3 a. The function w satisfies the equation (ex+2y)wy−2xwx=0.
3 b. The value of wxy is -2x(ex+2y)cos(ex+2xy) - 2ysin(ex+2xy)
4 . The material derivative of fatP is 0.
3. (a) To verify that the function w satisfies the equation (ex+2y)wy−2xwx=0, we can take the partial derivatives of w with respect to x and y, and then substitute them into the equation.
First, let's find the partial derivatives of w:
∂w/∂x = f'(ex+2xy)(ex+2y)
∂w/∂y = f'(ex+2xy)(2x)
Now, we can substitute these partial derivatives into the equation:
(ex+2y)(f'(ex+2xy)(2x)) - 2x(f'(ex+2xy)(ex+2y)) = 0
Simplifying this equation, we get:
2exf'(ex+2xy) + 4xyf'(ex+2xy) - 2exf'(ex+2xy) - 4xyf'(ex+2xy) = 0
This simplifies to 0 = 0, which is true. Therefore, the function w satisfies the equation (ex+2y)wy−2xwx=0.
(b) If f(u) = cosu, then we can find wxy by taking the partial derivative of w with respect to x and y, and then substituting f(u) = cosu:
∂w/∂x = f'(ex+2xy)(ex+2y) = -sin(ex+2xy)(ex+2y)
∂w/∂y = f'(ex+2xy)(2x) = -sin(ex+2xy)(2x)
Now, we can find wxy by taking the partial derivative of ∂w/∂x with respect to y:
wxy = ∂(∂w/∂x)/∂y = ∂(-sin(ex+2xy)(ex+2y))/∂y = -2x(ex+2y)cos(ex+2xy) - 2ysin(ex+2xy)
4. To find the material derivative of fatP, we can use the formula:
Df/Dt = ∂f/∂t + ∂f/∂x(dx/dt) + ∂f/∂y(dy/dt) + ∂f/∂z(dz/dt)
First, let's find the partial derivatives of f:
∂f/∂t = ysinx + ez
∂f/∂x = ty*cosx
∂f/∂y = tsinx
∂f/∂z = tez
Now, we can find the material derivative of fatP by substituting the point P = (π,1,0) and the derivatives of x, y, and z with respect to t:
Df/Dt = (1*sinπ + e^0) + (π*1*cosπ)(dx/dt) + (π*sinπ)(dy/dt) + (π*e^0)(dz/dt) = 0 + 0 + 0 + 0 = 0
Therefore, the material derivative of fatP is 0.
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The table below shows a cumulative frequency
distribution of runners' ages.
Cumulative Frequency Distribution
of Runners' Ages
Age Group. Total
20-29 8
20-39 18
20-49 25
20-59 31
20-69 35
According to the table, how many runners are in
their forties?
1) 25
2) 10
3) 7
4) 6
The average rate of change of f(x) = x² = x + 4 from x= 2 to x = 4 is
A)2
B)10
C)5
D)3
Answer: 7
Step-by-step explanation: The average rate of change of a function f(x) over an interval [a,b] is defined as the ratio of “change in the function values” to the "change in the endpoints of the interval"1. In other words, it is the slope of the line that passes through two points on the graph of f(x)2.
To find the average rate of change of f(x) = x² + x + 4 from x = 2 to x = 4, we can use this formula:
Average rate of change = [f(4) - f(2)] / (4 - 2)
First, we need to plug in x = 4 and x = 2 into f(x) and simplify:
f(4) = (4)² + (4) + 4 f(4) = 16 + 8 f(4) = 24
f(2) = (2)² + (2) + 4 f(2) = 4 + 6 f(2) = 10
Next, we need to subtract f(2) from f(4), and divide by (4 - 2):
Average rate of change = [24 - 10] / (4 - 2) Average rate of change = 14 / 2 Average rate of change = 7
A rectangular box with a volume of 60x^(6)y^(6)z^(10) width of 2x^(2)y^(3)z^(4) length of 10x^(3)y^(4)z^(5) units and a height, 2x^(2)y^(3)z^(4) units. Write an expression for its height
The expression for the height of the box is 3x^(1)y^(-1)z^(1) units.
The volume of a rectangular box is given by the formula V = lwh, where l is the length, w is the width, and h is the height. We are given the volume, width, and length of the box and we are asked to find the expression for its height.
Volume = 60x^(6)y^(6)z^(10)
Width = 2x^(2)y^(3)z^(4)
Length = 10x^(3)y^(4)z^(5)
Substituting the given values into the formula for volume, we get:
[tex]60x^{(6)}y^{(6)}z^{(10)} = (2x^{(2)}y^{(3)}z^{(4))}(10x^{(3)}y^{(4)}z^{(5))(h)[/tex]
Simplifying the right side of the equation, we get:
60x^(6)y^(6)z^(10) = 20x^(5)y^(7)z^(9)(h)
Dividing both sides of the equation by 20x^(5)y^(7)z^(9), we get:
h = (60x^(6)y^(6)z^(10))/(20x^(5)y^(7)z^(9))
Simplifying the fraction, we get:
h = 3x^(1)y^(-1)z^(1)
Therefore, the expression for the height of the box is 3x^(1)y^(-1)z^(1) units.
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The average salary of 36 employee was 4650 and the sample standard deviation was 165. Find the 90% confidence interval for the population mean salary. Select one: a. 4650 + 53.90 b. 4650 + 70.84 c. 4650 + 67.05 d. 4650 + 63.97 e. 4650 + 55.83 f. 4650 + 45.24 g. 4650 + 46.48 h. 4650 + 74.91
The 90% confidence interval for the population mean salary is 4650 + 45.24. The correct answer is option f.
To find the 90% confidence interval for the population mean salary, we need to use the formula:
CI = x ± z(s/√n)
Where:
- CI = confidence interval
- x = sample mean
- z = z-score for the desired confidence level
- s = sample standard deviation
- n = sample size
Plugging in the given values, we get:
CI = 4650 ± 1.645(165/√36)
CI = 4650 ± 1.645(165/6)
CI = 4650 ± 1.645(27.5)
CI = 4650 ± 45.2375
CI = 4650 ± 45.24
Therefore, the 90% confidence interval for the population mean salary is 4650 ± 45.24 or (4604.7625, 4695.2375).
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What is the value of e?
What is the value of f?
pls pls help if you can
The value of angle e is 55⁰.
The value of angle f is 55⁰.
What is the value of angle e?The value of angle e is calculated by applying the following principles of angles on a straight line.
The vertical angle between e⁰ and 100⁰ = 25⁰ ( vertically opposite angles are equal)
The value of angle e is calculated as;
e⁰ + 25⁰ + 100⁰ = 180⁰ ( sum of angles on a straight line )
e⁰ = 180⁰ - 125⁰
e⁰ = 55⁰
The missing base angle of the triangle on the same line as e is calculated as;
? + 110⁰ = 180⁰ (sum of angles on a straight line )
? = 180 - 110
? = 70⁰
The value of angle f is calculated as;
f⁰ + e⁰ + ? = 180⁰ ( sum of angles in a triangle )
f⁰ + 55⁰ + 70⁰ = 180
f⁰ = 180 - 125⁰
f⁰ = 55⁰
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A spinner is divided into five colored sections that are not of equal size: red, blue, green, yellow, and purple. The spinner is spun several times, and the results are recorded below:
Spinner Results
Color Frequency
Red 20
Blue 9
Green 19
Yellow 14
Purple 14
Based on these results, express the probability that the next spin will land on red or green or purple as a decimal to the nearest hundredth.
The spinner probability that the next spin will land either red or green or purple is found 0.70 as a decimal to the nearest hundredth.
Explain about the probability in spinner?The possibility or likelihood that a spinner could land on a specific value when spun is known as spinner probability. The spinner probability, for instance, would be 1/10, or 10%, if there were a spinner with 10 separate parts and you wanted to know the chance of landing on just one of them.Spinner Results
Color Frequency
Red 20
Blue 9
Green 19
Yellow 14
Purple 14
Total outcomes: 20 + 9 + 19 + 14 + 14
Total outcomes: 76
spinner probability = favorable outcome / total outcome
favorable outcome (red or green or purple) = 20 + 19 + 14 = 53
spinner probability = 53 / 76 = 0.69
spinner probability = 0.70
Thus, the probability that the next spin will land either red or green or purple is found 0.70 as a decimal to the nearest hundredth.
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What is the absolute deviation for 62 in the data set? {80, 75, 25, 62, 68}
0
5
6
62
The absolute deviation for 612 in the given data set is 14.8.
What is the absolute deviation?The average distance between each data value and the mean is known as the mean absolute deviation (MAD) of a data set.
A measure of variance in a data set is the mean absolute deviation.
We may determine how "spread out" the values in a data collection are by looking at the mean absolute deviation.
So, get the mean as follows:
80+75+25+62+68)/5
310/5
62
Now, get the distance of each number as follows:
-----------------
| 80 | 18 |
| 75 | 13 |
|+ 25 | 37 |
| 62 | 0 |
| 68 | 6 |
-----------------
Now, get the distance:
18+13+37+0+6/5
14.8
Therefore, the absolute deviation for 612 in the given data set is 14.8.
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What is the absolute deviation for 62 in the data set?
{80, 75, 25, 62, 68}
answer this for 50 points (riddle) if u get it right you aslo get brainliest
How do you make six an odd number?
Answer:
Remove the S and you have IX which is the roman numeral for 9
Answer:
How do you make six an odd number?Take away the S and you have IX that would make the roman numeral 9.
Step-by-step explanation:
You're welcome.
A person places $8290 in an investment account earning an annual rate of 6%, compounded continuously. Using the formula =V=Pe^rt, where V is the value of the account in t years, P is the principal initially invested, e is the base of a natural logarithm, and r is the rate of interest, determine the amount of money, to the nearest cent, in the account after 12 years.
Step-by-step explanation:
Using the formula V = Pe^(rt), where P is the principal initially invested, e is the base of a natural logarithm, r is the rate of interest, and t is the time in years:
V = Pe^(rt)
We are given that P = $8290, r = 0.06 (since the annual interest rate is 6%), and t = 12 (since we want to find the value of the account after 12 years). Therefore, we can plug in these values and solve for V:
V = 8290 * e^(0.06*12)
V = 8290 * e^(0.72)
V = $17,936.34
Therefore, the amount of money in the account after 12 years, to the nearest cent, is $17,936.34.
PLEASE ANSWER!!!! Jeremiah has collected 8 U.S. stamps and 12 international stamps. He wants to display them in identical groups of U.S. and international stamps, with no stamps left over. What is the greatest number of groups Jeremiah can display them in?
i will give brainliest.
Answer:
the greatest number of groups you can make is 4
Step-by-step explanation:
First we have to find the greatest number which can divide 8 and 12 exactly.
there is none but G.C.F is 4
That is, 8 U.S stamps can be displayed in 4 groups at 2 groups.
And 12 international stamps can be displayed in 4 groups at 3 groups
In this way, each of the 4 groups would have 2 U.S stamps and 3 international stamps. And all the 4 groups would be identical.
If f (x) = √x
Write the equation for the transformed functions "y1" and "y2"
y1 = 2f (x + 5) -3 Equation : _______
y2 = f(2(x - 5)) + 3 Equation : _______
Given the equation f(x) = x^2 - 9, write the equations for – f(x) and for f(–x)
f(x) = x^2 - 9
f(-x) = (-x)^2 - 9
y1 = 2√(x + 5) - 3
y2 = √(2(x - 5)) + 3
For the second question:
f(x) = x^2 - 9
f(-x) = (-x)^2 - 9
f(-x) = x^2 - 9
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A student uses the change of base formula on a logarithm expression and the result is the following formula: (log3)/(log8) What was the original formula?
The original logarithmic formula of the given expression is log₈3.
The logarithm has different properties that are fundamental to work with it and solve different problems. The change of base formula states that logₐb = (logₓb)/(logₓa), where x is any base.
In this case, the base x is not specified, so it can be any base. Using the change of base formula, we can rewrite the original formula as (logₓ3)/(logₓ8), which is equivalent to the given expression (log3)/(log8). Therefore, the original formula was log₈3.
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A study was conducted attempting to relate home ownership to family income. Twenty households were selected and family income was estimated, along with information concerning home ownership (y=1 indicates yes and y = 0 indicates no) the data are shown below
Household Income Home ownership status
1 38000 0
2 51200 1
3 39600 0
4 43400 1
5 47700 0
6 53000 0
7 41500 1
8 40800 0
9 45400 1
10 52400 1
11 38700 1
12 40100 0
13 49500 1
14 38000 0
15 42000 1
16 54000 1
17 51700 1
18 39400 0
19 40900 0
20 52800 1
Fit a logistic regression model to the response variable y. Use a simple linear regression model as the structure for the linear predictor.
Is the logistic regression model in part (a) adequate?
Provide an interpretation of the parameter Beta 1 in this model.
Yes, the logistic regression model is adequate to the response variable y. The parameter Beta 1 in this model measures the change in the log odds of the response variable when the independent variable (household income) increases by one unit. This is a measure of how the likelihood of home ownership changes in response to an increase in household income.
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i need help with this. can someone please help.
The area of the circle is 490. 625 cm²
How to determine the valueThe formula for calculate the circumference of a circle is expressed as;
C = 2πr
Given that;
C is the circumference of the circleπ is a constant valuer is the radius of the circleFrom the information given, we have that;
Substitute the value of the circumference
25π = 2πr
Divide by the coefficient of r
r = 12. 5cm
Area of a circle = πr²
Substitute the values
Area = 3.14 × (12.5)²
Find the square
Area = 490. 625 cm²
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which equation can be used to represent the distance d for the times given in the table rebeka chose A as the the correct answer.how did she get that answer
Answer:
Step-by-step explanation:
Pls give simple working
ch help video solution of the system of equations. 5x-y=44 -3x-y=-12 Submit Answer
The solution of the system of equations is x=7 and y=-9.
To find the solution of the system of equations, we need to solve for x and y. We can do this using the elimination method.
Multiply the first equation by 3 and the second equation by 5 to eliminate the x variable.
3(5x-y=44) -> 15x-3y=132
5(-3x-y=-12) -> -15x-5y=-60
Add the two equations together to eliminate the x variable.
15x-3y=132
-15x-5y=-60
-----------
-8y=72
Step 3: Solve for y.
-8y=72
y=-9
Substitute the value of y back into one of the original equations to solve for x.
5x-(-9)=44
5x+9=44
5x=35
x=7
Check the solution by substituting the values of x and y back into the original equations.
5(7)-(-9)=44
35+9=44
44=44
-3(7)-(-9)=-12
-21+9=-12
-12=-12
The solution is correct.
Submit the answer.
The solution of the system of equations is x=7 and y=-9.
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5000 people visit mueum during a month how many visitors were there for each age group. 18 and under, age 19 to 44, 45 to 64, age 65 and over
The number of degrees for each part of the museum visitors graph would be Age 18 and under: 108 degrees, Age 19 – 44: 180 degrees, Age 45 – 64: 54 degrees, and Age 65 and over: 18 degrees.
To find the number of degrees for each part of the museum visitors graph, we need to know the total number of visitors for that month. Let's assume that the total number of visitors for the month is 10,000.
Age 18 and under:
Let's assume that the number of visitors aged 18 and under is 3,000. To find the number of degrees for this section of the graph, we need to use the formula: (Number of visitors in the age group / Total number of visitors) x 360 degrees. So, (3000/10000) x 360 = 108 degrees.
Age 19 – 44:
Let's assume that the number of visitors aged 19-44 is 5,000. Using the same formula as above, we get (5000/10000) x 360 = 180 degrees.
Age 45 – 64:
Let's assume that the number of visitors aged 45-64 is 1,500. Using the same formula as above, we get (1500/10000) x 360 = 54 degrees.
Age 65 and over:
Let's assume that the number of visitors aged 65 and over is 500. Using the same formula as above, we get (500/10000) x 360 = 18 degrees.
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y = -15 x -5
11 x + y = -17
I need help with this.
Answer:
system of equation:
(x,y)
(3,-50)
Step-by-step explanation:
equation solving
If 270 students bought a cheese pizza or a pepperoni pizza, how many lunches were sold on Friday?
Option Percent
Cheese Pizza 50
Pepperoni Pizza 40
Fried Chicken 10
If each lunch costs $3.50, how much money will the cafeteria earn from all of the lunches?
The cafeteria earned $945 from all of the lunches sold on Friday.
What is Percentage?percentage, a relative value indicating hundredth parts of any quantity.
Let us calculate the number of students who bought each pizza.
Number of students who bought cheese pizza = 50% of 270 = 0.5 x 270 = 135
Number of students who bought pepperoni pizza = 40% of 270 = 0.4 x 270 = 108
Total lunches sold = Number of cheese pizzas sold + Number of pepperoni pizzas sold + Number of fried chicken sold
= 135 + 108 + 27
= 270
Therefore, a total of 270 lunches were sold on Friday.
To calculate how much money the cafeteria earned from all of the lunches, we can multiply the total number of lunches sold by the cost of each lunch:
Total earnings = Total lunches sold x Cost per lunch
= 270 x $3.50
= $945
Therefore, the cafeteria earned $945 from all of the lunches sold on Friday.
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Let a, b, and c be elements of a field F.Prove:
(1) If a + b = c + b, then a = c.
(2) If ab = cb and b doesn't equal 0, then a = c.
Answer:
The given statements can be proved as follows:
(1) If a + b = c + b, then a = c.
Proof:
- Start with the given equation: a + b = c + b
- Subtract b from both sides of the equation: a + b - b = c + b - b
- Simplify the equation: a = c
- Therefore, if a + b = c + b, then a = c.
(2) If ab = cb and b doesn't equal 0, then a = c.
Proof:
- Start with the given equation: ab = cb
- Divide both sides of the equation by b: ab/b = cb/b
- Simplify the equation: a = c
- Therefore, if ab = cb and b doesn't equal 0, then a = c.
In both cases, the elements a and c are equal when the given conditions are met.
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What is the vertex of 2(n+9)(n-6)
Answer:
(-1.5, -112.5)
Step-by-step explanation:
f(n) = 2(n + 9)(n - 6) = 0
n = -9, n = 6
½(6 - (-9)) = ½(15) = 7.5
-9 + 7.5 = -1.5
f(-1.5) = 2(-1.5 + 9)(-1.5 - 6)
= 2(7.5)(-7.5)
= -(15 × 7.5)
= -(75 + 37.5)
= - 112.5
Answer #14 using the picture
Answer:
(10,14)
Step-by-step explanation:
if you look there's a pattern
help me what is it i need urgent help
Answer:
#1 is equivalent
#2 is not equivalent
#3 is not equivalent
#4 is not equivalent
#5 is not equivalent
Hope this helps!
If y equals 12 when X equals 18 find X and why equals -16
Answer:
- 10
Step-by-step explanation:
If y =12 when x is 18
Find x when y =-16
What the relationship between x and y
y = x - 6 and x = y +6
Insert y(-16) into equation above
Y = X - 6
-16= X - 6
Collect like terms
Add 6 to both sides
-10 = X
The expression 5x − 7 represents the time it takes a commuter to travel in the morning to work. The expression 11x – 1 represents the time it takes a commuter to travel in the evening from work. What is the total travel time?
The expression that represents the total travel time is 16x − 8
Adding Polynomials:
To add polynomial expressions we need to add or subtract like terms and constant terms.
In the given problem to find the total travel time we need to add the time takes to travel to work and the time takes to travel from work to home.
Here we have
The expression represents the time takes to travel to work = 5x − 7
The expression represents the time takes to travel from work = 11x – 1
The total travel time = 5x − 7 + 11x – 1
=> 5x + 11x − 7 – 1
=> 16x − 8
Therefore,
The expression that represents the total travel time is 16x − 8
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A container built for transatlantic shipping is constructed in the shape of a right
rectangular prism. Its dimensions are 2.5 ft by 13.5 ft by 9 ft. If the container is
entirely full and, on average, its contents weigh 0.28 pounds per cubic foot, find the
total weight of the contents. Round your answer to the nearest pound if necessary.
The volume of a right rectangular prism is given by the formula below
[tex]V=whl[/tex]
Thus, in our case,
[tex]V=2.5\times13.5\times9=303.75[/tex]
The volume of the container is 1116.5ft^3.
Finally, multiply the total volume by the density given in the problem, as follows
[tex]weight=303.75\times0.28=85.05[/tex]
⇒ [tex]weight[/tex] ≈ [tex]85[/tex]
Rounded to the nearest pound, the answer is 85 pounds.
Write the polynomial as the product of linear factors. 9(x) = x4 - 2x3 + 5x2 8x + 4 g(x) = _________ List all the zeros of the function. (Enter your answers as a comma-separated list.) XE =__________
The polynomial as a product of linear factor [tex]9(x) = x4 - 2x3 + 5x2 8x + 4 g(x)[/tex] are g(x) = (x-2)(x-1)(x+1)(x+4) , all the zeros of function are 2,1,-1,-4.
In order to write the polynomial as a product of linear factors, we must first find its zeros. The zeros of a polynomial are the values of x that make the polynomial equal to zero. The way to find the zeros is to set the polynomial equal to zero, and solve for x.
For this particular polynomial, the equation would be: x^4 - 2x^3 + 5x^2 + 8x + 4 = 0.
We can solve this equation by factoring. When factoring, we look for common factors among the terms and group them together. After factoring, the equation becomes: (x - 2)(x - 1)(x + 1)(x + 4) = 0.
The zeros of the equation are x = 2, 1, -1, -4. This means that the polynomial can be written as the product of linear factors, which is (x-2)(x-1)(x+1)(x+4). The zeros of this function are x = 2, 1, -1, -4.
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The polynomial as the product of linear factors. 9(x) = x4 - 2x3 + 5x2 8x + 4 g(x) is a product of (x - 2)(x - 1)(x + 1)(x + 4) . All the zeros of function are x = 2,1,-1,-4.
In order to write the polynomial as a product of linear factors, we must first find its zeros. The zeros of a polynomial are the values of x that make the polynomial equal to zero. The way to find the zeros is to set the polynomial equal to zero, and solve for x.
For this particular polynomial, the equation would be: x^4 - 2x^3 + 5x^2 + 8x + 4 = 0.
We can solve this equation by factoring. When factoring, we look for common factors among the terms and group them together. After factoring, the equation becomes: (x - 2)(x - 1)(x + 1)(x + 4) = 0.
The zeros of the equation are x = 2, 1, -1, -4. This means that the polynomial can be written as the product of linear factors, which is (x-2)(x-1)(x+1)(x+4).
The zeros of this function are x = 2, 1, -1, -4.
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If you borrow $400 for 5 years at an annual interest rate of 3%, what is the total amount of money you will pay back?
Answer: $460
Step-by-step explanation:
To calculate the total amount of money you will pay back after borrowing $400 for 5 years at an annual interest rate of 3%, you can use the following formula:
Total amount = principal + interest
Principal = the amount you borrowed = $400
Interest = the amount of interest paid over the 5 years period.
To calculate the interest, we can use the formula:
Interest = Principal x Rate x Time
Rate = the annual interest rate expressed as a decimal = 0.03
Time = the time period in years = 5
So, the interest for the 5-year period is:
Interest = $400 x 0.03 x 5 = $60
Therefore, the total amount of money you will pay back is:
Total amount = $400 + $60 = $460
You will pay back $460 over five years if you borrow $400 at an annual interest rate of 3%.
Answer: To calculate the total amount of money to be paid back for borrowing $400 for 5 years at an annual interest rate of 3%, we need to use the formula for simple interest:
Simple interest = Principal x Rate x Time
where:
Principal = $400 (the amount borrowed)
Rate = 3% per year
Time = 5 years
Plugging these values into the formula, we get:
Simple interest = $400 x 0.03 x 5
= $60
Therefore, the total amount of money to be paid back is the sum of the principal and the simple interest:
Total amount = Principal + Simple interest
= $400 + $60
= $460
So, you will need to pay back a total of $460 after 5 years of borrowing $400 at an annual interest rate of 3%.
Step-by-step explanation:
6 ft – 9 inches to decimal feet.
The unit conversion the value is 6.75 ft.
What is unit conversion?The same feature is expressed in a different unit of measurement through a unit conversion. Time can be stated in minutes rather than hours, and distance can be expressed in kilometres rather than miles, or in feet rather than any other unit of length.
Here the given unit measurement is ,
=> 6ft 9 inches.
We know that, to convert inches into feet we need to divide by 100. Then,
=> 9 inches = 9/12 = 0.75 ft.
Now total measurement = 6+0.75 = 6.75 ft.
Hence the after unit conversion the answer is 6.75 ft.
To learn more about unit conversion refer the below link
https://brainly.com/question/13016491
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How to find the least common denominator of 1/6x + 3/8?
Answer:
Step-by-step explanation:
The least common denominator for 1/6 + 3/8 is 24
first find the multiples of 6 and 8
6= 6, 12, 18, 24, 30
8 = 8, 16, 24, 32
Next circle the number that appear in both
That number will be 24 which is the least common denominator.
