The vector of length four that points in the same direction as u = (1, 2, 2) is v = (4/3, 8/3, 8/3).
To determine the equation of the sphere with a radius of five units, we need the coordinates of its center.
From the given information, we know that the sphere intersects the zy-plane along the circle with the equation [tex](x - 1)^2 + (y + 4)^2 = 9[/tex].
The center of this circle can be found by setting x = 1 and y = -4 in the equation since the circle intersects the zy-plane.
Thus, the center of the sphere is (1, -4, 0).
Now, we can write the equation of the sphere using the center and the radius.
The equation of a sphere in 3D space is given by:
[tex](x - h)^2 + (y - k)^2 + (z - l)^2 = r^ 2[/tex]
where (h, k, l) represents the center coordinates and r represents the radius.
Substituting the values, we have:
[tex](x - 1)^2 + (y + 4)^2 + (z - 0)^2 = 5^2[/tex]
Simplifying the equation, we get:
[tex](x - 1)^2 + (y + 4)^2 + z^2 = 25[/tex]
Therefore, the equation of the sphere with a radius of five units and a center at a positive number is:
[tex](x - 1)^2 + (y + 4)^2 + z^2 = 25[/tex]
Now, let's determine a vector of length four that points in the same direction as u = (1, 2, 2).
To find a vector with the same direction, we can normalize vector u to have a length of 1 and then scale it by a factor of 4.
The normalization of a vector u is given by:
[tex]u_{normalized}[/tex] = u / ||u||
where ||u|| represents the magnitude or length of vector u.
Calculating the magnitude of vector u:
||u|| = [tex]\sqrt{(1^2 + 2^2 + 2^2)} = \sqrt{(1 + 4 + 4)} = \sqrt{9} = 3[/tex]
Now, we can normalize vector u:
[tex]u_{normalized}[/tex] = (1/3, 2/3, 2/3)
To get a vector of length four pointing in the same direction as u, we can scale the normalized vector by 4:
vector v = 4 *[tex]u_{normalized}[/tex]= (4/3, 8/3, 8/3)
Therefore, the vector of length four that points in the same direction as u = (1, 2, 2) is v = (4/3, 8/3, 8/3).
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8[x-y] use x=5 and y=2
Answer:
24
Step-by-step explanation:
Plugin the value of x and y
8[x + y] = 8*[5 - 2]
= 8 *3
= 24
Answer:
24
Step-by-step explanation:
8(5-2)
5-2=3
8 times 3=24
plz can i get brainliest:)
2 1/3% as a mixed number in simplest form
Answer:
71
Step-by-step explanation:
Round off 793.545 to one decimal
Answer:
793.6
Step-by-step explanation:
793.545=793.55=793.6
construct a rhombus with a 15 degree angle and sides equal to r.
Answer:
7uwwjwjwjwjai9qiwjwwiwjb2wuw8ejvewusikwvww
Determine the value of k for which the system has no solutions. k= I +y +4z I +2y-2z 4x +9y +kz = 0 = 1 = 6
The value of k for which the system has no solution is k = -16.
To determine the value of k for which the system has no solution, we can examine the system of equations:
x + y + 4z = 0 ...(1)
x + 2y - 2z = 0 ...(2)
4x + 9y + kz = 6 ...(3)
To have no solution, the system of equations must be inconsistent.
The coefficient matrix of the system is:
[tex]\left[\begin{array}{ccc}1&1&4\\1&2&-2\\4&9&k\end{array}\right][/tex]
The determinant of this matrix is given by:
|A| = (1 × 2 × k) + (1 × (-2) × 4) + (4 × 1 × 9) - (4 × 2 × 4) - (9 × (-2) × 1) - (k×1 ×1)
= 2k - 8 + 36 - 32 + 18 - k
= k + 16
For the system to have no solution, the determinant must be equal to zero:
k + 16 = 0
k = -16
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An overdetermined linear system Ax = b must be inconsistent for some vector b. Find all values of b_1,b_2, b_3, b_4, and b_5 for which the following overdetermined linear system is inconsistent:
x_1 - 3x_2=b_1
x_1 - 2x_2 = b_2
x_1 + x_2 = b_3
x_1 - 4x_2 = b_4
x_1 + 5x_2 = b_5
All possible values of b1, b2, b3, b4, and b5 for which the given overdetermined linear system is inconsistent are given by,
b T ≠ [1 + 3c1 - 4c3, 1 - 2c1 + c2 + 4c3, 1, 1 - 4c1 + 4c3, 1 + 5c1 + c2 - 3c3]T
for any constants c1, c2, and c3.
An overdetermined linear system Ax = b must be inconsistent for some vector b.
The given system is, x1 - 3x2 = b1 x1 - 2x2 = b2 x1 + x2 = b3 x1 - 4x2 = b4 x1 + 5x2 = b5
It can be written in matrix form as
Ax = b
where,
A = 1 -3 0 0 0 1 -2 1 0 -4 1 5
and,
x = x1 x2 and
b = b1 b2 b3 b4 b5
Since A has more rows than columns, so it's an overdetermined system.
In an overdetermined system, the matrix A does not have an inverse, thus we can't solve Ax = b exactly.
So, we have to use least-squares to get an approximate solution. However, the least-squares solution doesn't exist if and only if b is outside the column space of A.
i.e. there is no solution to the system Ax = b, so it's inconsistent.
The column space of A is the set of all linear combinations of the columns of A. Hence, we need to find the column space of A.
First, let's find the reduced row echelon form of A using Gaussian elimination.
Row 1 ÷ 11 -3 0 0 0 1 -2 1 0 -4 1 5
Row 2 -R1 + R2 0 1 0 0 0 1 -1 1 4 0 2
Row 3 -R1 + R3 0 4 1 0 0 0 3 1 -4 0 4
Row 4 -R1 + R4 0 -1 0 1 0 0 -1 5 4 0 5
Row 5 -R1 + R5 0 8 1 0 1 0 3 6 -3 0 10
Row 4 + 4R2 0 0 0 1 0 0 3 1 0 0 13
The RREF is given by, 1 0 0 0 -9/11 -3/11 5/11 -1/11 -4/11 0 0 19/11 0 1 0 0 3/4 1/4 -1/4 0 -3/4 0 2/4 0 0 0 0 0 0 0 0 0
The columns corresponding to the pivot columns form a basis for the column space of A, which is a subspace of R5. Hence, we can express the basis as, B = {b1, b2, b3, b4}, where
b1 = (1, 1, 1, 1, 1)b2 = (-3, -2, 1, -4, 5)
b3 = (0, 1, 0, 0, 1)
b4 = (-4, 4, -4, 4, -3)
Thus, the column space of A is spanned by these 4 vectors.
If b belongs to the column space of A, then the system Ax = b will be consistent, otherwise, it'll be inconsistent.
i.e. there is no solution to the system Ax = b.
The coefficients of b in terms of the basis B are given by,
B T b = [1, -3, 0, -4; 1, -2, 1, 4; 1, 1, 0, -4; 1, -4, 0, 4; 1, 5, 1, -3]b T
Thus, the system Ax = b is inconsistent when b is not in the column space of A.
i.e. when,
b T ≠ c1b1 + c2b2 + c3b3 + c4b4
for any constants c1, c2, c3, and c4.
Substituting the values of b1, b2, b3, and b4 in the above equation, we get,
1b1 + 0b2 + 0b3 + 0b4 ≤ 1 1b1 - 2b2 + 0b3 + 4b4 ≤ 1 1b1 + 1b2 + 0b3 + 0b4 ≤ 1 1b1 - 4b2 + 0b3 + 4b4 ≤ 1 1b1 + 5b2 + 1b3 - 3b4 ≤ 1
So, the values of b1, b2, b3, b4, and b5 for which the given system is inconsistent are given by,
b T ≠ [1, 1, 1, 1, 1]T + c1[-3, -2, 1, -4, 5]T + c2[0, 1, 0, 0, 1]T + c3[-4, 4, -4, 4, -3]T
for any constants c1, c2, and c3.
Hence, all possible values of b1, b2, b3, b4, and b5 for which the given overdetermined linear system is inconsistent are given by,
b T ≠ [1 + 3c1 - 4c3, 1 - 2c1 + c2 + 4c3, 1, 1 - 4c1 + 4c3, 1 + 5c1 + c2 - 3c3]T
for any constants c1, c2, and c3.
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A club consists of five men and seven women. A committee of six is to be chosen.
(a) How many committees of six contain three men
and three women?
(b) How many committees of six contain at least two men?
(a) To find the number of committees of six that contain three men and three women, we can use the concept of combinations.
The number of ways to choose three men out of five is given by the combination formula:
[tex]\({{5}\choose{3}} = \frac{5!}{3!(5-3)!} = 10\)[/tex]
Similarly, the number of ways to choose three women out of seven is given by:
[tex]\({{7}\choose{3}} = \frac{7!}{3!(7-3)!} = 35\)[/tex]
Since the choices for men and women are independent, we can multiply these two values to get the total number of committees with three men and three women:
[tex]\(10 \times 35 = 350\)[/tex]
(b) To find the number of committees of six that contain at least two men, we can consider two cases:
1. Committees with exactly two men:
The number of ways to choose two men out of five is [tex]\({{5}\choose{2}} = 10\)[/tex].
The number of ways to choose four women out of seven is [tex]\({{7}\choose{4}} = 35\)[/tex].
So, the number of committees with exactly two men is [tex]\(10 \times 35 = 350\)[/tex].
2. Committees with three men or more:
We have already calculated the number of committees with exactly three men and three women in part (a), which is 350.
To get the total number of committees with at least two men, we sum the results from the two cases: .
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find EG
please hurry i need help
Answer:
If both sides of the triangle are exactly equal (which can be assumed they are because of the right angle), then that means EF = FG
Given that information, we can determine that EF is 6.1.
Now, all you have to do is add 6.1 + 6.1 to get 12.2.
EG = 12.2.
Tong loaned Jody $50 for a month. He charged 5% simple interest for the month. How much did Jody have to pay Tong?
Answer:
$50(1.05) = $52.50
Jody had to pay $52.50 to Tong for the month.
200 PTS AND BRAINLIEST!!!!!!!!!!!!!!!!TYYYYY!!!!!!!!!!!!nEEDASAP
(a) Andre is planning on renting a new apartment, but he wants to stay within his budget on rent and utilities. Andre is looking at an apartment. The apartment costs $1450 per month, plus $250 for utilities. Will this apartment fit within Andre’s budget? Show your work and explain your reasoning.
(b) How much more money does Andre budget for savings than for groceries and utilities combined? Show your work. Write your answer as a dollar amount.
Answer:
Apartment 1 is his best option
Answer: Apartment 1
Step-by-step explanation:
Q6 What is the greatest number of parts of a circle What's the greatest number that can be formed by cutting the circle with 7 straight cuts? (NOTE: the parts do NOT have to be equal in size)
A) 14
B) 21
D) 29
C) 27
Please find the next fraction in this sequence. No files please
Find the largest possible constant r ∈ (0, 1) such that the function f : [0, r] → [0, r] defined by f(x) = x^2 is a (strict) contraction.
The largest possible constant r ∈ (0, 1) such that the function f : [0, r] → [0, r] defined by f(x) = x^2 is a strict contraction is r = 1/2.
To prove this, we need to show that there exists a positive constant k < 1 such that |f(x) - f(y)| ≤ k|x - y| for all x, y ∈ [0, r] with x ≠ y.
Let x, y ∈ [0, r] with x ≠ y. Then, we have:
|f(x) - f(y)| = |x^2 - y^2| = |(x - y)(x + y)| ≤ |x - y|(r + r) = 2r|x - y|
Therefore, if we choose k = 2r < 1, then |f(x) - f(y)| ≤ k|x - y| for all x, y ∈ [0, r] with x ≠ y.
Now, we need to find the largest possible constant r such that k < 1. We have k = 2r < 1, so r < 1/2.
Thus, the largest possible constant r ∈ (0, 1) such that f(x) = x^2 is a strict contraction is r = 1/2.
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Given y[n] = 3" u[n] - 1.5 8[n-1] = A) B) C) What is Y[z]? Draw the graph 3u[n] and 1.58 [n-1]. Simplify y[n] by first solving Y[z]. Use inverse Z for simplification.
The Z-transform of the given sequence y[n] = 3u[n] - 1.5 8[n-1] is Y[z] = (3z)/(z-1) - (1.5z)/(z-1). To simplify y[n], we need to find the inverse Z-transform of Y[z].
The Z-transform is a useful tool for analyzing discrete-time signals and systems. In this case, we are given the sequence y[n] = 3u[n] - 1.5 8[n-1], where u[n] is the unit step function and 8[n-1] represents the delayed unit impulse function.
To find the Z-transform of y[n], we can use the linearity property of the Z-transform. Since the Z-transform of u[n] is 1/(1-z^(-1)), and the Z-transform of 8[n-1] is z^(-1), we can substitute these values into the expression for y[n] to obtain:
Y[z] = 3(1/(1-z^(-1))) - 1.5(z^(-1))/(1-z^(-1))
To simplify Y[z], we need to combine the terms with a common denominator. By finding a common denominator of (1-z^(-1)), we can rewrite Y[z] as:
Y[z] = (3z - 1.5)/(z-1)
Now, to obtain y[n], we need to find the inverse Z-transform of Y[z]. The inverse Z-transform can be computed using techniques such as partial fraction expansion or by referring to Z-transform tables.
The simplified expression for y[n] would be:
y[n] = 3δ[n] - 1.5δ[n-1]
where δ[n] represents the discrete-time unit impulse function. The graph of y[n] would have an impulse of magnitude 3 at n = 0 and an impulse of magnitude -1.5 at n = 1.
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does anybody know the answer?
Answer:
y+48=90
y= 90-48
y= 42
Step-by-step explanation:
ok
by using the property
sum of two opposite interior angles of triangle is equal to exterior angle
A board game has a spinner divided into sections of equal size(10). Each section is labeled a number between 1-5. Which number is a reasonable estimate of the number of spins on a section labeled 5 over the course of 150 spins?
A(15)
B(25)
C(40)
D(60)
Consider the following IVP: x' (t) = -λx (t), x (0) = xo where λ=17 and x ER. What is the largest positive step size such that Heun's method is stable?
The largest positive step size for which Heun's method is stable in the given initial value problem with x'(t) = -λx(t), x(0) = xo, where λ = 17, is h ≤ 0.034.
Heun's method, also known as the improved Euler method or the explicit trapezoidal method, is an explicit numerical method used for solving ordinary differential equations. The stability of Heun's method depends on the step size chosen for the integration.
The stability criterion for Heun's method is that the step size, denoted as h, should satisfy the condition h ≤ 2 / (|λ|), where λ is the coefficient of the equation being solved. In this case, λ = 17.
Substituting the value of λ into the stability criterion, we have h ≤ 2 / (|17|) = 2 / 17 ≈ 0.1176. Therefore, the largest positive step size for stability is h ≤ 0.1176.
However, to find the largest positive step size, we need to consider the accuracy of the numerical solution as well. A smaller step size typically provides a more accurate solution. Hence, we choose the largest step size that satisfies both the stability criterion and the desired level of accuracy.
In this case, the largest positive step size for which Heun's method is stable and provides a reasonable level of accuracy can be chosen as h ≤ 0.034.
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2) to find [h ] or [h3o ] antilog(- ph)= [h ] therefore if ph = 4.0 [h ] = 1 x 10-4 [h3o ] = 10^ -ph if ph = 4.8 [h ] = 1.6 x 10-5 steps on my calculator
To find the concentration of H+ or H3O+ ions ([H+] or [H3O+]) given a pH value , you can use the formula:
[H+] = 10^(-pH)
Let's calculate the values for two different pH values: pH = 4.0 and pH = 4.8.
For pH = 4.0:
[H+] = 10^(-4.0)
[H+] ≈ 1 × 10^(-4)
Therefore, the concentration of H+ ions ([H+]) at pH 4.0 is approximately 1 × 10^(-4) or 0.0001.
For pH = 4.8:
[H+] = 10^(-4.8)
[H+] ≈ 1.6 × 10^(-5)
Therefore, the concentration of H+ ions ([H+]) at pH 4.8 is approximately 1.6 × 10^(-5) or 0.000016.
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Write down an expression for the perimeter of a rectangle. With length-L and width-W given your answer.
Answer:
[tex]2l + 2w[/tex]
Step-by-step explanation:
We know that a rectangle has four sides. And each pair of opposite sides are equal. We can set the length to [tex]l[/tex] and the width to [tex]w[/tex]. Since we know that both variables have a side that is equal to it, we know that the sum of the sides is[tex]l+l+w+w = 2l + 2w[/tex].
PLS HELP
I WILL MARK BRAINLIEST
IF YOU DONT KNOW DONT ANSWER
SOLVE FOR X IN THE 4 PROBLEMS
Answer:
# 1 the missing side is 11, 20+12+12= 44
Step-by-step explanation:
Answer:
1. 5
2. 4
3. 2
4. 10
Step-by-step explanation:
i did this already. so i knew the answers
Help please fast and don’t say files
Porter read 20% of the Star Wars book. He read 8 pages of the book. Write the number of pages in the Star Wars book
Answer: 40 pages in total
Explanation: Porter has read 20% of the book and this translates to 8 pages of the book.
20% = 8 pages
Assume the total number of pages is x:
20% * x = 8
0.2x = 8
x = 8 / 0.2
x = 40 pages in total
Answer:
40 pages in the book
Step-by-step explanation:
8/40 = .20
Assume IQ scores are normally distributed with a mean of 100 and standard deviation 10. Determine the percent probability that a randomly chosen person as an IQ LESS THAN 90
The distribution of IQ scores is normal, with a mean of μ = 100 and a standard deviation of σ = 10.
Percentage of probability that a randomly selected person will have an IQ less than 90.Solution:We have to find the probability that a randomly selected person will have an IQ less than 90.Using the Z-score formula:Z = (X - μ) / σWhereX = 90μ = 100σ = 10Putting the values into the equation we have:Z = (90 - 100) / 10Z = -1
Using the standard normal distribution table we find that the area to the left of the z-score -1 is 0.1587.That means:P(Z < -1) = 0.1587To find the percentage, we convert it to a percentage by multiplying by 100.0.1587 × 100 = 15.87%Therefore, the probability that a randomly selected person will have an IQ less than 90 is 15.87%.
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Someone please help me please
Answer:
Step-by-step explanation:
12 boxes of hay
Answer:48 bunches of hay
Step-by-step explanation:4 hunches for $9, therefore 108/9=12 and 4x12=48 bunches of hay
PLEASE HELPP !!! it’s due today
Answer:
18
Step-by-step explanation:
What type of function will fit these data points?
Linear
Quadratic
Exponential
None of the above
Answer: quadratic
Step-by-step explanation:
I guess if you can search up a picture of each function, you will be able to understand it. A linear graph is linear (as its name suggests), as a result, it never "goes back" to the value that has showed up. For example, the linear function y=3x+2. If you plug in a random value of y (let's say 10), then solve for x, you will only get 1 value of x. The data table that you have indicates that the y value of 0 has appeared twice. Thus, the function cannot be linear. The same thing applies to an exponential function. A quadratic function, however, can have to x values that result in the same y value. For example, the function y = [tex]x^2[/tex]-1. When you set y equal to zero....
0 = [tex]x^2[/tex]-1
0 = (x+1)(x-1)
x = -1 or 1.
Hope this helps :)
what is the equation of the quadratic graph witha focus of (3,4) and a directrtix of y=8
The equation of the quadratic graph is: [tex]y = \frac{1}{8}(x - 3)^2 + 4[/tex]
In a quadratic graph, the focus and the directrix determine the shape and position of the parabola. The focus (3,4) represents the vertex of the parabola, and the directrix y=8 is a horizontal line.
To evaluate the equation of the quadratic graph, we use the vertex form of a quadratic equation, which is [tex]y = a(x - h)^2 + k[/tex], where (h,k) represents the vertex.
The focus coordinates indicate that the vertex is at (3,4). Thus, h = 3 and k = 4.
Since the directrix is a horizontal line, its equation takes the form y = c, where c is a constant. In this case, the directrix equation is y=8, meaning the distance from the vertex to the directrix is 4 units (8 - 4 = 4).
Using the formula [tex]a =\frac{1}{4p}[/tex], where p is the distance from the vertex to the focus (or directrix), we find that [tex]p = \frac{4}{2} = 2[/tex].
Substituting the values into the vertex form equation, we get:
[tex]y = a(x - 3)^2 + 4[/tex]
To evaluate the value of a, we use the formula a = 1 / (4p), where p = 2. Substituting this value, we have:
[tex]a = \frac{1}{4*2} = \frac{1}{8}[/tex]
Therefore, the equation of the quadratic graph is:
[tex]y = \frac{1}{8} (x - 3)^2 + 4[/tex]
Hence, the equation of the quadratic graph with a focus of (3,4) and a directrix of y=8 is [tex]y = \frac{1}{8} (x - 3)^2 + 4[/tex].
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Find the value of the variables in the simplest form
Answer:
x = 3
Step-by-step explanation:
Using the tangent ratio in the right triangle and the exact value
tan60° = [tex]\sqrt{3}[/tex]
tan60° = [tex]\frac{opposite}{adjacent}[/tex] = [tex]\frac{x}{\sqrt{3} }[/tex] = [tex]\sqrt{3}[/tex] ( multiply both sides by [tex]\sqrt{3}[/tex] )
x = 3
Determine the area under the standard normal curve that lies to the right of (a) Z=0.24. (b) Z=0.02, (c) Z=-0.49, and (d) Z=1.89. (a) The area to the right of Z = 0 24 is (Round to four decimal places as needed.) (b) The area to the right of Z=0.02 is (Round to four decimal places as needed.) (c) The area to the right of Z=-0.49 is (Round to four decimal places as needed.) (d) The area to the right of 2 = 1.89 is (Round to four decimal places as needed) Textbook Statcrunch MACBOOK AIR esc 80 F3 888 F1 F4 0 FS 52 ! 1 $ 2 # 3 4 % 5 6 & 7
The answer to the questions is given in parts.
The standard normal distribution is a normal distribution of data that has been standardized so that it has a mean of 0 and a standard deviation of 1.
The area under the standard normal curve that lies to the right of various values of Z can be calculated using a table of standard normal probabilities, or by using a calculator or computer program. Here, we are given four values of Z and we need to determine the area under the standard normal curve that lies to the right of each value. We can use a standard normal table or a calculator to find these areas.
(a) The area to the right of Z = 0.24 is 0.4052 (rounded to four decimal places).
(b) The area to the right of Z=0.02 is 0.4901 (rounded to four decimal places).
(c) The area to the right of Z=-0.49 is 0.6879 (rounded to four decimal places).
(d) The area to the right of Z=1.89 is 0.0294 (rounded to four decimal places).
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$475 20 for 6 years at 6-% $787 for 3 years at 9% $131 70 for 6 yrs 8 months at 4-%. find the simple interest
The simple interest for each scenario is 1. $171, 2. $212.13, and 3. $35.15.
To calculate the simple interest for each scenario, we can use the formula: Interest = Principal × Rate × Time.
1. For $475 at 6% interest for 6 years:
Principal = $475, Rate = 6%, Time = 6 years.
Interest = 475 × 0.06 × 6 = $171.
2. For $787 at 9% interest for 3 years:
Principal = $787, Rate = 9%, Time = 3 years.
Interest = 787 × 0.09 × 3 = $212.13 (rounded to two decimal places).
3. For $131.70 at 4% interest for 6 years and 8 months (or 6.67 years):
Principal = $131.70, Rate = 4%, Time = 6.67 years.
Interest = 131.70 × 0.04 × 6.67 = $35.15 (rounded to two decimal places).
Therefore, the simple interest for each scenario is:
1. $171
2. $212.13
3. $35.15
What does "simple interest" mean?
Simple interest is a method for figuring out how much interest was paid on an amount of money during a specific time period at a specific rate. Simple interest has a fixed principle amount. Simple interest is a clear-cut and simple method for computing financial interest.
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The correct question would be as
Find the simple interest of each scenario:
1. Principal = $475, Rate = 6%, Time = 6 years.
2. Principal = $787, Rate = 9%, Time = 3 years
3. Principal = $131.70, Rate = 4%, Time = 6.67 years (6 years and 8 months)