determine the relative order of the metric prefixes of kilo-, micro-, centi-, and milli-. for the same base unit, choose... is less than choose... , which is less than choose... , which is less than choose... .

Answer:

1/2

Step-by-step explanation:

First, let's find the GCF of 3 and 6. In this case, it's 6.

Then, lets convert every fraction so that the denominator is 6. 1/6 already has this property, so let's convert 1/3. And to do that, we multiply the fraction by 2/2.

[tex]\frac{1}{3} * \frac{2}{2} = \frac{2}{6}[/tex]

Finally, we add the fraction of beads that are red and blue together, then simplify.

[tex]\frac{2}{6} + \frac{1}{6}=\frac{3}{6} =\frac{1}{2}[/tex]

Therefore, the answer is 1/2.

The complex number statement (3+8i)/(2+8i)=3/2 is false. The correct complex number statement is (3+8i)/(2+8i) = (24i + 6)/(-64i² + 16).

What is a complex number?

Complex numbers are those that are expressed as a+ib, where a and b are actual numbers and i is a imaginary number called a "iota."

The given complex number equation is (3+8i)/(2+8i)=3/2

Simplify both sides of the equation.

On the left side, there is -

= (3+8i)/(2+8i)

= (3+8i)/(2+8i) × (2-8i)/(2-8i)

= (6+24i-24i-64i²)/(2² + 2×8i-8i²)

= (-64i² + 24i + 6)/(-64i² + 16)

= (24i + 6)/(-64i² + 16)

On the right side, there is -

3/2 = 3/2

It can be seen that the simplified forms of both sides are not equal, so the original statement is false.

To make the statement true, change the right side of the equation to the left side simplified form.

(3+8i)/(2+8i) = (24i + 6)/(-64i² + 16)

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The measure of angle POQ is given as follows:

30º.

The area of a circle of radius r is given by π multiplied by the radius squared, as follows:

A = πr²

The radius of the circle in this problem is given as follows:

r = OP = 2.

(distance of the center O to point P on the circumference).

Hence the area of the circle is given as follows:

A = 4π.

The area of the sector is given as follows:

π/3.

The ratio of the area of the sector and the area of the circumference is given as follows:

(π/3)/(4π) = 1/12.

The entire area has a measure of 360º, hence the angle measure is obtained as follows:

1/12 x 360 = 30º.

The problem is given by the image presented at the end of the answer.

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The formula for g(x) is 2√(x-6) + 1. The solution has been obtained using the concept of translation.

What is translation?

In mathematics, a translation involves moving a shape up, down, left, or right. The translated shapes appear to be exactly the same size as the original ones; thus, the shapes are consistent with one another. Just one or more directions have been changed. There is no change to the shape because it is simply being moved from one location to another.

The object's shift in location can occur in a variety of ways, such as initially moving left, then turning right, and so on. Each point on the form will translate by the same number of units.

We are given f(x) = 2√x

Now, shifted down 1 unit, we get

2√x + 1

Also, shifted right 6 units, we get

2√(x-6)

So, g(x) = 2√(x-6) + 1

Hence, the formula for g(x) is 2√(x-6) + 1.

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Therefore , the solution of the given problem of gradient comes out to be

It has been established that line BC and AB are parallel option 2 is correct.

Pick two points on a graph that has a straight line as the direction. Change in y-coordinate divided by change in x-coordinate is the gradient of the line.In a visual representation, parallel lines encircle one another like train tracks. Parallel lines AB and CD go side by side.

Here,

Two lines that are parallel are denoted by the symbol ||. To indicate that AB is parallel to CD, we write AB||CD.

As a result, line segment AC's length is equal to the sum of line segments AB and BC.

The product of AB's and BC's slopes must be negative because AB and BC are obviously perpendicular to one another.

It has been established that line BC and AB are parallel.

Therefore , the solution of the given problem of gradient comes out to be

It has been established that line BC and AB are parallel.

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Answer:

$ balance = 120*(1+6.4%/4) ^4t

Step-by-step explanation:

principal =120

r =6.4% annual = 1.6% per quarter compounding every quarter

so 4 times a year, for t years

$ balance = 120*(1+6.4%/4) ^4t

The function that represents the balance of the investment account after t years is  [tex]y = 120 (1.016)^{4t[/tex].

The formula for calculating the balance of an investment account that earns compound interest is:

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

where:

A = the balance after t years

P = the principal (initial amount deposited)

r = the annual interest rate (as a decimal)

n = the number of times the interest is compounded per year

t = the time in years

In this case, we have:

P = $120

r = 6.4% = 0.064 (as a decimal)

n = 4 (compounded quarterly)

Therefore, the function that represents the balance y (in dollars) of the investment account after t years is:

[tex]y = 120 *(1 + \dfrac{0.064}{4})^{4t}[/tex]

Simplifying the expression inside the parentheses, we get:

[tex]y = 120 (1.016)^{4t[/tex]

This is the function that represents the balance of the investment account after t years.

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If a northbound train and a southbound train meet each other on parallel tracks heading in opposite directions. The speeds that the  two trains are traveling is:61 mph.

Speed of westbound train = x mph

Speed of eastbound train = x+16

2.5x + 2.5(x+16) = 265

2.5x + 2.5x + 40 = 265

Combine like terms

5x = 225

Divide both side by 5x

x =225/5

x = 45 mph

Westbound train = 45mph

Eastbound train =45 mph +16 mph = 61 mph

Therefore the speed is 61mph.

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The area of a rectangle with length x and width y is xy unit square.

An equation is an expression that shows the relationship between numbers and variables.

Area is the amount of space occupied by a figure. Area is usually calculated for a two dimensional shape while volume is calculated for a three dimensional shape.

A rectangle have equal opposite sides which are parallel to each other and all angles are at 90 degrees.

Area of rectangle = length * width

Area = x * y = xy

The area is xy unit square.

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The following are not characteristics of surveys :

A. The study involves on or more treatment groups and a control group

B. The study compared two or more treatments

A survey is a research technique used to gather data from a predetermined sample of respondents in order to learn more and acquire insights into a range of interesting topics.

There are Characteristics of an Effective Survey :

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The estimated area under the curve is 112.

We can use the left rectangle method to approximate the area under the curve f(x) = (2x)^2 + 6 from x = 2 to x = 4.

First, we need to choose a value of Δx = 0.5, which will determine the width of the rectangles. Then, we can find the height of each rectangle by evaluating the function at the left endpoints of the intervals.

For x = 2, the height of the first rectangle is f(2) = (2 * 2)^2 + 6 = 20.

For x = 2.5, the height of the second rectangle is f(2.5) = (2 * 2.5)^2 + 6 = 28.25.

For x = 3, the height of the third rectangle is f(3) = (2 * 3)^2 + 6 = 42.

For x = 3.5, the height of the fourth rectangle is f(3.5) = (2 * 3.5)^2 + 6 = 57.75.

For x = 4, the height of the fifth rectangle is f(4) = (2 * 4)^2 + 6 = 76.

Finally, we can multiply the height of each rectangle by its width (0.5) to find the area of each rectangle:

Area of first rectangle = 20 * 0.5 = 10

Area of second rectangle = 28.25 * 0.5 = 14.125

Area of third rectangle = 42 * 0.5 = 21

Area of fourth rectangle = 57.75 * 0.5 = 28.875

Area of fifth rectangle = 76 * 0.5 = 38

The total area under the curve can be estimated by adding up the areas of the rectangles:

Estimated area = 10 + 14.125 + 21 + 28.875 + 38 = 112

So the estimated area under the curve using the left rectangle method and Δx = 0.5 is 112.

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Answer:

Food supply would only last 4 days

Step-by-step explanation:

(20 cattle)(6 days) = 120 cattle·days      (this is the current feed supply on hand)

20 cattle + 10 = 30 cattle total

(120 cattle·days) / (30 cattle) = 4 days

a. The percentage of the original price that the consumers are going to pay will be 8.57%.

b. The markup percentage of the gallons will be

c. The percentage that decided not to sell the juice is 41.7%.

d. The price of each bottle will be:

a. The percentage of the original price that the consumers are going to pay will be:

= (3.50 - 3.20) / 3.50 × 100

= 0.30 / 3.50 × 100

= 8.57%

b. The markup percentage of the gallons will be:

= (3.20 - 2.70) / 2.70 × 100

= 0.50 / 2.70 × 100

= 18.5%

c. The percentage that decided not to sell the juice is:

= 125 / 300 × 100

= 41.7%

d. The price of each bottle will be:

= (1 - 15%) × $2.25

= $1.91

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The average rate of change on the interval [1, b] is:

r = 4*(b + 1)

For a function f(x), we define the average rate of change on the interval [a, b] as:

r = ( f(b) - f(a))/(b - a)

Here we have the function:

f(x) = 4x² - 7

And the interval is [1, b]

Then the average rate of change is:

r = (f(b) - f(1))/(b - 1)

r = ( 4b² - 7 - 4 + 7)/(b - 1)

r = (4b² - 4)/(b - 1)

r = 4*(b² - 1)/(b - 1)

r = 4*(b + 1)*(b - 1)/(b - 1)

r = 4*(b + 1)

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The entire expression is a sum with 0 factors

The coefficients are 1/2, 1 and 7

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

1/2x² + x + 7

The above expression is a quadratic expression and it has three terms

Quadratic expressions with real roots have 2 factors, otherwise it would have 0 factor

The expression 1/2x² + x + 7 does not have a real root because

b² < 2ac

Also in the expression, the coefficients are 1/2, 1 and 7

i.e. a = 1/2, b = 1 and c = 7

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Complete question

consider the expresson 1/2x² + x + 7.

complete 2 descriptions of the part of the expression

The entire expression is a sum with __ factors

The coefficients are _

On solving the given equation,  the value of x = -4.

Finding the value of f(x) =... or y =... that corresponds to a certain value of x is what it means to evaluate a function.

Simply swap out all instances of x with the value of x to do this. You must substitute a number for each variable and carry out the arithmetic operations in order to evaluate an algebraic expression.

Since 6 + 6 equals 12, the variable x in the example above is equal to 6. If we are aware of the values of our variables, we can substitute those values for the original variables before evaluating the expression.

Given,

[tex]\frac{2x + 4}{2} - \frac{3x}{x-2} = x\\ \\[/tex]

Taking LHS

⇒ [tex]\frac{2x^2 - 4x + 4x - 8 - 6x}{2(x-2)} \\[/tex]

⇒ [tex]\frac{x^2-4-3x}{(x-2)}[/tex]

Equating both sides,

⇒ [tex]\frac{x^2-4-3x}{(x-2)} = x\\x^2-4-3x = x^2 - 2x\\x = -4[/tex]

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To stop electrostatic discharge from harming your hardware, fasten a wrist strap to a painted metal surface.

Static electricity discharge can damage disc drives, media drives, electronic boards, and adapters. To avoid this harm, these gadgets are enclosed in antistatic bags. Take the following safety measures to guard against static electricity discharge harming this equipment.

To stop electrostatic discharge from harming your hardware, fasten a wrist strap to a painted metal surface.

Be sure to observe all electrical safety precautions while utilizing a wrist strap. For static control, a wrist strap is used. When you use or work on electrical equipment, your risk of getting shocked by electricity is not increased or decreased.

If you do not have a wrist strap, you must touch an unpainted metal surface of the system for at least 5 seconds before removing it from the ESD packing and installing or replacing components.

When you're ready to install the device in the system, take it out of the antistatic bag.

Touch the gadget to the system's metal frame while it is still in its antistatic bag.

Grasp the edges of the cards and boards. On the adaptor, stay away from the parts and gold connectors.

The complete question is :-

All of the following are precautions to prevent static electricity damage when replacing a control module EXCEPT:

Touch a known good ground before opening the package.

Wipe the terminals clean before installing the component.

Ground the package to a known good ground before opening.

See Electronic Service Precautions

Open the package only when ready to install the component.

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Event A is the event that the sum of the numbers rolled on the two dice is divisible by 4. Event B is the event that the sum of the numbers rolled on the two dice is less than 4.

(a) The total number of possible outcomes in this experiment is 36 (6 possible outcomes for each die).

(b) Outcomes in A are (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (8, 1), (8, 2), (8, 3), (8, 4), (8, 5), (8, 6), (12, 1), (12, 2), (12, 3), (12, 4), (12, 5), and (12, 6).

(c) Outcomes in B are (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), and (2, 6).

(d) P(A) is 5/36. (e) P(B) is 6/36. (f) P(A U B) is 11/36. (g) P(A n B) is 0 (there are no outcomes that belong to both A and B). (h) Events A and B are not independent because P(A U B) does not equal P(A) + P(B).

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The complex fifth roots of 3 are Z₀ = ∛3 (cos 0° + i sin 0°), Z₁ = ∛3 (cos 72° + i sin 72°), Z₂ = ∛3 (cos 144° + i sin 144°), Z₃ = ∛3 (cos 216° + i sin 216°) and Z₄ = ∛3 (cos 288° + i sin 288°)

The complex fifth roots of 3 can be found by finding the roots of the equation z^5 = 3. To find the roots in polar form, we first convert 3 to polar form:

3 = |3| (cos 0 + i sin 0) = 3 (cos 0 + i sin 0)

Next, we find the argument of the fifth roots, which is equal to 360°/5 = 72°. The polar form of the first root is then:

z₁ = ∛3 (cos 0° + i sin 0°) = ∛3 (cos 0° + i sin 0°)

The polar form of the other roots can be found by rotating the argument by 72°, 144°, 216°, and 288° respectively. The result is:

z₂ = ∛3 (cos 72° + i sin 72°)

z₃ = ∛3 (cos 144° + i sin 144°)

z₄ = ∛3 (cos 216° + i sin 216°)

z₅ = ∛3 (cos 288° + i sin 288°)

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Note that the next number in the progression or sequence or pattern is 35,585.

Note that a number pattern is a series of numbers that adheres to a set of criteria. The rule might be mathematical, such as adding a constant value to each item in the series, or it can be based on some other logical relationship.

The next term in the sequence or pattern is derived as follows:

3x5 -3 = 12

5x12 -5 = 55

12x55 -12 = 648

55x648 -12 = 35,585

Therefore, it is correct to state that 35,585 is the next term in the sequence

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If the slope of the line has two negative signs, then it is positive.

The slope of a linear function represents the rate of change of the linear function.

If we have two points on the linear function, then the slope is obtained by the division of the change in the output by the change in the input.

If both changes are negative, then we are dividing two negative numbers, meaning that the slope is positive, as the quotient of two negative numbers is positive.

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Answer:

2

Step-by-step explanation:

If a pizza feeds 3 kids and you need to feed 6 kids, the scale factor for calculating how many pizzas you need would be 2.

A scale factor is used to determine the relationship between the size of an object and a similar or corresponding object. In this case, you have a pizza that can feed 3 kids and you need to feed 6 kids, so the relationship between the number of kids that can be fed by one pizza and the number of kids you need to feed is 6/3 = 2

So, you need to double the quantity of pizza to feed 6 kids, therefore the scale factor would be 2.

Answer:

8.75 miles

Step-by-step explanation:

using the equation for the line of best fit

y = 0.8x + 0.75

substitute x = 10 into the equation

y = 0.8(10) + 0.75 = 8 + 0.75 = 8.75

the equation predicts she will run 8.75 miles in week 10

The  linear equations which are not  in one variable are x²-5x+6=0 and  x³ = x

What is linear equation ?

Linear equation can be defined as the equation in which highest degree is one.

Given Equations are

x^2 - 5x + 6 = 0

Here the highest degree is greater than one so it is not a linear equation

And x

x^3  = x

x^3 - x =  0

Here also the highest degree is greater than one so it is not a linear equation

3x-4=0

Here the highest degree is equals to one so it is  a linear equation

7x-6x = 3 + 9x

x = 3+9x

8x+3 = 0

Here the highest degree is equals to one so it is  a linear equation

Therefore , The  linear equations which are not  in one variable are x²-5x+6=0 and  x³ = x

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well, there are a total of "x" stamps in his collection, which oddly enough is 100%.

now, we know that 34% of "x" is 204, what the heck is "x"?

[tex]\begin{array}{ccll} amount&\%\\ \cline{1-2} x & 100\\ 204& 34 \end{array} \implies \cfrac{x}{204}~~=~~\cfrac{100}{34} \\\\\\ \cfrac{ x }{ 204 } ~~=~~ \cfrac{ 50 }{ 17 }\implies 17x=10200\implies x=\cfrac{10200}{17}\implies x=600[/tex]

X intercept: 10/5

Y intercept: 0

To find the intercepts of a linear equation, you need to set either the X or Y variable to 0 and then solve for the other variable.

For example, if you have the equation 5x - 2y = 10, you can set X equal to 0 and solve for Y to find the Y intercept, or you can set Y equal to 0 and solve for X to find the X intercept.

X Intercept:

To find the X intercept, set Y equal to 0 and solve for X.

5x - 2(0) = 10

5x = 10

x = 10/5

X intercept = 10/5

Y Intercept:

To find the Y intercept, set X equal to 0 and solve for Y.

5(0) - 2y = 10

-2y = 10

y = -10/2

Y intercept = 0

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The derivative of the summation with lower limit that equal to 0 is [tex]$$y^{\prime}=\sum_{n=0}^{\infty}(-1)^n \cdot \frac{(2 n+1)}{(n !)^2} \cdot\left(\frac{x}{3}\right)^{2 n}$$[/tex].

From the given data,

The derivative of the summation from n equals 0 to infinity of the quotient of the product of negative 1 raised to the nth power and x raised to the quantity 2 times n plus 1 power and the product of 3 to the 2 times n power and the square of n factorial is  [tex]$y=\sum_{n=0}^{\infty} \frac{(-1)^n x^{2 n+1}}{3^{2 n} \cdot(n !)^2}$[/tex]

Derivatives can be classified into different types based on their order such as first and second order derivatives.

Derivative⇒ [tex]$y^{\prime}=\sum_{n=0}^{\infty} \frac{(-1)^n \cdot(2 n+1)(x)^{2 n}}{3^{2 n} \cdot(n !)^2}$[/tex]

⇒ [tex]$$y^{\prime}=\sum_{n=0}^{\infty}(-1)^n \cdot \frac{(2 n+1)}{(n !)^2} \cdot\left(\frac{x}{3}\right)^{2 n}$$[/tex].

Therefore, the summation with lower limit that equal to 0 is [tex]$$y^{\prime}=\sum_{n=0}^{\infty}(-1)^n \cdot \frac{(2 n+1)}{(n !)^2} \cdot\left(\frac{x}{3}\right)^{2 n}$$[/tex]

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Answer:

...

Step-by-step explanation:

To find the number of calories Davis burns while running for 30 minutes, we can use the information that he burns 1080 calories running for 2.5 hours.

We know that 30 minutes is 1/120 of 2.5 hours. So, we can divide the total number of calories by 120 and we get the calorie burn for 30 minutes:

1080 calories / 120 = 9 calories

The number of calories Davis burns while swimming for 30 minutes can be determined by using the equation y = 266x, where x represents the number of hours he swims. We know that he swims for 30 minutes, or 1/120 hours, so we can plug that value into the equation:

y = 266(1/120) = 2.216 calories

So, Davis burns 7 calories more running for 30 minutes than swimming for 30 minutes, assuming the rates remain constant.

The vector representing the force of sphere 2 on sphere 1 is[tex]$\frac{2}{5} \vec{i}$[/tex] because the charge of sphere 2 is twice that of sphere 1 and they are separated by a distance of 5 cm.

Step 1: Calculate the charge of sphere 2. Since it is twice that of sphere 1, the charge of sphere 2 is 2 x the charge of sphere 1.

Step 2: Calculate the distance between the two spheres. This is given as 5 cm.

Step 3: Calculate the force. The force is equal to the product of the charges of the two spheres divided by the square of the distance between them.

Step 4: Construct the vector. The vector representing the force of sphere 2 on sphere 1 is[tex]$\frac{2}{5} \vec{i}$.[/tex]

The vector representing the force of sphere 2 on sphere 1 is [tex]$\frac{2}{5} \vec{i}$[/tex], as the charge of sphere 2 is twice that of sphere 1 and they are separated by a distance of 5 cm. This is calculated by multiplying the charges of the two spheres and dividing by the square of the distance between them.

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Two boys, Zach and Carlos, go for a walk. Zach begins walking 20 seconds earlier than Carlos and they both walked the same distance.

How to calculate the distance?

To calculate the distance that the boys walked, you would need to know their speed and the duration of their walk.

Speed is typically measured in units such as miles per hour (mph) or kilometers per hour (kph). If you know the speed of the boys and the duration of their walk in hours, you can calculate the distance they walked using the formula: distance = speed x time.

The sentence is describing a scenario in which two boys, Zach and Carlos, go for a walk. It is stated that Zach begins walking 20 seconds earlier than Carlos. This means that Zach starts his walk 20 seconds before Carlos does. It is also stated that both boys had walked exactly the same distance. This means that despite Zach starting his walk 20 seconds earlier than Carlos, they both covered the same distance during their walk.

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Answer:

[tex]a. \;\; \boxed{\$ 530.60}[/tex]

[tex]b.\; \;\boxed{ \$552.45}[/tex]

[tex]c,\;\;\boxed{\$610.60}[/tex]

[tex]d. \;\; \boxed{\$745.91}[/tex]

Step-by-step explanation:

All four cases deal with compound interest on the same amount of $500 and same interest rate of 2%.

The only difference is in the frequency of compounding and time

Compound Interest Formula

[tex]\boxed{A = P\left(1 + \dfrac{r}{n}\right)^{n\cdot t}\\\\}[/tex]

where

In this particular problem we have
P = $500
r = 2% = 0.02

These are common for all parts of the question
Only n an t are different for each of the question sub-parts

Part a

Compounding is done annually (once a year) for 3 years

n = 1
t = 3 years
n · t = 3

[tex]A = 500\left(1 + \dfrac{0.02}{1}\right)^3\\\\A = 500\left(1.02\right)^3\\\\A = 500(1.061208)\\\\A=\boxed{\$ 530.60}[/tex]

For accuracy of calculations, I will not compute and store the exponent part, I will perform the calculations in one shot

Part b

Here the compounding is done quarterly (4 times a year) for 5 years

n = 4
t = 5 years
nt = 4 · 5 = 20

[tex]A = 500\left(1 + \dfrac{0.02}{4}\right)^{20}\\\\\\A = 500(1.005)^{20}\\\\A =\boxed{ \$552.45}[/tex]

Part c

Compounding done monthly(12 times a year) for 10 years

n = 12

t = 10

nt = 120

[tex]A = 500\left(1 + \dfrac{0.02}{12}\right)^{120}\\\\\\A = \boxed{\$610.60}[/tex]

Part d

First let's figure out what continuous compounding means

[tex]\fbox{\begin{minipage}[t]{1\columnwidth \fboxsep - 2\fboxrule}%\textsf{What is continuous compounding?} \\\textsf{Continuous compounding is the mathematicallimit that compound interest can reach if it's calculated and reinvestedinto an account's balance over a theoretically infinite number ofperiods. While this is not possible in practice, the concept of continuouslycompounded interest is important in finance. (Investopedia)}\}%\end{minipage}}[/tex]

The formula for continuous compounding can be determine by using the standard formula for periodic compounding and taking limits as
[tex]n \rightarrow \infty[/tex]

Therefore, for compounding continuously , the formula can be derived from
[tex]\lim _{n\to \infty } P\left(1 + \dfrac{r}{n}\right)^{nt}\\\\[/tex]

One of the limit formulas states

[tex]\lim _{x\to \infty } \left(1 + \dfrac{a}{x}\right)^{x} = e^a\\\\[/tex]

Therefore
[tex]\lim _{n\to \infty } \left(1 + \dfrac{r}{n}\right)^{n} = e^r\\\\[/tex]

So for the continuous compounding case, the formula is
[tex]\boxed{A = P \cdot e^{rt}}[/tex]

Here we have
r = 0.02
t = 20 years
rt = 20(0.02) = 0.4

Plugging in P = 500, and rt = 0.4 we get
[tex]A = 500 \cdot e^{0.4}\\\\A = \boxed{\$745.91}[/tex]

Answer:

a) $530.60

b) $552.45

c) $610.60

d) $745.91

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Compound Interest Formula}\\\\$ A=P\left(1+\frac{r}{n}\right)^{nt}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\ \phantom{ww}$\bullet$ $P =$ principal amount \\ \phantom{ww}$\bullet$ $r =$ interest rate (in decimal form) \\ \phantom{ww}$\bullet$ $n =$ number of times interest is applied per year \\ \phantom{ww}$\bullet$ $t =$ time (in years) \\ \end{minipage}}[/tex]

Given:

Substitute the given values into the compound interest formula and solve for A:

[tex]\implies A=500\left(1+\dfrac{0.02}{1}\right)^{1 \cdot 3}[/tex]

[tex]\implies A=500\left(1.02\right)^{3}[/tex]

[tex]\implies A=500(1.061208)[/tex]

[tex]\implies A=530.604[/tex]

Therefore, the value of the account after 3 years if interest is compounded annually is $530.60 (nearest cent).

Given:

Substitute the given values into the compound interest formula and solve for A:

[tex]\implies A=500\left(1+\dfrac{0.02}{4}\right)^{4 \cdot 5}[/tex]

[tex]\implies A=500\left(1.005\right)^{20}[/tex]

[tex]\implies A=500(1.10489557...)[/tex]

[tex]\implies A=552.447788...[/tex]

Therefore, the value of the account after 5 years if interest is compounded quarterly is $552.45 (nearest cent).

Given:

Substitute the given values into the compound interest formula and solve for A:

[tex]\implies A=500\left(1+\dfrac{0.02}{12}\right)^{12 \cdot 10}[/tex]

[tex]\implies A=500\left(1.0016666...\right)^{120}[/tex]

[tex]\implies A=500(1.22119943...)[/tex]

[tex]\implies A=610.599716...[/tex]

Therefore, the value of the account after 10 years if interest is compounded monthly is $610.60 (nearest cent).

[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Continuous Compounding Interest Formula}\\\\$ A=Pe^{rt}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\\phantom{ww}$\bullet$ $P =$ principal amount \\\phantom{ww}$\bullet$ $e =$ Euler's number (constant) \\\phantom{ww}$\bullet$ $r =$ annual interest rate (in decimal form) \\\phantom{ww}$\bullet$ $t =$ time (in years) \\\end{minipage}}[/tex]

Given:

Substitute the given values into the continuous compounding interest formula and solve for A:

[tex]\implies A=500e^{0.02 \cdot 20}[/tex]

[tex]\implies A=500e^{0.4}[/tex]

[tex]\implies A=500(1.49182469...)[/tex]

[tex]\implies A=745.912348...[/tex]

Therefore, the value of the account after 20 years if interest is compounded continuously is $745.91 (nearest cent).