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Work Shown:
Recall that the imaginary number i has the definition of [tex]i = \sqrt{-1}[/tex]
List out the first few powers of i
i^0 = 1i^1 = ii^2 = -1i^3 = -ii^4 = 1I'm skipping steps when I listed out those terms, so let me know if you need to see them.
The process repeats every 4 items. This means we'll divide the exponent by 4 to look at the remainder.
Let's do that for the first exponent
12/4 = 3 remainder 0
The remainder 0 tells us that i^12 = i^0 = 1
Also,
22/4 = 5 remainder 2
The remainder 2 means i^22 = i^2 = -1
-------------------
In short: i^12 = 1 and i^22 = -1
which means 6i^12 + 7i^22 = 6(1) + 7(-1) = -1
The variables y and x have a proportional relationship, and y = 12 when x = 5.
What is the value of y when x = 8?
Answer:
y = 19.2
Step-by-step explanation:
We know there is relationship between y and x, so according to the formula [tex]y = kx[/tex]
k = 12/5
we get one function y = 2.4x
when x = 8, input it into the function,
y = 2.4×8 = 19.2
1/2(x+4)=1/2x+1 is there a solution
Solve |5x+1|<-2 and write the solution in interval notation.
If you meant to say [tex]|5x+1| < -2[/tex], then there are no solutions.
This is because the |5x+1| is never negative. This applies to the result of any absolute value function. Absolute value represents distance on a number line. Negative distance is not possible.
---------------------------------------------------------------
If you meant to say [tex]|5x+1| \le 2[/tex], then the steps are shown below
[tex]|5x+1| \le 2\\\\-2 \le 5x+1 \le 2\\\\-2-1 \le 5x+1-1 \le 2-1\\\\-3 \le 5x \le 1\\\\-3/5 \le 5x/5 \le 1/5\\\\-3/5 \le x \le 1/5\\\\-0.6 \le x \le 0.2\\\\[/tex]
The interval notation would be [-3/5, 1/5] in fraction form
That is equivalent to [-0.6, 0.2] in decimal form.
Use square brackets to include each endpoint.
Describe the end behavior of each polynomial.
(a) The leading coefficient is positive and the degree is odd, so:
As x approaches infinity, y approaches infinity.As x approaches negative infinity, y approaches negative infinity.(b) The leading coefficient is negative and the degree is even, so:
As x approaches infinity, y approaches negative infinity.As x approaches negative infinity, y approaches negative infinity.explain how to write 50,000 using exponents
Explanation:
Place the decimal point between the first two digits to get 5.0
Then we must move the decimal point 4 spots to the right to get back to 50,000
So that's where the "4" comes in with the 10^4
This means 50,000 = 5 * 10^4 which is in scientific notation.
The answer in scientific notation is 5 times 10^4
The exponent of four tells you to multiply 10 by itself 4 times.
10 x 10 = 100 x 10 = 1000 x 10 = 10000
And 10000 times 5 is equal to 50,000.
A business has a budget of $720 to spend on their marketing campaign. Each month, x, the business spends $40 on different advertisements. The amount of money remaining in the budget
for marketing can be modeled by the function Mx)=-40x+720. Based on the graph of the linear function M(x) and the context of the problem, what is the domain?
O [10, 18]
O [10, 720]
0 [10, 8]
O [R]
A reasonable domain for this function is [0, 18)
Domain of a functionThe domain of a function is a value of the independent variable of the function for which it exists.
Given the equation below:
M(x)= -40x+720
Since the function represents the The amount of money remaining in the budget for marketing then the function f(x) must be greater than zero;
−40x + 720 > 0
-40x> -720
x < 720/40
x < 18
This shows that a reasonable domain for this function is [0, 18)
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Erica is participating in a road race. The first part of the race is on a 5.2-mile-long straight road oriented at an angle of 25∘ north of east. The road then turns due north for another 6.0 mi to the finish line.
In miles, what is the straight-line distance from the starting point to the end of the race?
The distance between the starting point(initial) and the ending point(last) is known as displacement is 9.46miles.
What is displacement?
Displacement is defined as the change in position of an object. It's a vector volume and has a direction and magnitude. It's represented as an arrow that points from the starting position to the final position. For illustration- If an object moves from A position to B, also the object's position changes. Distance is a scalar volume that refers to" how important ground an object has covered" during its stir. Displacement is a vector volume that refers to" how far out of place an object is" it's the object's overall change in position. Displacement is a defense medium that involves an individual transferring negative heartstrings from one person or thing to another.In the first(1st) part of the race displacement of Erica is
d1 = 5.2 miles at 25(twenty-five) deg North of East
x(ax) and y-component of 1st(first) part of the race will be:
d1 = 5.2*cos 25(twenty five) deg i + 5.2*sin 25(twenty five) deg j
d1 = (4.71 i + 2.20 j) miles (this id d1)
In the second(2nd) part of the race
d2 = 6(six) miles towards north,
So
d2 = (6 j) miles (this is d2)
Now net(accurate) displacement of Erica will be, Using Vector addition(summation):
d = d1(4.71 i + 2.20 j) + d2(6j)
d = (4.71 i + 2.20 j) +(sum) (6 j)
d = 4.71 i +(sum) 8.20 j
So Magnitude of net(accurate) displacement will be:
|d| = [tex]\sqrt{4.71^2+8.20^2}[/tex]
|d| = 9.46 miles(nine.four-six) = distance between starting(initial) point and ending point
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Juan and Mahdi are to run a race, but Juan can run 18 meters per second, whereas Mahdi can run 6 meters per second. If Mahdi has a 228 -meter head start, how long will it be before the faster runner catches the slower runner, if they begin at the same time? Round your answer to the nearest whole number.
Answer:
19 seconds
Step-by-step explanation:
You want to know the time it takes for Juan, running 18 m/s, to overtake Mahdi, running 6 m/s, if Mahdi has a 228 meter head start.
Closing timeThe time to close the gap between the runners will be ...
time = distance/speed
time = (228 m)/(difference in speeds) = (228 m)/(18 m/s -6 m/s)
time = (228 m)/(12 m/s) = (228/12) s = 19 s
It will take Juan 19 seconds to catch Mahdi.
__
Additional comment
The fastest human runner will not exceed 12.3 meters per second, especially over a 300+ meter distance.
In general, a
person is 1% shorter in the evening than in the
morning. Use your height to write a conditional
that uses this fact.
The conditional statement is that; If generally a person is 1% shorter in the evening than in the morning, then my height is about 1.62326 m in the evening.
How to write conditional statements?
First, we will calculate how much is one percent of your height and then subtract that percentage from the total height to get your" height in the evening".
For example:
If my height is 1.64 m, then;
0.01 * 1.64 m = 0.0164 m
Step 2 is;
1.64m - 0.0164m = 1.62326 m
Now, using these height, let's write a conditional that uses the described fact:
Conditional:
If generally a person is 1% shorter in the evening than in the morning, then my height is about 1.62326 m in the evening.
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Describe the features of a Cartesian Plane in terms of its axes, the direction of the axes and its centre.
Answer:
A Cartesian Plane is a plane in Geometry which is an flat surface of infinite size. It has two number lines that intersect at right angles at their zeros. One number line is horizontal, and the other number line is vertical. The point of intersection is the zero of each line. In the horizontal line, the numbers increase to the right. In the vertical line, the numbers increase up. The horizontal line is called the x axis. The vertical line is called the y axis. The center of the Cartesian Plane is the intersection of the two axes, the two number lines, and is called the origin. Points on the Cartesian Plane are described with two numbers in parentheses separated by a comma. The position of the point in the Cartesian Plane is given by the first number, called the x-coordinate, telling us the number of units to the right (positive) or left (negative) of the origin and by the second number, called the y-coordinate, telling us the number of units up (positive) or down (negative) from the origin.
Answer 2 and 3. I’ll give Brainlyest and a lot of points.
Answer:
Please add the graph for these problems so I can solve your questions? Thank you!
Step-by-step explanation:
Question
The value of y varies directly as a, and y = 2 when x = 8. Find the equation represents this relationship.
Optionally you can replace 0.25 with the fraction 1/4
===================================================
Explanation:
The variable y varies directly with x, which means we have the template of y = kx for some constant k.
Plug in x = 8 and y = 2. Then solve for k
y = kx
2 = k*8
8k = 2
k = 2/8
k = 1/4
k = 0.25
Therefore, we go from y = kx to y = 0.25x
-------------
Check:
Plugging in x = 8 should get us to y = 2
y = 0.25x
y = 0.25*8
y = 2
The answer is confirmed.
SOLVE. y''+3y'+2y=4e^x cos3x
Solve the homogeneous equation
[tex]y'' + 3y' + 2y = 0[/tex]
Its characteristic equation is
[tex]r^2 + 3r + 2 = (r + 1) (r + 2) = 0[/tex]
with roots at [tex]r=-1[/tex] and [tex]r=-2[/tex], hence the characteristic solution is
[tex]y_c = C_1 e^{-x} + C_2 e^{-2x}[/tex]
For the nonhomogeneous equation, I'll use variation of parameters. We're looking for a solution of the form
[tex]y = u_1 y_1 + u_2 y_2[/tex]
to the equation
[tex]y'' + a(x) y'' + b(x) y = f(x)[/tex]
such that
[tex]\displaystyle u_1 = - \int \frac{y_2f(x)}{W(y_1,y_2)} \, dx[/tex]
[tex]\displaystyle u_2 = \int \frac{y_1 f(x)}{W(y_1,y_2)} \, dx[/tex]
The Wronskian [tex]W(y_1,y_2)[/tex] of the two fundamental solutions [tex]y_1=e^{-x}[/tex] and [tex]y_2=e^{-2x}[/tex] is
[tex]W(y_1,y_2) = \begin{vmatrix} y_1 & y_2 \\ {y_1}' & {y_2}' \end{vmatrix} = -e^{-3x}[/tex]
Then we have
[tex]\displaystyle u_1 = - \int \frac{e^{-2x} \cdot 4e^x \cos(3x)}{-e^{-3x}} \, dx = 4 \int e^{2x} \cos(3x) \, dx[/tex]
[tex]\displaystyle u_2 = \int \frac{e^{-x} \cdot 4e^x \cos(3x)}{-e^{-3x}} \, dx = -4 \int e^{3x} \cos(3x) \, dx[/tex]
Recall Euler's identity,
[tex]e^{(a+bi)t} = e^{at} (\cos(bt) + i \sin(bt))[/tex]
Then we have the general antiderivative
[tex]\displaystyle \int e^{(a+bi)t} \, dt = \frac1{a+bi} e^{(a+bi)t} + C = \frac{a-bi}{a^2+b^2} e^{(a+bi)t} + C[/tex]
Taking the real parts of both sides, we have
[tex]\displaystyle \mathrm{Re}\left\{\int e^{(a+bi)t} \, dt \right\} = \mathrm{Re}\left\{\frac{a-bi}{a^2+b^2} e^{(a+bi)t} + C\right\} \\\\ \int\,\mathrm{Re}\left\{e^{(a+bi)t}\right\} \, dt = \frac{e^{at}}{a^2+b^2} \mathrm{Re}\left\{(a-bi)(\cos(bt) + i \sin(bt))\right\} + C \\\\ \int e^{at} \cos(bt) \, dt = \frac{e^{at}}{a^2+b^2} (a\cos(bt)+b\sin(bt)) + C[/tex]
so that
[tex]\displaystyle u_1 = 4 \int e^{2x} \cos(3x) \, dx = \frac{4e^{2x}}{13} (2\cos(3x) + 3 \sin(3x))[/tex]
and
[tex]\displaystyle u_2 = -4 \int e^{3x} \cos(3x) \, dx = -\frac{2e^{3x}}3 (\cos(3x) + \sin(3x))[/tex]
We've found
[tex]y = u_1 y_1 + u_2 y_2[/tex]
[tex]\displaystyle y = \frac{4e^x}{13} (2\cos(3x) + 3 \sin(3x)) - \frac{2e^x}3 (\cos(3x) + \sin(3x))[/tex]
[tex]\displaystyle y = \frac2{39} e^x (5\sin(3x) - \cos(3x))[/tex]
Then the general solution to the differential equation is
[tex]\boxed{y(x) = C_1 e^{-x} + C_2 e^{-2x} + \frac2{39} e^x (5\sin(3x) - \cos(3x))}[/tex]
(a) Write a formula for the distance between the points (x,y) and (4,6)
(b) If the distance
between above points is 9 units, write an equation.
Divide the following and then check by multiplying.
48,845
405
Answer:
The given Divisor = 405 and Dividend = 48849
The Quotient is 120 and the Remainder is 249
Hope it helps you
Solve the nonlinear inequality. Expr
x³-9x>0
(-3,0) U (3,00)
Graph the solution set.
How would I graph this on a number line ?
Answer:
x∈(-3;0)∪(3;+∞).
Step-by-step explanation:
1) if to solve the inequality x³-9x>0, then
x(x-3)(x+3)>0, and solution is
x∈(-3;0)∪(3;+∝).
2) the graph is provided in the attachment (it is marked with red colour).
13. What value of b would make the function f (x)- Ibal stretch horizontally by a
factor of 4 and reflect across the y-axis?
Answer:
give me those points boy=o >:)
Step-by-step explanation:
While horizontal and vertical shifts involve adding constants to the input or to the function itself, a stretch or compression occurs when we multiply the parent function
f
(
x
)
=
b
x
by a constant
|
a
|
>
0
. For example, if we begin by graphing the parent function
f
(
x
)
=
2
x
, we can then graph the stretch, using
a
=
3
, to get
g
(
x
)
=
3
(
2
)
x
and the compression, using
a
=
1
3
, to get
h
(
x
)
=
1
3
(
2
)
x
.
Aviva's pay stub shows gross earnings of $662.30 for a week. Her regular rate of pay is $10.80 per hour for a 35 hour week and overtime is paid time and a half regular pay. How many Hours did she work
Number of hours Aviva's work for gross earnings of $662.30 for a week with regular rate of $10.80 per hour for 35 hour week and 1.5 of regular pay for overtime is 52.6hours.
Given,
Gross earning for week=$662.30
Regular rate per hour for 35hour week=$10.80
Overtime rate=1.5×(10.80)
For 'h' number of hours
(10.80×35)+(10.80×1.5) h=$662.30
⇒378 +16.20h=662.30
⇒h=284.30/16.20
⇒h≈17.6hours
Number of hours Aviva's work=35 +17.6
=52.6hours
Therefore, number of hours Aviva's work for gross earnings of $662.30 for a week with regular rate of pay $10.80 per hour for a 35 hour week and 1.5 of regular pay for overtime is 52.6hours.
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If the measure of A is 28 less than three times the measure of B. A and B are a linear pair. Find the measure of A and B.
Measure of linear pair A and B are 86 and 38 respectively.
Let the measure of B = x
And measure of A = 3x – 28
As A and B are linear pair. Therefore their sum will be equal to 180
That is numerically,
A + B = 180
Putting the required values of A and B we will get
3x – 28 + x = 180
4x = 180-28
4x = 152
x = 38
Hence the measure of A = 3*38 – 28 = 86
And the measure of B = x = 38
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Whats (500 x 2) divided by 2?
Answer:
500
Step-by-step explanation:
500*2=1000
1000/2=500
Suppose the scores on a test given to all juniors in a school district are normally distributed with a mean of 71 and a standard deviation of 5. Find the percent of juniors whose score is at least 71. % of juniors scored at least 71.
The percent of juniors whose score is at least 71 % is 69.15 %.
We are given that:
Test of the students are normally distributed.
Mean = μ = 71
Standard deviation = σ = 5
We need to find the percent of juniors whose score is at least 71 %.
Now, we know that:
z = ( x - μ) / σ
z = 100 - 71 / 5
z = 29 / 5
z = 5.8
So, percentage = 69.15 %
Therefore, we get that, the percent of juniors whose score is at least 71 % is 69.15 %.
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Keith works at a garage. In the last week, the garage worked on 105 vehicles.
Out of these 35 were MOTS.
What percentage of the work was MOTS? Give your answer to 2 decimal places.
In the garage 33.33 percentage of work were MOTS.
Percentage is defined as per-cent, that is out of hundred. The percentage is calculated by dividing the acquired value by the total value and multiplying the result by 100.
Now in the given problem,
Total number of vehicles on which Keith worked is 105.
The number of MOTS among the 105 is 35.
Now we have to calculate the percentage of the MOTS out of a total 105 vehicles.
Let us use the formula=(number of MOTS ÷ total vehicles)×100
Therefore required percentage=[tex]\frac{35}{105}\times 100\%[/tex]
or, percentage=33.333...%
Therefore we can conclude that the total percentage of MOTS were approximately 33.33%(rounded to 2 decimal places).
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find the area of 8in 5in 4in 4in 10in 6in
By decomposing the figure into simpler ones, we conclude that the area of the irregular figure is 108 square inches.
How to find the area of the irregular figure?
Here we have an irregular figure that can be decomposed into 3 simpler figures, these are:
A rectangle of 6 inches by 8 inches (the top one).A rectangle of 10 inches by 4 inches (the one in the middle).A rectangle of 5 inches by 4 inches (the one at the right).Remember that the area of a rectangle is the product between the two dimensions, so the areas of these 3 rectangles are:
a = 6in*8in = 48in^2
a' = 10in*4in = 40in^2
a'' = 5in*4in = 20in^2
Adding these areas we get:
48in^2 + 40in^2 + 20in^2 = 108 in^2
The area of the irregular figure is 108 square inches.
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Is (0, 3) a solution to the equation y=x+3? (1 point)
O yes
Ono
Answer: Yes, it is.
Step-by-step explanation:
given: x is 0 and y is 3. Plug in the equation and solve
y=x+3
3=0+3
3=3
Yes, it is a solution
Find the least numbers which must be subtracted from the following number to make them perfect square
42029
Step-by-step explanation:
[tex] 3\sqrt{8} + \sqrt{114} [/tex]
i need some help please help
3x + 1<10
The value of x from the inequality is x < 3
What is an inequality?An inequality is defined as a statement of an order relationship between two numbers or algebraic expressions.
Given the inequality;
3x + 1 < 10
To determine the solution, we take the followings steps;
collect like terms
3x < 10 - 1
Find the difference between the terms
3x < 9
Make 'x' the subject of formula
x < 9/ 3
x < 3
Thus, the value of x from the inequality is x < 3
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Find the solutions to x² = 28.
A. x=+7√2
OB. x=+2√/14
OC. x=+14√2
X
OD. x=+2√7
X
Answer:
x=+2√7
Step-by-step explanation:
(100 points!) What is the slope-intercept form for the following equation?
x + 5y = 12
Questions:
x= 5y + 12
y= -1/5x + 12/5
x + 5y = 12x
5y= −x + 12
Answer:
the slope intercept form is 5y= -x+12
Find the domain. Y= 1-7x / 8-|3x-15|
The domain is written in the form { x : x ≠ 7/3 }
Given data
Y= 1-7x / 8-|3x-15|
How to solve for the domainThe domain refers to the values of x, from the equation we solve for the possible values that will not make the equation undefined.
For the equation to be undefined the denominator should be zero hence we solve for the value of x that will make the equation, to have a denominator equal to zero and the domain will be any other values of x except the value that gives the denominator equal to zero,
The denominator is 8-|3x-15|
solving for the value of x that gives the equation that makes the equation equal to zero
8-|3x-15| = 0
3x - 15 = -8
3x = -8 + 15
3x = 7
x = 7/3
The domain is written in the form { x : x ≠ 7/3 }
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2. Which statement correctly describes the
perpendicular bisector construction below?
The statement correctly describes the perpendicular bisector construction below The construction of JK to bisect GH: Option D is the correct answer
What is perpendicular bisector?The term perpendicular bisector refers to the line that divides another line into two equal parts, making an angle of 90 degrees. The term is made up of two words, perpendicular and bisector.
Perpendicular refers to angle 90 degrees, while the bisector refers to dividing into two equal parts.
In the picture, the main line is GH while JK is the line constructed to bisect the main line
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