\\\ \cfrac{6(4-\sqrt{2})}{16 - 2}\implies \cfrac{6(4-\sqrt{2})}{14}\implies \cfrac{3(4-\sqrt{2})}{7}\implies \cfrac{12-3\sqrt{2}}{7}[/tex]

The width of rectangle  = 90 m

And, The length of rectangle = 110 m

A rectangle is a two dimension figure with 4 sides, 4 corners and 4 right angles. The opposite sides of the rectangle are equal and parallel to each other.

Given that;

The perimeter of a rectangle is 400 meters.

And, The length is 20 meters longer than the width.

Now, Let the width of rectangle = x

Then, The length of rectangle = x + 20

We know that;

⇒ Perimeter of rectangle = 2 ( Length + width )

Substitute values ,

⇒ 400 = 2 (x + x + 20)

⇒ 400 = 2 (2x + 20)

⇒ 200 = 2x + 20

⇒ 2x = 200 - 20

⇒ 2x = 180

⇒ x = 90

Thus, The width of rectangle = x = 90 m

And, The length of rectangle = x + 20 = 90 + 20 = 110 m

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Two other points on this line are (1, 1.5) and (2, 3)

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

This means that the slope (m) of the line is

m = 6/4

Evaluate

m = 1.5

For the other points, we have

f(2) = 2 * 1.5 = 3

f(1) = 1 * 1.5 = 1.5

So, we have (1, 1.5) and (2, 3)

Hence, the points are (1, 1.5) and (2, 3)

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Answer:

i got u i have your email

Step-by-step explanation:

Answer:

19 < x < 53

Step-by-step explanation:

given 2 sides of a triangle then the third side x is in the range

difference of 2 sides < x < sum of 2 sides , that is

36 - 17 < x < 36 + 17

19 < x < 53

The number of years, t , after which the value of the company's new equipment will be less than 7500 is 2.64 years

To solve this problem, we can use the formula for exponential growth:

V(t) = V₀ * (1 + r)^t

Given that the parameters are;

To determine time t when the value of the equipment will be less than $7500.

Let's set up an inequality;

V(t) < 7500

Substituting the formula for V(t), we get:

V₀ * (1 + r)^t < 7500

Substitute the values

(1 + 0.159)^t < 0.5

t * log(1 + 0.159) < log(0.5)

t > log(0.5) / log(1 + 0.159)

Using a calculator, we get:

t > 2.64

Hence, the value is 2.64 years.

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The equation in slope-intercept form is y = 10x + 41.

Mathematically, the point-slope form of a straight line can be calculated by using this mathematical expression:

y - y₁ = m(x - x₁) or y - y₁ = (y₂ - y₁)/(x₂ - x₁)(x - x₁)

Where:

Based on the information provided, we can logically deduce the following data points on the line:

Points on x-axis = (-3, -4).

Points on y-axis = (11, 1).

At point (-3, 11), a linear equation of this line can be calculated in slope-intercept form as follows:

y - 11 = (1 - 11)/(-4 + 3)(x + 3)

Y - 11 = 10(x + 3)

y = 10x + 30 + 11

y = 10x + 41.

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The equation cos (π/6 + β) is solved to get

From the question it was given that sine β = -0.8

solving for β

sine β = -0.8

β = arc sine ( -0.8 )

β = -0.9273

β ≈ -π/3

substituting the value into cos (π/6 + β)

cos (π/6 + β)

cos (π/6 - π/3)

cos (π/6 - π/3)

= √3 / 2

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The Total Surface Area for given prism will be 166 [tex]cm^2[/tex].

The entire surface area of a rectangular prism may be computed by adding the areas of its six sides. A rectangular prism's surface area may be determined using the following formula:

[tex]Surface Area = 2(l*w) + 2(l*h) + 2(w*h)[/tex]

where:

l = length of the rectangular prism

w = width of the rectangular prism

h = height of the rectangular prism

Using this formula for given problem,

Given,

l=5cm

w=4cm

h=7cm

[tex]\text{Total Surface area}=2.(5.7)+2.(4.5)+2.(4.7)\\\\=70+40+56\\\\=166\;cm^2[/tex]

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The number of years the interest was compounded quarterly is given by the equation b = 9.098 years

Compound interest is interest based on the initial principle plus all prior periods' accumulated interest. The power of compound interest is the ability to generate "interest on interest." Interest can be added at any time, from continuously to daily to annually.

The formula for calculating Compound Interest is

A = P ( 1 + r/n )ⁿᵇ

where A = Final Amount

P = Principal

r = rate of interest

n = number of times interest is applied

b = number of time periods elapsed

Given data ,

Let the amount after b years be A = $ 26,725.00

The number of years be = b

The number of times the interest is applies n = 4 ( quarterly )

Let the principal amount be P = $ 7,000

A = P ( 1 + r/n )ⁿᵇ

Substituting the values in the equation , we get

t = ln ( A/P ) / n [ ln ( 1 + r/n ) ]

t = b = n ( 26,725.00 / 7,000.00 ) / ( 4 × [ ln ( 1 + 0.15/4 ) ] )

On simplifying the equation , we get

b = ln ( 26,725.00 / 7,000.00 ) / ( 4 × [ ln ( 1 + 0.0375 ) ] )

b = ln ( 3.81785714 ) / 4 [ ln ( 1.0375 ) ]

On further simplification , we get

b = 9.098 years

Hence , the number of years the interest was applied is 9 years 1 months

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Answer:

The surface area of a cylinder can be calculated using the formula:

Surface area = 2πr^2 + 2πrh

Where r is the radius and h is the height of the cylinder.

Plugging in the values we have:

Surface area = 2π(8^2) + 2π(8)(8) = 2π(64) + 2π(64) = 128π

Rounding to the nearest thousandth, we have:

Surface area = 128π = 403.23 in^2 (rounded to the nearest thousandth)

So the surface area of the cylinder with height 8 in and radius 8 in is 403.23 in^2.

Step-by-step explanation:

The support vector of the input dsolve('Dy = -3*y') is "dsolve(diff(y,t) == -3*y)".

A symbol is a mathematical object that represents a mathematical entity, such as a number, a variable, or a function. In mathematical software, symbols are used to represent variables, which can be used in mathematical expressions to perform computations.

In the past, character vectors or strings were used to represent symbols in certain mathematical software programs.

In the future release of this software, support for character vectors or strings will be removed. Instead, the program will require users to use the "syms" function to declare variables as symbols.

As an example of how this might be used in practice, consider the differential equation

=> "dy/dt = -3y".

In the past, this equation might have been entered into a mathematical software program using a character vector or string to represent the variable "y".

However, in the future release of this software, the user will need to use the "syms" function to declare "y" as a symbol. The equation would then be entered using the "diff" function, like this:

=> "dsolve(diff(y,t) == -3*y)".

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Answer:

Step-by-step explanation:

a) The number of people showing up at the bar during any given minute is a Poisson process with an average rate of 1 person per minute. Let X be the number of people who show up during a random minute. Then X follows a Poisson distribution with parameter λ = 1. The probability of at least 3 people showing up in a random minute is:

P(X >= 3) = 1 - P(X < 3) = 1 - (P(X = 0) + P(X = 1) + P(X = 2))

We can use the Poisson probability mass function to find P(X = k) for k = 0, 1, 2:

P(X = k) = (e^-λ) * (λ^k) / k!

Plugging in λ = 1, we get:

P(X = 0) = e^-1 / 0! = 1/e

P(X = 1) = e^-1 * 1^1 / 1! = 1/e

P(X = 2) = e^-1 * 1^2 / 2! = 1/2e

So, the probability of at least 3 people showing up in a random minute is:

P(X >= 3) = 1 - (P(X = 0) + P(X = 1) + P(X = 2)) = 1 - (1/e + 1/e + 1/2e) = 1 - (2/e + 1/2e) = 1 - (5/2e)

b) The probability of having to wait at least 3 minutes for the next person to show up is equal to the probability of having 0, 1, or 2 people show up during the first three minutes. Using the same calculation as above, we find:

P(X <= 2) = P(X = 0) + P(X = 1) + P(X = 2) = (1/e) + (1/e) + (1/2e) = (2/e + 1/2e) = (5/2e)

So, the probability of having to wait at least 3 minutes for the next person to show up is 1 - (5/2e) = 1 - P(X <= 2).

c) If he has already waited for 1 minute, the probability of having to wait at least 3 more minutes for the next person to show up is equal to the probability of having 0 or 1 people show up during the next 2 minutes. Using the same calculation as above, we find:

P(X <= 1) = P(X = 0) + P(X = 1) = (1/e) + (1/e) = 2/e

So, the probability of having to wait at least 3 more minutes for the next person to show up is 1 - (2/e) = 1 - P(X <= 1).

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1) From Given output confidence level, is 95% level of significance = 1-0.95

10.05

2) null hypothesis : M = 150000

=

3) Alternative hypothesis Ha: [M> 150000]

4) since P value = 0.033 is less than d=0.05 Reject null hypothesis.

5) There is sufficient evidence to conclude that the mean life of transmissions is over 1500 comity

6) 95% CI = â ± E

-1519±1.129

=(151.2-1.129, 151.2+1.129)

=(150.071, 152.329)

7) Since M=150 fall out side of the C. We Shoul retect null hypothesis.

8) The null hypothesis value fall outside of the Confidence Interval.

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The range is the set of y-coordinates of the function.

As per graph we can see:

Add all together to get the range:

As interval the range is:

If Mrs. Imperiale's credit card has an APR of 13.2%. The amount she would  save in finance charges Round to the nearest ten dollars is $80.

Finance charge if bought on the first day

Finance charge = (31 × 7400 ÷ 31) × (0.132 ÷ 12)

Finance charge = 7400  ×  0.011

Finance charge= $81.40

Finance charge if bought on the last day

Finance charge = (1 × 7400 ÷ 31) × (0.132 ÷ 12)

Finance charge = 238.709677 ×  0.011

Finance charge= $2.625

Finance charge = $2.63

Savings = $81.40 - $2.63

Savings = $78.77

Savings = $80 ( Approximately)

Therefore the savings is $80.

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Answer:

Step-by-step explanation:

3/2*2/3(6y)=30*3/2

6y=45

6y/y=45/y

y=7.5

Answer:

YOU L HAHA

Step-by-step explanation:

LOS ER

The value of k is,

The probability distribution as a table:

r   1     2          3

P(X)   0    1/15  2/15

Probability is a mathematical term, which can be defined as the ratio of the number of favorable outcomes to the total number of outcomes of an event. The possibility that an event will occur is measured by probability.

Probability of Event = Favorable Outcomes/Total Outcomes = X/n

To find the value of k, we can use the fact that the sum of the probabilities for all possible values of X must equal 1.

Therefore, we have:

P(X = 1) + P(X = 2) + P(X = 3) = k(4(1)(1²) + 4(2)(2²) + 4(3)(3²))

                                              = 24k

Since we also know that P(X = 1) = 0, we have:

P(X = 1) + P(X = 2) + P(X = 3) = P(X = 2) + P(X = 3) = 0.8

Substituting this into the equation above, we get:

0.8 = 24k

k = 0.8/24 = 1/30

Therefore, the probability distribution can be written as:

r 1 2 3

P(X) 0 1/15 2/15

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The area of the wall is 275 ft square

The number of gallon to cover the wall approximately is 3

the price of the paint is $73.5

1. The area of the wall is given as l * w

this is the value that is used to solve for the area of a rectangle

the length is given as 10 ft

the width is given as 22 ft

the area would be 10 * 22

= 220 ft square

find the area of the triangle = rectangular part is given as height = 15 ft - 10 ft = 5 ft

base = 22 ft

area = 1/2 b h

= 0.5 * 22 * 5

= 55

total area = 220 + 55 = 275 ft square

2. If 1 gallon of paint covers 105 ft

the amount that would cover 275 ft would be gotten when we cross multiply

1 = 105 ft

x  = 275 ft

275 ft = 105 x

x = 275 / 105

x = 2.619

x ~ 3 gallons of paint

3. If one gallon is 24.5 dollars, the total amount that you would purchase is 3 * 24.5

= $73.5

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The composite function has its value to be g o h(x) = -20x^2 - 30x

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

h(x) = 4x + 6

g(x) = -5x

Using the above as a guide, we have the following:

g o h(x) = g(x) * h(x)

substitute the known values in the above equation, so, we have the following representation

g o h(x) = (4x + 6) * -5x

Evaluate

g o h(x) = -20x^2 - 30x

Hence, the value is g o h(x) = -20x^2 - 30x

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