The output column, S, of the given input-output table was created from the student ID 21444368 using an algorithm where if the student ID digit is an even integer, a 0 is placed in the corresponding row of column S, and if the student ID digit is an odd integer, a 1 is placed in the corresponding row of column S.
Student ID: 81725225
P Q R S
1 1 1 0
1 1 0 1
1 0 1 1
0 1 0 0
0 0 1 1
0 1 1 0
0 0 0 1
Student ID: 81725225
P: The first digit of my student ID is 8 so it is an even integer, therefore I place a 0 in row 1 of column P.
Q: The second digit of my student ID is 1 so it is an odd integer, therefore I place a 1 in row 2 of column Q.
R: The third digit of my student ID is 7 so it is an odd integer, therefore I place a 1 in row 3 of column R.
S: The fourth digit of my student ID is 2 so it is an even integer, therefore I place a 0 in row 4 of column S.
The resulting table is:
P Q R S
1 1 1 0
1 1 0 1
1 0 1 1
0 1 0 0
0 0 1 1
0 1 1 0
0 0 0 1
The output column, S, of the given input-output table was created from the student ID 21444368 using an algorithm where if the student ID digit is an even integer, a 0 is placed in the corresponding row of column S, and if the student ID digit is an odd integer, a 1 is placed in the corresponding row of column S. The same algorithm was applied to my student ID 81725225, resulting in the table shown above.
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if a pizza feeds 3 kids, but you need to feed 6, what would the scale factor be for calculating how many pizzas you need?
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.
You use beads to make design. Of the beads, 1/3 are red and 1/6 are blue. The rest are white. What fraction of the beads are red red or blue?
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.
A rectangle has length x and width y. Select all the statements that must be true.
The area of a rectangle with length x and width y is xy unit square.
What is an equation?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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use left rectangles and delta (x) = 0.5 to approximate the area under f(x) = (2x)^2 +6 from x = 2 to x=4 . give one decimal place in your answer.
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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A farmer has enough food to feed 20 animals in his cattle for 6 days. How long
would the food last if there were 10 more animals in his cattle?
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
Please answer this question
Answer:
Step-by-step explanation:
7x9=63
because 7 down times 9 across
lan collects stamps. He put 34% of his stamp collection in an album. If there are
204 stamps in the album, how many stamps does he have in his collection?
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]
Which two of the following are not characteristics of surveys?
A. One or more treatment groups and a control group a
the study.
B. Two or more treatments are compared in the study.
C. The result of the study are analyzed statistically.
D. Data are gathering during the course of the study.
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
What is a surveyA 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 :
measurable survey objectivessound research design.effective survey question designsound sampling strategy, when needed effective survey response strategy. meaningful data summary. effective data display and reporting.Learn more about survey at:https://brainly.com/question/27396612
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1. (04.03 MC)
If a household is saving 35% of their total bi-weekly gross pay of $1,546.00, determine how many months it will take to save a 20% down payment and 2.5% for closing costs for a
property with a purchase price of $195,450.00. (2 points)
Answer:
38
Step-by-step explanation:
Find all the complex roots. Leave your answer in polar form with the argument in degrees. The complex fifth roots of 3. Zo= ___ (cos ___º + i sin ___º)z₁=___(cos___º + i sin ___º)z2=___(cos___º + i sin___º) z3=___ (cos___º + i sin___º) z4=___(cos___º + i sin___º)
The complex fifth roots of 3 are Z₀ = ∛3 (cos 0° + i sin 0°), Z₁ = ∛3 (cos 72° + i sin 72°), Z₂ = ∛3 (cos 144° + i sin 144°), Z₃ = ∛3 (cos 216° + i sin 216°) and Z₄ = ∛3 (cos 288° + i sin 288°)
The complex fifth roots of 3 can be found by finding the roots of the equation z^5 = 3. To find the roots in polar form, we first convert 3 to polar form:
3 = |3| (cos 0 + i sin 0) = 3 (cos 0 + i sin 0)
Next, we find the argument of the fifth roots, which is equal to 360°/5 = 72°. The polar form of the first root is then:
z₁ = ∛3 (cos 0° + i sin 0°) = ∛3 (cos 0° + i sin 0°)
The polar form of the other roots can be found by rotating the argument by 72°, 144°, 216°, and 288° respectively. The result is:
z₂ = ∛3 (cos 72° + i sin 72°)
z₃ = ∛3 (cos 144° + i sin 144°)
z₄ = ∛3 (cos 216° + i sin 216°)
z₅ = ∛3 (cos 288° + i sin 288°)
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the charge of sphere 2 is twice that of sphere 1. which vector below shows the force of 2 on 1 if the distance
The vector representing the force of sphere 2 on sphere 1 is[tex]$\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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I need the answers for this whole side work must be shown or you get reported please don’t make me waste my points
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:
How to calculate the percentage?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 point A, B and C are (9,8),(12,4) and (4,-2).
1.) Find the gradient of the line through a and b.
2.) The equation of the line through C which is parallel to AB.
3.) Calculate the length of the line segment AB and bc.
4.) Show that AB is perpendicular to Bc.
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.
How do you find the gradient of a line A and B?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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Find the average rate of change of the function on the interval specified for real number b.
f(x) = 4x2 − 7 on [1, b]
The average rate of change on the interval [1, b] is:
r = 4*(b + 1)
How to find the average rate of change?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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Find 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. Write your answer as a summation with lower limit of summation equal to 0.
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]
The process of finding the derivative is called differentiation. The inverse process is called anti-differentiation. Let’s find the derivative of a function y = f(x). It is the measure of the rate at which the value of y changes with respect to the change of the variable x. It is known as the derivative of the function “f”, with respect to the variable x.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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Help please
Determine whether the statement is true or false. If the statement is false, make the necessary changes) to produce a true statement.
Image attached
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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Joseph buys 11 bottles of apple juice at the corner store for a total cost of $9.68. If
each bottle costs the same amount, how much is 12 bottles of juice?
Answer: $
Submit Answer
Answer:
10.56
Step-by-step explanation:
Solution set
2x + 4 / 2 - 3x / x - 2 = X
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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Which of the following are not linear equations in one variable?
A.x²-5x+6=0
B. x³ = x
C. 3x-4=0
D. 7x-6x = 3 + 9x
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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all of the following are precautions to prevent static electricity damage when replacing a control module except
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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what is the answer!??
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
Zach had been walking for
distance.
when both boys had walked exactly the same
Place the correct value to complete the sentence.
Two boys, Zach and Carlos, go for a walk. Zach begins walking 20 seconds earlier
than Carlos
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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Can a slope have two negative signs
If the slope of the line has two negative signs, then it is positive.
How to obtain the slope of a linear function?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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Davis burns 1,080 calories when he runs for 2.5 Chours.)The number of calories he burns while
swimming Y)can be described using the
equation y = 266x, where Prepresents the number of hours Davis swims. How many more calories will Davis burn running for 30 minutes) than swimming for 30 minutes assuming the rates remain constant
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.
In circle O O, O P = 2 OP=2 and the length of ⌢ = 7 9 π PQ ⌢ = 9 7 π. Find the area shaded below. Express your answer as a fraction times π π.
The measure of angle POQ is given as follows:
30º.
What is the area of a circle?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.
Missing InformationThe problem is given by the image presented at the end of the answer.
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1) Identify the quadrant in which the point belongs. (-17,-87)
The point (-17, -87) belongs to the third quadrant
How to determine the quadrant of the pointFrom the question, we have the following parameters that can be used in our computation:
Point = (-17, -87)
In the above point, we have
x = -17 and y = -87
This means that the x and the y coordinates are negative
The quadrant where the x and the y coordinates are negative is the third quadrant
Hence, the point is in the third quadrant
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Let f(x) = 2√x.
If g(x) is the graph of f(x) shifted down 1 units and right 6 units, write a formula for g(x).
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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consider the expresson 1/2x to the second power +x+7. complete 2 descriptions of the part of the expression
The entire expression is a sum with 0 factors
The coefficients are 1/2, 1 and 7
How to describe the expressionFrom 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 _
A northbound train and a southbound train meet each other on parallel tracks heading in opposite directions. The northbound train travels 16 miles per hour faster than the southbound train. After 2.5 hours, they are 265 miles apart. At what speeds are the two trains traveling?
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.
How to find the speedSpeed 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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NO LINKS!!! NOT MULTIPLE CHOICE!!
59. You put $500 into a bank account that earns 2% interest. Determine each of the following.
a. The value of the account after 3 years if compounded annually.
b. The value of the account after 5 years if compounded quarterly.
c. The value of the account after 10 years if compounded monthly.
d. The value of the account after 20 years if compounded continously.
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]
Part aGiven:
P = $500r = 2% = 0.02t = 3 yearsn = 1 (annually)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).
Part bGiven:
P = $500r = 2% = 0.02t = 5 yearsn = 4 (quarterly)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).
Part cGiven:
P = $500r = 2% = 0.02t = 10 yearsn = 12 (monthly)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 $610.60 (nearest cent).
Part d[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Continuous Compounding Interest Formula}\\\\$ 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:
P = $500r = 2% = 0.02t = 20 yearsSubstitute 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).