[tex]\dfrac{6\sqrt 3~ \text{cis} \left( \dfrac{7 \pi} 6 \right)}{3 \sqrt 5 ~\text{cis} \left( \dfrac{\pi}{3}\right)}\\\\\\=\dfrac{2\sqrt 3 \cdot e^{i\tfrac{7\pi}{6}}}{\sqrt 5\cdot e^{i \tfrac{\pi}{3}}}\\\\\\=\dfrac{2 \sqrt 3}{\sqrt 5} \cdot \left(e^i \right)^{\tfrac{7\pi}{6} - \tfrac{\pi }{3}}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~;\left[\text{cis}~ \theta = e^{i \theta}\right]\\\\\\=\dfrac{2 \sqrt{3}}{\sqrt 5} \cdot \left(e^i\right)^{\tfrac{5\pi}6}\\\\\\=\dfrac{2\sqrt{3}}{\sqrt 5}\cdot e^{i\tfrac{5 \pi}6}\\\\\\[/tex]
[tex]=\dfrac{2\sqrt{3}}{\sqrt 5} \left[ \cos \left(\dfrac{5\pi}{6}\right) + i \sin \left(\dfrac{5\pi}{6}\right) \right]~~~~~~~~~~~~~~~~~~;\left[e^{i\theta} = \cos \theta + i \sin \theta} \right]\\\\\\=\dfrac{2\sqrt{3}}{\sqrt 5}\left[ \cos \left(2 \times \dfrac{\pi}2 -\dfrac{\pi}{6} \right) + i\sin \left(2 \times \dfrac{\pi}2 -\dfrac{\pi}{6} \right) \right]\\\\\\=\dfrac{2\sqrt{3}}{\sqrt 5} \left[ -\cos \left(\dfrac{\pi}{6}\right) + i \sin \left(\dfrac{\pi}{6}\right) \right]\\\\\\[/tex]
[tex]=\dfrac{2\sqrt{3}}{\sqrt 5}\left( -\dfrac {\sqrt{3}}{2} + i \cdot \dfrac 12 \right)\\\\\\=-\dfrac{3}{\sqrt 5} + i \dfrac{\sqrt 3}{\sqrt 5}[/tex]
Answer:
Step-by-step explanation:
Just took the test:
6sqrt3 cis (7pie/6)/ 3sqrt5 cis (pie/3)=
2sqrt15/5 cis (5pie/6)
Expand & simplify
3(t+5)+5(t−6)
Answer:
8t - 15
Explanation:
⇒ 3(t+5)+5(t−6)
distribute inside parenthesis
⇒ 3(t) + 3(5) + 5(t) + 5(-6)
multiply the variables
⇒ 3t + 15 + 5t - 30
collect like terms
⇒ 3t + 5t + 15 - 30
add/ subtract like terms
⇒ 8t - 15
Solution: 8t-15
Detailed explanation:
First the problem asks us to expand
[tex]\tt 3(t+5)=3\cdot t +3\cdot5=3t+15[/tex]
Similarly
[tex]\tt 5(t-6)=5\cdot t -5\cdot 6=5t-30[/tex]
Therefore
[tex]\tt 3t+15+5t-30[/tex]
Upon combining like terms
[tex]\tt 8t+15-30[/tex]
Upon doing this further
[tex]\tt 8t-15[/tex]
What statement is true for f(x) = 3.2^x?
Answer:
Step-by-step explanation:
On a coordinate plane, a line goes through (negative 2, negative 4) and (2, 2). a point is at (negative 3, 1). what is the equation, in point-slope form, of the line that is parallel to the given line and passes through the point (−3, 1)? y – 1=negative three-halves(x 3) y – 1=negative two-thirds(x 3) y – 1= two-thirds(x 3) y – 1= three-halves(x 3)
The equation, in point-slope form, of the line that is parallel to the given line and passes through the point is y - 1 = 3/2(x+3)
Equation of a lineA line is the distance between two points. The equation of a line in point-slope form is expressed as:
y - y0 = m(x-x0)
Given the coordinates (-2, -4) and (2, 2) on the line, the slope is expressed as:
Slope = 2-(-4)/2-(-2)
Slope = (6)/4
Slope = 3/2
Find the equation of the line
y -(1) = 3/2(x-(-3))
y - 1 = 3/2(x+3)
Hence the equation, in point-slope form, of the line that is parallel to the given line and passes through the point is y - 1 = 3/2(x+3)
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Answer:
d
Step-by-step explanation:
The solution to an inequality is graphed on the number line.
A number line going from negative 5 to positive 5. A solid circle appears between positive 4 and positive 5. The number line is shaded from the circle to negative 5.
What is another way to represent this solution set?
{x | x < 4.5}
{x | x ≤ 4.5}
{x | x > 4.5}
{x | x ≥ 4.5}
The correct answer is option B which is {x | x ≤ 4.5}
What is a number line?
A number line is a representation of a graduated straight line used to represent real numbers in introductory mathematics.
Because the arrow is pointing to the left, the answer to the solution set needs to be less than the number the dot is on.
A solid dot on a number means that that number is included in the solution set.
In the picture the solid dot is on 4.5, so the solution set needs to be equal to or less than 4.5.
Therefore the correct answer is option B which is {x | x ≤ 4.5}
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The first step in determining the solution to the system of equations, y = -x2 - 4x - 3 and y = 2x + 5, algebraically is to set the two equations equal as -x2 - 4x -3 = 2x + 5. What is the next step?
Answer:
See below
Step-by-step explanation:
Next step is to gather 'like' terms
-x2 - 4x -3 = 2x + 5.
x^2 + (4x + 2x) + (3+5) = 0 then simplify
x^2 + 6x + 8 = 0 then factor
(x+4)(x+2) = 0
so either of these terms = 0
x+4 = 0 means x = -4
x+ 2 = 0 means x = - 2
Answer:
d
Step-by-step explanation:
Combine like terms onto one side of the equation.
Choose the correct simplification of the expression (2x7y)2(y5)3.
The correct simplification of the expression is 4x¹⁴y¹⁷
Given,
The expression is (2x⁷ y)² (y⁵)³
To distribute the exponent, apply the power rule (ab)ⁿ = aⁿbⁿ
Implement the product rule to (2x⁷y)².
Here, we have a = 2x⁷ , b = y and n = 2
=(2x⁷)² y²(y⁵)³
= 2²(x⁷)² y²(y⁵)³
= 4(x⁷)² y²(y⁵)³
Now exponents should be multiplied while using the product rule.
The product rule is (aˣ)ⁿ = aˣⁿ
We apply this to (x⁷)² and (y⁵)³
So the expression becomes,
=4x⁷·² y² (y⁵·³)
= 4x¹⁴ y² y¹⁵
∵ In exponents we have a rule that if bases are same then powers are added i.e, aˣ aⁿ = aˣ⁺ⁿ
∴ 4x¹⁴ y¹⁵ ⁺ ²
= 4x¹⁴ y¹⁷
Hence we get the answer as 4x¹⁴ y¹⁷
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Oliver wants to buy apples and blueberries to make a fruit tart. Apples cost $3.50 per pound and blueberries cost $3.25 per pound. How many pounds of fruit does he buy if he buys 2 pounds of apples and 3 pounds of blueberries? How many pounds of fruit does he buy if he buys xx pounds of apples and y pounds of blueberries?
Total pounds, 2 pounds of apples and 3 pounds of blueberries:
Total pounds, xx pounds of apples and y pounds of blueberries:
Need answer quickly as possible!
The total pounds of fruits bought if he buys 2 pounds of apples and 3 pounds of blueberries is $16.75.
The total pounds of fruits bought if he buys x pounds of apples and y pounds of blueberries is $3.50x + $3.25y.
What is the total pounds of fruits bought?The total pounds of fruits bought = (pounds of apples x cost per pound) + (pounds of blueberries x cost per pound)
($3.50 x 2) + (3.25 x 3)
$ 7 + 9.75 = $16.75
($3.50 x x) + (3.25 x y)
$3.50x + $3.25y
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How many factors does 16 have
Answer:
5
Step-by-step explanation:
1,2,4,8,16 total 5 factors
assume that y varies inversely with x. If y= 1.6 when x= 0.5, find x when y= 3.2
Answer:
x = 0.25
Step-by-step explanation:
If y varies inversely with x, then:
[tex]y \propto \dfrac{1}{x} \implies y=\dfrac{k}{x} \quad \textsf{(for some constant k)}[/tex]
Given:
y = 1.6 when x = 0.5Substitute the given values into the found equation and solve for k:
[tex]\implies 1.6=\dfrac{k}{0.5}[/tex]
[tex]\implies k=1.6(0.5)[/tex]
[tex]\implies k=0.8[/tex]
Therefore:
[tex]y=\dfrac{0.8}{x}[/tex]
To find the value of x when y = 3.2, substitute y = 3.2 into the found equation and solve for x:
[tex]\implies 3.2=\dfrac{0.8}{x}[/tex]
[tex]\implies x=\dfrac{0.8}{3.2}[/tex]
[tex]\implies x=0.25[/tex]
a recursive formula for the sequence 18,9,4.5,... is
Answer:
A1 = 18
A(n) = A (n-1) -2
Step-by-step explanation:
N represents the number in a sequence and the difference in the sequence is -2 since each time the number is listed in the sequence it is divide by 2.
Match each pair of equivalent values.
1 .
1
square root of 16
2 .
12
square of 8
3 .
4
4 2
4 .
64
1 2
5 .
16
6 .
2
Answer:
1 - [tex]1^{2}[/tex]
12 - ?
4 - square root of 16
64 - [tex]8^{2}[/tex]
16 - [tex]4^{2}[/tex]
2.828 - square root of 8
Step-by-step explanation:
I need more terms to match
Square root of 16 is 4, square root of 8 is 2, but for an answer it would be , 4² is 16, 1² is 2, number 5 would be 64 and number 6 would be 12.
What are mathematical values?A mathematical value can generally be any specific mathematical object. This is most frequently a number in elementary mathematics, such as a real number like or an integer like 42. Every integer or even other mathematical object set to a variable or constant is that object's value.
A mathematical expression's value is the outcome of a computation it describes when its variables and constants are given values. The amount that the function assumes for these argument numbers is the result of a function, assuming the value(s) provided to its argument(s).
Square root of 16 is 4, square root of 8 is 2, but for an answer it would be , 4² is 16, 1² is 2, number 5 would be 64 and number 6 would be 12.
Therefore, Square root of 16 is 4, square root of 8 is 2, but for an answer it would be , 4² is 16, 1² is 2, number 5 would be 64 and number 6 would be 12.
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Omar is going to an amusement park. The price of admission into the park is $25, and once he is inside the park, he will have to pay $5 for every ride he rides on. How much money would Omar have to pay in total if he goes on 8 rides? How much would he have to pay if he goes on r rides?
Cost with 8 rides:
Cost with r rides:
Need answer quickly!
The cost of 8 rides will be equal to $65 and the equation for the r rides will be equal to $25 + $5(r).
What is an equation?Mathematical expressions with two algebraic symbols on either side of the equal (=) sign are called equations.
This relationship is illustrated by the left and right expressions being equal to one another. The left-hand side equals the right-hand side is a basic, straightforward equation.
As per the given information in the question,
Price of admission into the park = $25
Price to pay for every ride in the park = $5
(a)
Then, the price to pay for 8 rides will be,
$25 + $5(8)
= $25 + $(40)
= $65
(b)
Then, the equation for the price to pay for r rides will be equal to,
$25 + $5(r)
Here, r is the number of rides.
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Please help me I am stuck
15. 3x-5=3
16. 3(x-2) = 4(x + 3)
17. +3=
18. 2==
19. 4-2-3
X+2
20.-2-3-3
21. 쭉 = 큭
22. 2x-5-1=3-x
3. 2x + ==3(x-1)
4. 10x-1=x+4
5. k=8k-2
6. -x + 2 = 3x-27
4
7. 1-6x = -x +
B.
9. 2-(x + 1) = 2 + 5x
D. 2(1-x) = 10 - (x + 2)
. 9x - (4x - 5) = 3x - 11
x-9=3(x + 1)
-xx-2)-4=x+6▸
+ 2x
6(5-4x) = -4(5x + 11)
Answer:
tgnjfgznjE XVNJ
Step-by-step explanation:
rtjtejWEJAREfnndzjt
Answer:
15. 2.66
16. - 18
17. Impossible
18. Impossible
19. - 1
20. - 8
21.
The blades of a windmill turn on an axis that is 30 feet from the ground. The blades are 10 feet long and complete two rotations every minute.
Write a sine model, y = asin(bt) + k, for the hight (in feet) of the end of one blade as a function of time t (in seconds). assume the blade is pointing to the right when t = 0 and that the windmill turns counterclockwise at a constant rate.
Options are displayed in the picture
Answer:
D: y= 10sin([tex]\frac{\pi }{15}t[/tex]) + 30
Step-by-step explanation:
30 feet above the ground is the k value.
10 feet long is how long the blades reach up and down which is also known as the amplitude.
The rest is given in all answers.
If 10% off is 20 what is 23% off?
A farmer has only goats and hens in his farm,on a particular day,he sold 12 animals (goats and hens) and the total number of their legs was 32. how many goats and how many hens did he sell that day?
The number of goats sold is 4 and hens sold is 8.
What is an Equation ?An equation is a mathematical statement which equates two algebraic expression with an equal sign.
Let the total number of goats sold be g
Let the total number of hens sold is h
g + h = 12
The total no. of legs of a goats sold will be 4g
and the total number of legs of a hens sold will be 2h
and it is given that
4g +2h = 32
The system of equation formed is
g + h = 12
4g +2h = 32
Multiplying equation (1) by 2 and then (1) - (2)
-2g = -8
g = 4
Therefore the number of goats sold is 4 and hens sold is 8.
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En una tienda comercial anuncia descuentos sucecivos del 30% y 20% en todos los electrodomésticos, ¿Que descuento único equivale?
Answer:
0
Step-by-step explanation:
The _____ _____, denoted , is given by the formula _____, where x is the number of individuals with a specified characteristic in a sample of n individuals.
The sample proportion denoted [tex]\hat{p}[/tex] is given by the formula [tex]\hat{p} = \frac{x}{n}[/tex], where, x is the number of individuals with a specified characteristic in a sample of n individuals.
What is a sample proportion?A sample proportion can be defined as the proportion of individuals in a sample that have a specified characteristic or trait.
Mathematically, sample proportion can be calculated by using this formula:
[tex]\hat{p} = \frac{x}{n}[/tex]
Where:
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Graph the function.
For the function whose graph is shown below, which is the correct formula for the function?
Answer:
[tex]\displaystyle{y=-3x-1}[/tex]
Step-by-step explanation:
I have two methods: standard-learning which will explain overall mathematically - basically formula, etc. - while the other one trick-tips method will rely on choices rather mathematical calculations but with explanation on why and how.
Standard-LearningFirst, let’s pick two points that are integers since they are easily to be found via graphs. We see that (-1,2) and (0,-1) is a part of the graph so now we finally have picked two points for our calculations.
When we have finally picked two points, we will be using them for slope calculation. This part is crucial for finding an equation of a line because a line has to contain slope in its equation.
To calculate the slope, we use rise over run formula which is:
[tex]\displaystyle{\dfrac{\Delta y}{\Delta x} = \dfrac{y_2-y_1}{x_2-x_1}}[/tex]
Note that [tex]\displaystyle{\Delta}[/tex] is delta - in both mathematic and physics, it means changes in.
So, we will be using those two points that we have picked to substitute in the slope formula. Let’s determine that [tex]\displaystyle{(x_2,y_2)=(-1,2)}[/tex] and [tex]\displaystyle{(x_1,y_1)=(0,-1)}[/tex].
Therefore, our slope will be:
[tex]\displaystyle{m=\dfrac{2-(-1)}{-1-0}}\\\\\displaystyle{m=\dfrac{2+1}{-1}}\\\\\displaystyle{m=-3}[/tex]
Next, find the y-intercept. From the graph, we see that the graph intercepts at (0,-1) which is y-intercept so we will only use y-value. Hence, b = -1.
From the slope-intercept form:
[tex]\displaystyle{y=mx+b}[/tex]
Substitute m (slope) = -3 and b (y-intercept) = -1. Hence, the equation of line is:
[tex]\displaystyle{y=-3x-1}[/tex]
Fast Way (Trick-Tips)Relying on choices and knowledge of graph, notice that there are slopes of 3 and -3.
Keep in mind that if slope is negative, it will decrease down and if slope is positive, it will increase up.
Therefore, cut off all choices with positive slopes which are 1 and 3. We are now left with 2 and 4.
Next, look at the graph and tell its y-intercept. It intercepts y at y = -1 so we take value of -1.
So choice 4 is cut off and now we have choice 2 as the answer.
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Raya buys a van for £9500 plus VAT at 20%
Raya pays a deposit for the van.
She then pays the rest of the cost in 12 equal payments of £665 each month.
Find the ratio of the deposit Ray pays to the total of the 12 equal payments.
Give your answer in its simplest form.
Step-by-step explanation:
yo thara bhai joginder tu he ek bndr
find the new prices after the given percentage increases.
a] £35 increased by 10%
b} £105.99 increased by 12%
Answer:
a] 38.5$
b] 118.71$
Step-by-step explanation:
A)35$ was increased by 10%
So:
10% of 35$
10/100 x 35 ($)
3.5$
So 35$ was increased by 3.5$ and the new price is 38.5$
B)105.99$ was increased by 12%
So:
12% of 105.99$
12/100 x 105.99$
=12.7188$
Let's round it to the nearest hundredths:
12.72$
So 105.99$ was increased by 12.72$ and the new price is 118.71$
HOPE IT HELPS :D
Tip: "of" means multiplication in Maths.
Gerald purchased a rectangular plot of land. the length of the plot is 20 feet more than the width. the cost of the land was $12 per square foot. gerald also had a fence put around the entire perimeter of the plot, at a cost of s8 per linear foot. the total amount he spent on both the land and the fence was $10,560. part a write an equation in two variables for the perimeter, p, and an equation in two variables for the area, a, of the plot of land where x is the width of the plot in feet. provide evidence to support your equations
Answer:
P =2(x+ y) [tex]ft[/tex].
A =xy [tex]ft^{2}[/tex]
Step-by-step explanation:
perimeter of rectangle (P)= 2(L+B
Area of rectangle (A) = L×B
Given,
width = 'x'
length = y = x+20
therefore using above formula
P = 2(x+20 + x) = 2(2x+20)
A = (x+20)(x)
cost of land per square foot = $12
cost of land per linear foot = $8
total cost of land = $10560
now,
P×8 + A×12 = 10560
2(2x+20)×8 + (x+20)(x)×12 = 10560
32x + 320 + 12x²+ 240x = 10560
12x² + 272x = 10240
12x² + 272x - 10240 = 0
solving this quadratic equation using Discriminant method
x = [-b ±√(b² - 4ac)]/2
where b is the coefficient of x
a is the coefficient of x²
and c is the constant term
by replacing values in the above formula
x = 19.5 , x = -42.24
as the area cannot be negative therefore -42.24 is neglected
hence x = 19.5ft
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A 4-digit combination lock can be opened only if a correct combination of digits from 0-9 is chosen. Find the probability of guessing the correct combination if: none of the digits are repeated, any of the digits can be repeated, any of the digits can be repeated but the 3 and 8 are not in the combination.
The probabilities are as follows:-
if none of the digits is repeated 0.504If any of the digits can be repeated 1If any of the digits can be repeated but the 3 and 8 are not in the combination 0.24.What is probability?Probability is defined as the ratio of the number of favourable outcomes to the total number of outcomes in other words the probability is the number that shows the happening of the event.
The probability will be calculated as follows:-
Total combinations are = 10⁴ = 10000
A) if none of the digits are repeated:-The combinations will be 10 x 9 x 8 x 7 = 5040
Probability = 5040 / 10000
Probaility = 0.504
B) If any of the digits can be repeated:- The combinations will be 10 x 10 x 10 x 10 = 10000
Probabilityy = 10000 / 10000 = 1
C )If any of the digits can be repeated but the 3 and 8 are not in the combination the combinations will be 7 x 7 x 7 x 7 = 2401
Probability = 2401 / 10000 = 0.24
Therefore if none of the digits is repeated 0.50.If any of the digits can be repeated 1.If any of the digits can be repeated but the 3 and 8 are not in the combination 0.24.
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In 20 years charlie will be three times as old as he is now.how old is charlie is he now
Answer:
10 years old
Step-by-step explanation:
let his age be x , then in 20 years
x + 20 = 3x ( subtract x from both sides )
20 = 2x ( divide both sides by 2 )
10 = x
Charlie is 10 years old
In AABC, what is the value of x?
Answer:
x = 6.5
Step-by-step explanation:
* since it's a triangle ALL the angles inside should give you 180° * Equate everything to 180°
8x + 10x - 10°+ 10x + 8° = 180°( sum of angles on a triangle)
28x - 2° = 180° * group like terms*
28x = 182°
x = 6.5
What are the roots of the polynomial equation x cubed minus 5 x 5 = 2 x squared minus 5? use a graphing calculator and a system of equations. round noninteger roots to the nearest hundredth.
The solution to the equation of the polynomial function x³-5x+5 = 2x²-5 is (c) –2.24, 2, 2.24.
What is Polynomial Equation?
A polynomial is an expression consisting of indeterminates and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.
Here, The function is given as:
x³-5x+5 = 2x²-5
Split the function.
So, we have:
y = x³-5x+5
y = 2x²-5
Using a graphing calculator, we have the x coordinate at the point o intersection of both equations to be
x = –2.24, 2, 2.24
Hence, the solution to the equation is (c) –2.24, 2, 2.24
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48 girls, 41 boys, and 5 teachers sat in the gym. Write the ratio of girls to boys in three ways.
I took the text last month
i need help i don’t know how to solve these!
Its actually pretty simple, all you do is graph both inequalities and wherever both of the shaded areas overlap is where all of the solutions will be. Hope this helps
I WILL GIVE BRAINIEST
Line AB and Line CD bisect each other at point E. If AE = 3x - 10y + 123, BE = -2x - 7y + 122, CE = 9x + 10y - 171, and DE = 3x + 20y - 249, what are the values of x and y?
The values of x and y are 7 and 12 respectively
Solving simultaneous equationsFrom the question, we are to determine the values of x and y
From the given information,
Line AB and Line CD bisect each other at point E
Then, we can conclude that
AE = BE
and
CE = DE
From the given information,
AE = 3x - 10y + 123
BE = -2x - 7y + 122
CE = 9x + 10y - 171
DE = 3x + 20y - 249
Then, we can write that
3x - 10y + 123 = -2x - 7y + 122
Simplifying
3x +2x -10y + 7y = 122 -123
5x -3y = -1 ----------- (1)
Also,
9x + 10y - 171 = 3x + 20y - 249
Simplifying
9x -3x +10y -20y = -249 + 171
6x -10y = -78 --------- (2)
Now, solve equations (1) and (2) simultaneously
5x -3y = -1 ----------- (1)
6x -10y = -78 --------- (2)
Multiply equation (1) by 10 and equation (2) by 3
10 × ( 5x -3y = -1
3 × ( 6x -10y = -78
50x -30y = -10 ----------- (3)
18x -30y = -234--------- (4)
Then, subtracting equation (4) from equation (3), we get
50x -18x -30y-(-30y) = -10-(-234)
32x -30y + 30y = -10 + 234
32x = 224
∴ x = 224/32
x = 7
Put the value of x into equation (1), to find y
5x -3y = -1
5(7) -3y = -1
35 -3y = -1
35 + 1 = 3y
36 = 3y
∴ y = 36/3
y = 12
Hence, the values of x and y are 7 and 12 respectively
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