Evaluate the following in polar and standard forms. Select all answers that apply. [3(cos(27°) isin(27°))]^5
![Evaluate The Following In Polar And Standard Forms. Select All Answers That Apply. [3(cos(27) Isin(27))]^5](https://i5t5.c14.e2-1.dev/c-qa-images/contents/attachments/FXRqCFKFxelVy5GlPzLEmDDWQK2My8m4.png)
Answers
if z = a ( cos x + i sin x ) ⇒⇒⇒ ∴ z^n = a^n ( cos nx + i sin nx)
Part (1)
∴ [ 3 (cos 27° + i sin 27°) ]⁵ = (3⁵) ( cos 5*27° + i sin 5*27°)
= 243 ( cos 135° + i sin 135°) ⇒⇒⇒⇒ Polar form
=================================
Part (2)
The angle 135° in the second quadrant ⇒ sin is (+) and cos is (-)
and its reference angle = 180° - 135° = 45°
sin 45° = cos 45°= 1/√2 = (√2)/2
∴ sin 135° = (√2)/2 and cos 135° = -(√2)/2
∴ 243 ( cos 135° + i sin 135°) = 243 [ -(√2)/2 + i (√2)/2 ]
= - 243(√2)/2 + 243 (√2)/2 i ⇒⇒⇒⇒ standard form
===========================================
The correct answers are options (1) and (3)
option (1) ⇒⇒⇒ 243 ( cos 135° + i sin 135°) ⇒⇒⇒⇒ Polar form
option (3) ⇒⇒⇒ - 243(√2)/2 + 243 (√2)/2 i ⇒⇒⇒⇒ standard form
Related Questions
Make b the subject of the formula p=2a+b
Answers
p-2a=b
-b -b
p-b=2a
-p -p
-b=2a-p
/-1 /-1
b=-2a+p
hope this helps =)
Assume we have two lines which are perpendicular. If one line has a slope of 3, what is the slope of the other line? Enter your answer as an integer or fraction in lowest terms.
Answers
The slope of a line perpendicular to a line with a slope of 3 is -1/3. This is because perpendicular lines have negative reciprocal slopes.
Explanation:In mathematics, when two lines are perpendicular, their slopes are negative reciprocals of each other because the product of their slopes equals to -1. This means that if one line has a slope of 3, the slope of the other line would be the negative reciprocal of 3, which is -1/3. So, the slope of the other line is -1/3.
To understand this further, let's delve into the figure of Slope and the Algebra of Straight Lines. Here, 'm' represents the slope which is the rise over the run and 'b' is the y-intercept, which is the point at which the line crosses the y-axis. Considering the fact that the slope of the line is 3, there is an increase of 3 in the y-coordinate (or vertical axis) for every increase of 1 in the x-coordinate (or horizontal axis).
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A triangle has vertices A (1, 5) B(2, 8) and C(2,0). It moves 3 units right and 5 units down.
HELP QUICK PLEASE
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X is left to right, so moving 3 units right, you would add 3 to very X coordinate.
Y is moving up or down, moving 5 units down, means you would subtract 5 from every Y coordinate.
The new coordinates would be:
A(4,0) B(5,3) and C(5,-5)
Marcia has $84 less than three times as much as sue. together they have $132. how much money does marcia have?
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and then let 3x-84 = marcia
therefore
(x)+(3x-84)=132
4x-84=132
4x=216
x=54
however just solving for x doesnt completely solve. it only shows sues money because x = sue
for marcia
3x-84
3(54)-84
162-84
78
therefore Marcia has 78$.
To find out how much money Marcia has, a system of linear equations is set up and solved. Sue has $54, which means Marcia has $78 since she has $84 less than three times Sue's amount. They together have $132.
Explanation:The question involves solving a system of linear equations to determine how much money Marcia has if she has $84 less than three times as much as Sue. Together they have $132. To solve this, let's name the amount Sue has as S, and consequently, Marcia has 3S - $84. Adding both amounts, we get the equation S + (3S - 84) = $132.
Combining like terms gives us 4S - 84 = $132. We solve for S by first adding 84 to both sides, which yields 4S = $216. Dividing both sides by 4 gives us S = $54. Knowing Sue's amount, we can compute Marcia’s by multiplying Sue’s amount by three and then subtracting 84, so 3 x $54 - $84 = $162 - $84 = $78. Therefore, Marcia has $78.
A cross section of a right triangular prism is created by a plane cut through the points shown and is also perpendicular to the opposite base.
what is the most specific name of the shape representing the cross section?
A. triangle
B. Rectangle
C. Trapezoid
D. Paralellogram
Answers
9514 1404 393
Answer:
B. Rectangle
Step-by-step explanation:
The cross section will be a rectangle, as illustrated in the attachment.
The correct answer is B. Rectangle.
To determine the shape of the cross section, we need to understand the properties of a right triangular prism and the nature of the plane cut. A right triangular prism has two congruent right triangles as its bases, and the lateral faces are rectangles.
When a plane cut is made through points that are vertices of the base right triangle and is perpendicular to the opposite base, the cross section will intersect all the lateral edges of the prism. Since the lateral edges are perpendicular to the base, the cross section will have four right angles, making it a quadrilateral.
Furthermore, because the cut is perpendicular to the opposite base, the cross section will have two pairs of parallel sides. One pair of parallel sides comes from the intersection with the two lateral faces that are parallel to each other, and the other pair comes from the intersection with the base and the top face of the prism, which are also parallel.
Given these properties, the most specific name for the shape of the cross section is a rectangle, as it is a quadrilateral with four right angles and opposite sides that are parallel and equal in length.
Therefore, the correct answer is B. Rectangle.
you are purchasing a house 12 years from now the estimated purchase price is 171,600.00 you want to make a 20%down payment how much do you have to save per month to reach your goal
Answers
We need to save 20% of 171600; this is 0.2(171600) = $34320
We have 12 years to save for this; 12 months per year for 12 years is 144 months:
144x = 34320
Divide both sides by 144:
144x/144 = 34320/144
x = 238.33
Anyone know this geometry question?
Answers
No combo of any of those angles add up to exactly 180 to suggest that they are parallel lines.
Joe will go to the swimming pool on 20 different days this month. • a one-day pass to the pool is $2.25. • a monthly pass to the pool is $30.00. how much money will joe save by buying a monthly pass
Answers
What is the selling price of an item if the original cost is $784.50 and the mark up on the item is 6.5 percent?
Answers
from my calculations it should be $835.49
Answer:
30
Step-by-step explanation:
Which expression is equivalent to x+2+ [4x - x2+6x+8\x+4]
Answers
Answer: 4x
Use the distance formula to find the distance between (−8, 2.5) and (0, −4.5). 1. Substitute coordinates: 2. Simplify parentheses: 3. Evaluate powers: 4. Simplify. What is the distance between (–8, 2.5) and (0, –4.5)? Round to the nearest hundredth. d ≈
Answers
We have to use the distance formula to find the distance between (−8, 2.5) and (0, −4.5).
The distance formula states:
" For the given points [tex] A(x_{1},y_{1}) [/tex] and[tex] B(x_{2},y_{2}) [/tex], the distance between A and B is given as:
AB = [tex] \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}} [/tex]
We have to find distance between (−8, 2.5) and (0, −4.5). So, [tex] x_{1}= -8, y_{1}= 2.5 , x_{2}=0 , y_{2}=-4.5 [/tex]
AB = [tex] \sqrt{(0+8)^{2}+(-4.5-2.5)^{2}} [/tex]
AB = [tex] \sqrt{64+49} [/tex]
AB = [tex] \sqrt{113} [/tex]
AB = 10.630 units
By rounding to the nearest hundredth, we get
Distance between A and B = 10.63 units.
Suppose you cut a small square from a square of fabric as shown in the diagram. Write an expression for the remaining shaded area. Factor the expression. Type your answer below.
Answers
1. You have the area of the larger square shown in the figure above, is:
A=x²
2. Then, you have the area of the smaller square shown in the figure above, is:
A=3²
A=9
3. Therefore, the expression for the remaining shaded area is:
A=x²-9
A=(x-3)(x+3)
Each of the walls of a room with square dimensions has been built with two pieces of sheetrock, a smaller one and a larger one. the length of all the smaller ones is the same and is stored in the variable small. similarly, the length of all the larger ones is the same and is stored in the variable large. write a single expression whose value is the total area of this room. do not use the pow function. submit
Answers
1. You have the following information given in the problem above:
- The room has square dimensions and it has been built with two pieces of sheetrock, a smaller one and a larger one.
2. Therefore, let's call
x: the smaller one.
y: the larger one.
3. Then, you have that the lenght of the wall is the sum of the smaller one and the larger one:
x+y
4. So, the area of the room is:
(x+y)(x+y)
(x+y)²
Therefore, the answer is: (x+y)²
36 ants including the queen created a new colony. after 9 monthe there are now 766 ants. at 17 months how many ants will live in the conoly
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I am really bad at math can someone please help and explain ?
Answers
root (10) ^ ((3/4) x)
Rewriting we have:
root (10) ^ ((1/4) 3x)
By power properties we can rewrite the expression as:
(8 ^ root (10)) ^ (3x)
Answer:
The equivalent expression is:
(8 ^ root (10)) ^ (3x)
option 4
Wich statement about the following equation is true ? 3x^2-8x+5=5x^2
Answers
The product of two consecutive integers is 156. what are the integers?" what type of equation should she create to solve this problem?
Answers
Answer:
12 and 13 or
- 13 and - 12
A quadratic equation r² + r - 156 = 0
Step-by-step explanation:
Let the first number be r, it means that the second will be r + 1.Given that the product of two consecutive integers is 156. we may interpret that to mean that
r(r + 1) = 156
r² + r - 156 = 0
r² + 13r - 12r -156 = 0
(r + 13)(r - 12) = 0
r = -13 0r 12
if r = -13, r + 1 = -12
If r = 12, r + 1 = 13
A ________________ event has a 100% chance of occurring. Example: Picking a red ball from a bag of only red balls.
Answers
We have been given that probability of the even is 100%. In other words we can say that the probability is 1. It means that the event will certainly occur.
Now, we know that the highest value of probability is 1 and lower value is 0.
Mathematically, we can represents it as
[tex]0\leq P(E)\leq 1[/tex]
If the probability is 0 then the event will never occur.
And if the probability is 1 then event will definitely occur.
Therefore, A certain event has a 100% chance of occurring.
The graph of f(t) = 3 • 2^t shows the value of a rare coin in year t. What is the meaning of the y-intercept?
A. When it was purchased (year 0), the coin was worth $3.
B. In year 1, the coin was worth $6.
C. When it was purchased (year 0), the coin was worth $2.
D. Every year the coin is worth 3 more dollars.
Answers
Answer:
The answer is the option A
When it was purchased (year [tex]0[/tex]), the coin was worth [tex]\$3[/tex]
Step-by-step explanation:
we know that
The y-intercept of a function is the value of the function when the value of the independent variable is equal to zero
In this problem we have
[tex]f(t)=3(2^{t})[/tex]
the independent variable is the variable t
so
Find the y-intercept
For [tex]t=0[/tex] ------> year zero
Find the value of the function
[tex]f(0)=3(2^{0})=\$3[/tex]
That means ----> When it was purchased (year [tex]0[/tex]), the coin was worth [tex]\$3[/tex]
what is 68 less than 5 times a number is equal to the number?
Answers
The number here is will be represented as under -
Let the number be x.
Thus, the 5 times the number will be = 5x.
Now, it says, the number is 68 less than 5 times a number.
The equation so formed will be -
5x - 68 = x
4x = 68
x = 68.
Thus, the number will be 17.
Let us re-check the number,
(5 × 17) - 68 = 17
85 - 68 = 17
17 = 17. Thus, the number will be 17.
Pamela drove her car 99 kilometers and used 9 liters of fuel. She wants to know how many kilometers shee can drive with 12 liters of fuel. How many kilometers can Pamela drive with 12 liters of fuel?
Answers
Answer:
132 km
Step-by-step explanation:
1. she can drive 99 km (kilometers) with 9 liters of fuel, 99/9 is 11
2. if she wants to find out how many kilometers she can drive with 12 liters of fuel, then just multiply 12x11 which is 132
What is the equation of a parabola with a focus (-2,4) and directrix y = 0?
Answers
The equation of the parabola with focus (-2,4) and directrix y = 0 is (x + 2) = 8(y - 2). This is derived by finding the vertex and applying the general equation for a vertically oriented parabola.
To find the equation of a parabola with a given focus and directrix, we use the definition that a parabola is the set of all points that are equidistant from the focus and the directrix. In this case, the focus is at (-2,4) and the directrix is y = 0. The vertex of the parabola will thus be halfway between the focus and the directrix, which means it will lie on the line y = 2 (since the focus has a y-coordinate of 4 and the directrix is at y = 0).
The standard form equation of a vertically oriented parabola with its vertex at the origin is x = 4py, where p is the distance from the vertex to the focus (or to the directrix). For a parabola that is shifted to have a vertex not at the origin, the equation is (x-h)² = 4p(y-k) where (h,k) is the vertex.
In this scenario, because the vertex is not at the origin and the parabola opens upwards (since the directrix is below the focus), we'll adjust the standard form equation to take these factors into account. The vertex (h,k) is (-2,2), the focus is (-2,4), thus p = 2. Therefore, the equation is (x + 2)² = 4*2*(y - 2)
Use the graph below to answer the question that follows:
What are the amplitude, period, and midline of the function?
Amplitude: 4; period: 2π; midline: y = −1
Amplitude: 8; period: 2π; midline: y = 1
Amplitude: 8; period: π; midline: y = −1
Amplitude: 4; period: π; midline: y = 1
Answers
Amplitude:4 (total height of the graph divide 2)
period: 2π ( one full oscillation, 5π/2-π/2)
midline: y= -1 ( total height uses 8 boxes in
graph....midline would be equal
boxes on both side)
Answer:
Amplitude : 4
Period: 2π
Mid-line: y=-1
A is correct
Step-by-step explanation:
We are given graph of periodic function.
We need to find amplitude, period and midline of the function.
From graph:
Maximum value = 3
Minimum value = -5
[tex]\text{Mid-Line }=\dfrac{Max+Min}{2}[/tex]
[tex]\text{Mid-Line }=\dfrac{3-5}{2}=-1[/tex]
The mid-line equation, y=-1
Amplitude: Distance between mid-line and maximum value
Amplitude = 3-(-1) = 4
The amplitude of given graph is 4
Period: It is time when graph repeat their path.
In the given graph repeat their in 2π of interval.
The period of graph is 2π
In 2010, a city's population was 1,405,233 and it was decreasing at a rate of 1.1%. At this rate when will the city's population fall below 1,200,000?
a. 2024
b. 2027
c. 2036
d. 2049
Answers
P(t) = 1405233 * (1 - 0.011)^(t)
models the population at t years after 2010. Then, when P(t) = 1,200,000, we have
1200000 = 1405233 * (0.989)^t
(0.989)^t = 1200000/1405233
t = log(1200000/1405233)/log(0.989)
t = 14.27 years
This means 14.27 years after 2010. Therefore, the answer to this question is 2024.
A log function is a way to find how much a number must be raised in order to get the desired number. The population of the city will be 1,200,000 in the year 2024.
What is Logarithm?A log function is a way to find how much a number must be raised in order to get the desired number.
[tex]a^c =b[/tex]
can be written as
[tex]\rm{log_ab=c[/tex]
where a is the base to which the power is to be raised,
b is the desired number that we want when power is to be raised,
c is the power that must be raised to a to get b.
For example, let's assume we need to raise the power for 10 to make it 1000 in this case log will help us to know that the power must be raised by 3.
The decrease in the population of the city can be written as,
[tex]\rm Final\ population=(Initial\ population)\times (1-r)^n[/tex]
[tex]1,200,000=1,405,233(1-0.011)^n\\\\0.8539=(0.989)^n\\\\\rm log(0.8539)=n\ log(0.989)\\\\n= \dfrac{log(0.8539)}{log(0.989)}\\\\n=14.279 \approx 14.3[/tex]
Since the initial population was 1,405,000 in the year 2010, then the population of the 1,200,000 will be 14 years later.
Thus, the population of the city will be 1,200,000 in the year 2024.
Learn more about Logarithm:
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Jake rented a kayak at $26 for 3 hours. If he rents the same kayak for 5 hours, he has to pay a tota rent of $42. Write an equation in the standard form to represent the total rent (y) that Jake has to pay for renting the kayak for X hours
Answers
Which point is an x-intercept of the quadratic function f(x) = (x – 8)(x + 9)? (0,8) (0,–8) (9,0) (–9,0)
Answers
___________________________________________________________
Explanation:
___________________________________________________________
Given the quadratic function in "factored form" ;
→ with "y" substituted for: "f(x)" — as follows:
___________________________________________________________
→ " y = (x − 8)(x + 9) " ;
→ Find the "x-intercept" of the equation ;
→ {among the answer choices given} ;
___________________________________________________________
Note: The "x-intercept(s)" of an equation refer(s) to the coordinates of the point(s) on the graph of the equation at which the graphed equation crosses the "x-axis".
In other words, the "x-intercept(s)" of an equation refer(s) to the solution of the equation at which: " x = 0 " .
___________________________________________________________
At this point, let us consider our given answer choices:
___________________________________________________________
Note:
____________________________________________________________
Consider the first 2 (two) given answer choices:
____________________________________________________________
Choice: [A]: " (0, 8) " ;
Choice: [B]: " (0, -8) " .
____________________________________________________________
→ Both of these are INCORRECT ;
→ {since these 2 (two) answer choices have "non-zero" values as
"y-coordinates" .}.
Note that by definition, all "x-intercepts" MUST have "y-coordinates" with a value of "0" {zero}.
→ Both of these are INCORRECT ; since these 2 (two) answer choices have "non-zero" values as "y-coordinates".
→ Note that by definition, all "x-intercepts" MUST have "y-coordinates" with a value of "0" {zero}.
Note that:
___________________________________________________________
Choice: [A]: " (0, -8)" ; has "- 8" — [ not: "0" ] — as a: "y-coordinate" ;
and that :
Choice: [B]: " (0, 8) " ; has " 8 " — [not: "0" ] — as a: "y-coordinate".
____________________________________________________________
This narrows our answer choices to the last 2 (two) remaining choices:
____________________________________________________________
Choice: [C]: " (9, 0) " ; AND:
____________________________________________________________
Choice: [D]: " (- 9, 0) " .
____________________________________________________________
Note that both of them could be "x-intercepts" ; since both of them have
values of "0" {zero} as "y-coordinates" .
____________________________________________________________
→ Let us examining EACH of the remaining 2 (two) answer choices. It does not matter the order, but let us start with: Choice: [C]: " (9, 0) " .
→ Consider the original equation:
____________________________________________________________
→ " y = (x − 8)(x + 9) "
→ Note the answer choice given for Choice: [C]: " (9, 0)" .
→ This means that when we plug in "9" for "x" , we should get "0" for "y" ;
→ Let us plug in these values for "x" to see if "0" (for "y") holds true:
→ 0 =? (9 − 8)(9 + 9) ?? ;
→ 0 =? (1) ( 18) ?? ;
→ 0 ≠ 18 ;
→ As such: Choice: [C]: " (9, 0)" — is INCORRECT.
____________________________________________________________
→ At this time, we may assume that: "Choice [D]: " (-9, 0)" — the only remaining answer choice is the correct answer.
→ However, we shall examine this "answer choice" appropriately; as follows:
____________________________________________________________
→ Consider the original equation:
____________________________________________________________
→ " y = (x − 8)(x + 9) "
→ Note the answer choice for: [C]: " ( - 9, 0)" .
→ This means that when we plug in "-9" for "x" , we should get "0" for "y" ;
→ Let us plug in these values for "x" ; to see if "0" (for "y") ; holds true:
→ 0 =? (9 − 8)(- 9 + 9) ?? ;
→ 0 =? (1) ( 0) ?? ;
→ 0 =? 0 ?? ;
→ 0 = 0 ! Yes!
___________________________________________________________
As such:
__________________________________________________________
→ The correct answer is: Choice: [D]: " (-9, 0)" .
___________________________________________________________
When it comes to reimbursement packages, company a offers $175 plus $0.55 per mile. company b offers $200 plus $0.45 per mile. what number of miles driven would make the two reimbursement packages equal?
Answers
y = 0.55x + 175
y = 0.45x + 200
We equate both equations:
0.55x + 175 = 0.45x + 200
We clear the value of x:
0.55x - 0.45x = 200 - 175
0.10x = 25
x = 250 miles
Answer:
A number of thousands driven that would make the two reimbursement packages equal is:
x = 250 miles
Answer:250
Step-by-step explanation:
You spin a spinner 48 times. Out of the 48 trials, the pointer lands 6 times on section two. What is the experimental probability of the pointer landing on section two?
1/3
1/27
1/8
1/7
Answers
Answer:
1/8
Step-by-step explanation:
Algebra question ( Matrices and Determinants ) 20 points
Answers
The correct answer is option A
Answer:
.
Step-by-step explanation:
Kendra has a painting canvas that is 20 inches wide by 37 inches high. She painted a red rose, which covered 50% of the canvas.
What was the area of the canvas which was covered by the red rose?
Answers
A = (20) * (37)
A = 740 in ^ 2
We can make the following rule of three:
740in ^ 2 -----> 100%
x --------------> 50%
Clearing x we have:
x = (50/100) * (740)
x = 370 in ^ 2
Answer:
the area of the canvas which was covered by the red rose was:
x = 370 in ^ 2
The area of the canvas covered by the red rose is 370 square inches, calculated by multiplying the width and height of the canvas to find the total area and then taking 50% of that area.
The task is to calculate the area of a canvas that was covered by a red rose painting, which took up 50% of the total canvas area. To find the area covered by the red rose, we first need to calculate the total area of the canvas. The total area (A) of a rectangle is calculated by multiplying its width (W) by its height (H), which gives us the formula:
A = W x H
In this case, Kendra's canvas is 20 inches wide and 37 inches high. Using the formula:
A = 20 inches x 37 inches = 740 square inches
Now, since the red rose covers 50% of the canvas, we simply take 50% of the total area to find the area covered by the red rose:
Area covered by the red rose = 50% of 740 square inches = 0.50 x 740 square inches = 370 square inches
Therefore, the area of the canvas covered by the red rose is 370 square inches.