The general solution for the non-homogeneous equation is the sum of the homogeneous solution and the particular solution::
y(x) = c1 * cos(√-17i * x) + c2 * sin(√-17i * x) + (3/64) * e^8x
where c1 and c2 are arbitrary constants.
The method of undetermined coefficients is used to find a particular solution for a non-homogeneous linear ordinary differential equation of the form:
dy/dx + p(x)y = g(x)
where p(x) and g(x) are known functions. To use this method, we first find the general solution of the corresponding homogeneous equation:
dy/dx + p(x)y = 0
The general solution of this homogeneous equation can be written as:
y = c1 * y1(x) + c2 * y2(x)
where c1 and c2 are arbitrary constants and y1(x) and y2(x) are the linearly independent solutions of the homogeneous equation.
Next, we guess a particular solution yp of the non-homogeneous equation of the form:
yp = A * f(x)
where A is an unknown constant and f(x) is a function of x that matches the form of the forcing function g(x).
For the given equation: y'' + 17y = 3e^8x
The corresponding homogeneous equation is: y'' + 17y = 0
The characteristic equation is: r^2 + 17 = 0
The roots are: r = ±√-17i
So the general solution of the homogeneous equation is:
y = c1 * cos(√-17i * x) + c2 * sin(√-17i * x)
where c1 and c2 are arbitrary constants.
For the non-homogeneous equation, we guess a particular solution of the form:
yp = A * e^8x
where A is an unknown constant. Substituting this into the non-homogeneous equation, we have:
yp'' + 17yp = A * 8^2 * e^8x = A * 64e^8x
Comparing the right-hand sides of the two equations, we see that yp'' + 17yp = A * 64e^8x = 3e^8x, so A = 3/64.
Therefore, a particular solution for the non-homogeneous equation is:
y = yp = (3/64) * e^8x
The general solution for the non-homogeneous equation is the sum of the homogeneous solution and the particular solution:
y = c1 * cos(√-17i * x) + c2 * sin(√-17i * x) + (3/64) * e^8x
where c1 and c2 are arbitrary constants.
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select all expressions that are equivalent to 2to the power of 4 / ((3.2 x 0.8) + 1.44)
The value of the given expression will be 4.
Expression in maths is defined as the relation of numbers variables and functions by using mathematical signs like addition, subtraction, multiplication and division.
Given that the expression is 2⁴ / ( 3.2 x 0.8 0 + 1.44 ). SOlve the expression as below,
E = 2⁴ / ( 3.2 x 0.8 ) + 1.44 )
Solve the denominator by mathematical operation.
E = 16 / ( 2.56 + 1.44 )
Divide the numerator by denominator,
E = 16 / 4
E = 4
Therefore, the solution of the expression is 4.
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Find The Value Of X.
Answer:
Step-by-step explanation:
The segments of the triangles are proportional.
Add all the measurements on the left to get the complete length of the
left side - 9 + 3 + 12 = 24
Form a proportion with a smaller triangle still using the left side,
this time 9 + 3 = 12
Use the bottom measurement of 22 to complete the proportion:
You have:
24/22 = 12/x
cross multiply - 12 × 22 = 264
divide by 24 = 11
x = 11
I need help on this asap!!!!
Alan makes the statement about the intersection point of a system of inequalities. Explain why Alan’s statement is incorrect.
The intersection point is always a solution to a system of
inequalities because that is where the two lines meet
Answer: No, Alan's statement is incorrect!
Step-by-step explanation:
The intersection point of a system of inequalities is not always a solution. It only becomes a solution if the two lines meet at the same point and the lines are not parallel. If the two lines are parallel, then they will not intersect and thus the intersection point will not be a solution.
Step-by-step explanation:
if there is an intersection point (the lines are not parallel), there is still the question, if the points on the line are included in the solution or not.
if not, then the intersection point is not part of the solution either.
this is controlled by the absence of presence of the "or equal" part of the inequality.
>= greater than or equal
<= less than or equal
if the inequality is only
> greater than
< less than
the lines are just delimiters, but their points are not part of the solution. and therefore not the intersection point either.
which expression cannot be factored using real numbers?
x^3+1
x^6+8
x^8-2
x^8-1
The quadratic-like polynomial x⁶ + 8 has no real roots, that is, it has two complex roots. (Correct choice: B)
What polynomial cannot be factored using real numbers?
In this problem we must determine what polynomial has no real roots. From real algebra we remember that any polynomial of the form x²ⁿ + a², where a is a positive real number, has no real numbers. That is, the factor form of the polynomial is shown below:
(xⁿ + i a) · (xⁿ - i a)
In other words, any polynomial of the form x²ⁿ + a², a > 0, has two complex roots. Now, by direct comparison, we conclude that polynomial x⁶ + 8 has no real roots.
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THIS IS AREA OF A TRIANGLE
area is 14ft²
hight is 6ft
whats base???
The base of the triangle with area 14ft² and height 6ft is 4⅔ feet
What is area of a triangle?A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted by triangle ABC.
The area of a triangle is given by ;
1/2bh
Area = 14ft²
height = 6ft²
therefore;
14 = 1/2 b × 6
28 = 6b
divide both sides by 6
b = 28/6
b = 14/3
b = 4⅔ ft
therefore the base of the triangle is 4⅔ feet.
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A charge 5. 00 nc is placed at the origin of an xy-coordinate system, and a charge -2. 00 nc is placed on the positive x-axis at x = 4. 00 cm. A third particle, of charge 6. 00 nc is now placed at the point x = 4. 00 cm , y = 3. 00 cm.
The x-component of the total force exerted on the third charge by the other two charges is equal to k * (30.00 nC * 4.00 cm / 5.00 cm + 12.00 nC).
Force is a physical quantity that measures the influence of one object on another. It is a vector quantity, meaning it has both magnitude and direction.
Let's use Coulomb's Law to calculate the x-component of the total force exerted on the third charge by the other two charges. The x-component of the total force is just the sum of the x-components of the forces exerted by each of the two charges.
First, we'll calculate the force exerted by the 5.00 nC charge at the origin. The direction of this force will be towards the third charge, so the angle θ is the arctangent of the ratio of the y-coordinate of the third charge and the x-coordinate of the third charge. That is, θ = arctan(3.00 cm / 4.00 cm). Using Coulomb's Law, we have:
F = k * q1 * q2 / r²,
where k is the Coulomb constant, q1 and q2 are the magnitudes of the charges, and r is the distance between them. For this case,
=> r = √(4.00 cm² + 3.00 cm²) = 5.00 cm.
Thus,
F = k * 5.00 nC * 6.00 nC / 5.00 cm^2 = k * 30.00 nC.
Fx = F * cos(θ) = k * 30.00 nC * cos(arctan(3.00 cm / 4.00 cm))
Fx = k * 30.00 nC * 4.00 cm / 5.00 cm.
Next, we'll calculate the force exerted by the -2.00 nC charge on the x-axis. This force will be in the negative x-direction, so the angle θ is 180 degrees. Thus,
F = k * -2.00 nC * 6.00 nC / 4.00 cm^2 = -k * 12.00 nC.
Fx = F * cos(180) = -k * 12.00 nC * cos(180) = k * 12.00 nC.
Finally, we add up the x-components of the forces exerted by each charge to get the total x-component of the force:
Fx = k * 30.00 nC * 4.00 cm / 5.00 cm + k * 12.00 nC
Fx = k * (30.00 nC * 4.00 cm / 5.00 cm + 12.00 nC).
Complete Question:
A charge of 5.00 nC is placed at the origin of the xy - coordinate system, and a charge of -2.00 nC is placed on the positive x-axis at x = 4.00 cm. A third particle, of charge 6.00 nC is now placed at the point x = 4.00 cm, y = 3.00 cm.
Find the x-component of the total force exerted on the third charge by the other two.
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Each student in a math class has to buy the items in the table. Write an expression in simplest form that represents the
total cost for x students.
Item
Calculator
Notebook
Pack of pencils
11.55
9.65x+0.65x+1.25x
11.55x
x(9.65+0.65+1.25)
Cost
$9.65
$0.65
$1.25
The expression in simplest form that represents the total cost for x students is 11.55x, the correct option is C.
What is an expression?Expression in mathematics can be defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Given that a student in a math class has to buy the items in the table.
Now,
x(9.65+0.65+1.25)
=9.65x+0.65x+1.25x
=11.55x
Therefore, the expression in simplest form will be 11.55x.
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Someone please help me!!
Answer:
84 units
Step-by-step explanation:
The darkest shaded face has 12 full units. Times that by the width or the lightest shade area (7) you get the answer 84 units.
12(7) = 84
Leo replaces the strings on some guitars. He puts 6 new strings on Each guitar. He uses 18 strings. How many guitars does leo put Strings on?
The number of guitars in which leo put Strings on is found to be "3.
Explain about the division of the number?Multiplication has been the opposite of division.
If 3 groups of 4 add up to 12, then 12 split into three sections of equal size comes in 4 in each group. Producing equal groups or determining how many people constitute each group after a fair distribution is the basic objective of division.The stated question is-
On some guitars, Leo swaps off the strings.
Each guitar receives six new strings from him.
His 18 strings are used.
Number of guitars = Total strings / number of strings used in each guitar
Number of guitars = 18/ 6
Number of guitars = 3
Thus, the number of guitars in which leo put Strings on is found to be "3.
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What are the 5 most common Pythagorean triples?
The 5 most common Pythagorean triples are ( 3 , 4 , 5 ) , ( 5 , 12 , 13 ) ,
( 6 , 8 , 10 ) , ( 9 , 12 , 15 ) , and ( 15 , 20 , 25 ) .
Pythagorean triples:
Any three positive integers that completely satisfy the Pythagorean theorem is considered a Pythagorean triple number. According to the theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides of a right triangle. A Pythagorean triple consists of the three sides of a right triangle. In this article, you will learn how to create multiple Pythagorean triples.
A Pythagorean triple consists of the three natural numbers a, b, c such that a² + b² = c². These triples are usually written as (a, b, c), examples
(3, 4, 5) are well known. If (a, b, c) is a Pythagorean triple, then (ka, kb, kc) is also a Pythagorean triple for all natural k. A primitive Pythagorean triplet is a triplet in which a, b, and c are coprime (i.e., have no common divisor greater than 1). For example, (3, 4, 5) is a basic Pythagorean triple, but (6, 8, 10) is not.
A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle and is necessarily a right triangle.
This name comes from the Pythagorean theorem, which states that the side lengths of each right triangle satisfy the formula a² + b² =c². A Pythagorean triple gives the length of three integers of the sides of a right triangle. However, right triangles whose sides are not integers do not form a Pythagorean triple. For example, a triangle with sides a = b = 1 and c = √2 but, (1,1,2) is not a Pythagorean triple because √2 is not an integer. Also, the numbers 1 and √2 do not have a common multiple because √2 is irrational.
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solve the equation ab=bc for A, a, assuming that a, b, and c are square matrices and b is invertible.
The equation AB = BC for A is A = BC[tex]B^{-1}[/tex].
As per the given data:
The equation AB = BC for A is given.
Also, we have to assume that A, B, and C are square matrices and that B is invertible.
Here we have to solve the equation AB = BC for A.
If the determinant of matrix A is non-zero, then the inverse of that matrix exists.
The inverse of matrix A is denoted by [tex]$A^{-1}$[/tex]. If a matrix is invertible, then the condition [tex]$A A^{-1}=I=A^{-1} A$[/tex] will be true.
Any matrix A multiplied by the identity matrix I will produce the same matrix A as the output, i.e., AI = IA = A.
Since B is invertible there exists [tex]B^{-1}[/tex]
[tex]\Rightarrow A B B^{-1}=B C B^{-1} \quad\left[/tex]
Solve the obtained equation for A by using the properties B[tex]B^{-1}[/tex]
[tex]$\Rightarrow A I=B C B^{-1}$[/tex] where I is the identity matrix of same order.
A = BC[tex]B^{-1}[/tex] because A I = I A = A
Therefore, the required matrix A = BC[tex]B^{-1}[/tex] .
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015 Chapter 1: Perimeter and Area in the Coordinate Plane > Section Exercises 1.4> Exercise 8
8
Find the perimeter of the polygon with the vertices Q(-3, 2), R(1, 2), S(1,-2), and T(-3,-2)
The perimeter is
units
The perimeter of the polygon with vertices Q(-3, 2), R(1, 2), S(1,-2), and T(-3,-2) is 16 units.
What is the perimeter?Perimeter is a circumferential measure of the figure, for example, the perimeter of a square is the sum of all its sides.
Here,
To find the perimeter of a polygon, we need to add up the lengths of all its sides. In this case, we have a quadrilateral with vertices Q(-3, 2), R(1, 2), S(1,-2), and T(-3,-2).
We can use the distance formula to find the lengths of each side:
Length of QR = √[(1 - (-3))² + (2 - 2)²] = 4
Similarly,
Length of RS= 4
Length of ST= 4
Length of TQ= 4
Therefore, the perimeter of the polygon is 4 + 4 + 4 + 4 = 16
So the perimeter of the polygon with vertices Q(-3, 2), R(1, 2), S(1,-2), and T(-3,-2) is 16 units.
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The answer to this question?
Answer: 0.429
Step-by-step explanation: Use a calculator and put the fraction and hit enter to get it in decimal form. You should get a number like 0.42857142857 but the problem asks us to round to the nearest thousandth. Since there is a 5 in the ten-thousandth place, we must round up. Therefore, the final answer will be 0.429.
Here’s another…23+18w=-21+14w
Answer: -11
Step-by-step explanation:
what should a good residual plot look like if the regression line fits the data well?
A good residual plot should have the points randomly scattered around the horizontal line with no discernible pattern.
This indicates that the model is well fit to the data. The residuals should also be equally distributed on either side of the horizontal line. This can be seen as the sum of the residuals being equal to 0, or mathematically expressed as the sum of the residuals (e) being equal to 0, where [tex]e = y - ŷ[/tex], where y is the observed value and ŷ is the predicted value. The equation can be expressed as Σe=0.
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i don’t understand how to get answer
Answer:
You probably know that it is a good idea to brush your teeth every day. What does this do that helps prevent tooth decay?
Step-by-step explanation
4 divided by an eighth
Answer:
32
Step-by-step explanation:
The average of your first 6 tests is 82. If your test average after your seventh test is an 80. 5, what did you score on the seventh test?.
The score on the seventh test is 71.5
"Information available from the question"
The average of your first 6 tests is 82.
If your test average after your seventh test is an 80. 5
To find the score on the seventh test.
Now, According to the information:
Average of 6 test is 82
We know that :
Average = Sum of all test / Total no. of test => 82 = Sum of all test / 6
Sum of all test = 492
Let the seventh test be 'x'
So, Sum of seventh test is = 492 + x
Average = [tex]\frac{492+x}{7}[/tex]
80.5 = [tex]\frac{492+x}{7}[/tex]
=> 80.5 × 7 = 492 + x
x = (80.5 × 7) - 492
=> x = 563.5 - 492
x = 71.5
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(-2,1) (-8,-5) round nearest tenth
The slope of line from the points given is 1
What is the slope of lineThe slope of a line is a measure of how steep the line is and the direction it is going. In mathematical terms, it is defined as the ratio of the change in the y-coordinate (rise) to the change in the x-coordinate (run) between two points on the line. Slope is often represented by the letter "m". If (x₁, y₁) and (x₂, y₂) are two points on the line, the slope can be calculated using the formula:
m = (y₂ - y₁) / (x₂ - x₁)
In this problem, we can take the two points and calculate the slope.
m = (y₂ - y₁) / (x₂ - x₁)
Substituting the values into the formula to find the slope;
m = -5 - 1 / -8 - (-2)
m = 1
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Complete Question; Calculate the slope of the line (-2, 1) and (-8, -5)
Help for those questions please
The evaluation of the expressions are shown below
How to evaluate the expressionsTo determine the solution to the expressions, we make use of laws of indices and exponents
Expression 1 (a) and (b)
(27)^(-4/3) ÷ (1/64)^(2/3) = (3^3)^(-4/3) ÷ (1/4^3)^(2/3)
(27)^(-4/3) = (3)^(-4) ÷ (1/4)^(2)
(27)^(-4/3) = 1/81 ÷ 1/16
So, we have
(27)^(-4/3) = 16/81
Next, we have
(32)^(2/5) - (-125)^(-1/3) = (2^5)^(2/5) - (-5^3)^(-1/3)
(32)^(2/5) - (-125)^(-1/3) = (2)^(2) - (-5)^(-1)
(32)^(2/5) - (-125)^(-1/3) = 4 + 1/5
(32)^(2/5) - (-125)^(-1/3) = 21/5
Equation (2)
Here, we have
3(8/27)^(-2/3) = 2^x/3^y
So, we have
3(3/2)^(2) = 2^x/3^y
So, we have
3^3/2^2 = 2^x/3^y
By comparison, we have
x = -2 and y = 3
Equation (3)
Here, we have
3^x(9^[1 - x]) = 27^[x + 2]
So, we have
3^x(3^2[1 - x]) = 27^[x + 2]
This gives
3^(x + 2[1 - x]) = 3^3[x + 2]
By comparison, we have
x + 2[1 - x] = 3[x + 2]
So, we have
x + 2 - 2x = 3x + 6
Evaluate
4x = -4
x = -1
Equation (4)
Here, we have
(5^x)^2x = 25(1/125)^x
So, we have
(5)^2x^2 = 5^[2 - 3x]
By comparison, we have
2x^2 = 2 - 3x
So, we have
2x^2 + 3x - 2 = 0
Evaluate
x = -2 and x = 1/2
Equation (5)
Here, we have
2^x+1 + 2^3-x = 10
Here, we make use of a graphing tool
So, we have
x = 0 and x = 2
Expression (6)
Here, we have
√16^x+2
Express 16 as 2^4
So, we have
√16^x+2 = √2^4[x+2]
Take the square roots
√16^x+2 = 2^2[x+2]
Open the bracket
√16^x+2 = 2^2x+4
Hence, the expression is 2^2x+4
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can someone help me with this question the letter a b or c or d
Answer:
The answer is D
Step-by-step explanation:
because u need to divide the y and x and when u do it to all of them u will get the same ratio which is 4
how to find focus of parabola
(h, k+1/4a) are the coordinates for the focus of parabola. And the standard form is (x - h)2 = 4p (y - k).
To get the focus of a parabola, you must be aware that the equation of a parabola in a vertex form is y=a(xh)2+k, where a denotes the slope of the equation. The center of the parabola is at (h, k+1/4a), according to the formula. The focal point of the parabola is its axis. The centre of a parabola is 'a' units away from the vertex. The vertex and the focus are positioned on the parabola's axis, which you may identify using the equation for the parabola.
The axis of the parabola with a second-degree x equation is the y axis, or a line parallel to it, whereas the axis of the parabola with a second-degree y equation is the x axis, or a line parallel to it.
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Gracie deposited $20,000 into an account that compounds interest weekly at a rate of 1.56%. Each week, her bank withdraws $200 from the account and puts that amount into her mortgage account. How long will it take until the account has a balance of $0? Round to the nearest tenth of a year.
It takes 2 years for the account balance to come down to 0.
What is the compound interest?Compound interest is the interest on savings calculated on both the initial principal and the accumulated interest from previous periods.
The formula used to find the compound interest = [tex]A=P(1+\frac{r}{100})^{nt}[/tex].
Given that, principal =$20,000, rate of interest =1.56%, n=52 and W=$200.
Use the formula, [tex]P=W\frac{1-(1+\frac{r}{n})^{-nt} }{\frac{r}{n} }[/tex]
[tex]20000=200\frac{1-(1+\frac{0.0156}{52})^{-52t} }{\frac{0.0156}{52} }[/tex]
[tex]100=\frac{1-(1+0.0003)^{-52t}}{0.0003}[/tex]
[tex]0.03=1-(1.0003)^{-52t}[/tex]
[tex]0.97=(1.0003)^{-52t}[/tex]
Take log on both the side, we get
-52t·log(1.0003)=log0.97
So, t=2 years
Therefore, it takes 2 years for the account balance to come down to 0.
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If 3/(x + 2) = (y + 3)/2 and x and y are integers, what is the sum of all possible values of x?
Answer:
X = 6/ y+3 - 2
and
Y -3x/x+2
Step-by-step explanation:
hope this helps
The original cost of a shirt is $40. A store manager marks up the cost by 20% to get
the selling price. During a sale, the store manager marks down the cost of the shirt
by 20%. What is the percent change from the original cost to the sale price? Is this
a percent increase or a percent decrease? Show your work.
Answer:
Step-by-step explanation:
Ans:percentage increased by 20
Gross profit/Cost of sales=20percent
Gross profit=20percent x cost of sales
Sales=120percent x cost of sales
Sales=120x40
=$48
48-40=$8
8/40x100percent=20percent
Answer: Decrease? not 100% sure
Step-by-step explanation:
a positive divided by a negative become?
A positive divided by a negative always results in a negative quotient.
Let's use the example 8 / (-2) again. We can write this expression as:
8 / (-2) = 8 / (-1) * 2
We can then simplify the expression on the right-hand side by multiplying -1 and 2, which gives us:
8 / (-1) * 2 = -8 / 2
Now, we can divide 8 by 2, which gives us:
-8 / 2 = -4
So we see that the quotient is negative.
We can generalize this to any positive number divided by a negative number. Let's say we have a positive number a and a negative number -b. We can write the expression as:
a / (-b) = a / (-1) * b
Multiplying -1 and b gives us:
a / (-1) * b = -a / b
So we see that the quotient is negative, since -1 times b is negative, and dividing a positive number by a negative number results in a negative quotient.
Therefore, we can conclude that when a positive number is divided by a negative number, the result is always negative.
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Please help
7f=$51.80
find f
Answer:
7.40
Step-by-step explanation:
51.8 divided by 7 is 7.4
i need to find x. please show the work aswell
Explanation:
The missing angle between the 107 and (4-2x) is 55 due to the vertical angles theorem.
The three angles mentioned add to 180 because they form a straight line.
Here's what the steps would look like:
107+55+(4-2x) = 180
166-2x = 180
-2x = 180-166
-2x = 14
x = 14/(-2)
x = -7
Given an integer representing a 10-digit phone number, output the area code, prefix, and line number using the format (800) 555-1212.
Here we have a program written in C++ language. Each // symbol is a relative comment explaining the reason for each command line (and is not included during the program execution).
The Program:
#include <iostream>
//c++ build in library
using namespace std;
//main code body starts here
int main()
{
//declare variable to store phone numbers,its area code, prefix and line number.
long pnumber; //declare long variable
int ac, prefix, lnumber;
//declare integer variables, where ac: Area Code and lnumber is the Line Number
cout<<"Enter a 10-digit Phone Number: "<<endl;
//cout command prints on screen the desired message
cin>>pnumber;
//cin command enables the user to interact with the programm and enter information manually
//main body to obtain the desired output starts below
//each 'division' is used to allocate the correct value at the correct
//since prefix is used to get the desired output of ( ) -
ac = pnumber/10000000;
prefix = (pnumber/10000)%1000;
lnumber = pnumber%10000;
//main body ends here
cout<<"Based on your 10-digit number"<<endl;
cout<<"below you have (AreaCode), Prefix, and - line number "<<endl;
cout<<"("<<ac<<")"<<""<<prefix<<"-"<<lnumber<<endl;
//Prints on screen the desired output of the long 10 digit Number
return 0; //ends program
}
-------------------------------------------------------------------------------------------------------
The Output Sample:
Enter a 10-digit Phone Number:
8019004673
Based on your 10-digit number
below you have (Area Code), Prefix, and - line number
(801) 900-4673
....Programm finished with exit code 0
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Please help me. I am inept when it comes to math. thx
Note that the proof required is given based on the AA Similarity Postulate.
Similar triangles: If two triangles are similar then their corresponding angles are equal.
By the transitive property of equality if a = b, and b= c then a = c.
AA postulate of similarity states that when two corresponding angles of two triangles are equal then they are called similar to each other.
Given ΔABC ~ ΔRST; and
ΔDEF ~ ΔRST
To Prove: ΔABC ~ DEF Note that:
ΔABC ~ ΔRST - Given
ΔDEF ~ ΔRST
- Given
∠A = ∠R, ∠D = ∠R - Definition of Similar Triangles
∠C = ∠T, ∠F = ∠T
∠A= ∠C, ∠D = ∠F - Transitive property of Equality; Thus,
ΔABC ~ Δ DEF (AA Similarity Postulate)
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