There are two real roots for the quadratic equation x² - 8 · x + 13 = 0, contained in the number x = 4 ± √2.
How to find the roots of a polynomial by completing the square
In this question we must apply algebraic handling to simplify a quadratic equation and find the roots that satisfy the expression. Completing the square consists in transforming part of the equation into a perfect square trinomial, and then we clear for x:
x² - 8 · x + 13 = 0
x² - 8 · x + 16 = 3
(x - 4)² = 3
x - 4 = ± √2
x = 4 ± √2
There are two real roots for the quadratic equation x² - 8 · x + 13 = 0, contained in the number x = 4 ± √2.
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Select the correct answer from each drop-down menu.
(0.1² +20+15, z < 10
0.25³ +k, z 210
f(x)==
a) =
If the left-hand limit of f(z) is equal to the right-hand limit of f (x) as x approaches 10, the limit of f (x) as x approaches 10 is
the value of k is
and
The limit of f(x) as x approaches 10 is -25, and the value of k is -25.
For the left-hand limit,
lim(x->10-) f(x)
= lim(x->10-) (0.1x² + 20x + 15)
For the right-hand limit,
lim(x->10+) f(x)
= lim(x->10+) (0.25x³ + k)
Since we want the left-hand limit to be equal to the right-hand limit
lim(x->10-) f(x) = lim(x->10+) f(x)
lim(x->10-) (0.1x² + 20x + 15)= lim(x->10+) (0.25x³ + k)
0.1(10)² + 20(10) + 15= 0.25(10)³ + k
10 + 200 + 15 = 0.25(1000) + k
225 = 250 + k
k = 225 - 250
k = -25
Therefore, the limit of f(x) as x approaches 10 is -25, and the value of k is -25.
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Gillian's Restaurant has an ice-cream counter where it sells two main products, ice cream and frozen yogurt, each in a variety of flavors. The restaurant makes one order for ice cream and yogurt each week, and the store has enough freezer space for 115 gallons total of both products.
A gallon of frozen yogurt costs $0.75 and a gallon of ice cream costs SO.93, and the restaurant budgets $90 each week for these products. The manager estimates that each week the restaurant sells at least twice as much ice cream as frozen yogurt. Profit per gallon of ice cream is $4.15. and profit per gallon of yogurt is $3.60.
a. Formulate a linear programming model for this problem.
b. Solve this model by using graphical analysis
The constraints involve three variables (Y, I, and Z), it is not feasible to graphically represent the entire Feasible region in three dimensions.
The linear programming model for decision variables, objective function, and constraints.
Decision Variables:the number of gallons of frozen yogurt purchased each week as Y and the number of gallons of ice cream purchased each week as I.
Objective Function:
The objective is to maximize the total profit. The profit from frozen yogurt can be calculated as 3.60Y, and the profit from ice cream can be calculated as 4.15I. Thus, the objective function is:
Maximize Profit: Z = 3.60Y + 4.15I
Constraints:
1. Budget constraint: The total cost of frozen yogurt and ice cream should not exceed the weekly budget of $90.
Cost Constraint: 0.75Y + 0.93I ≤ 90
2. Freezer space constraint: The total gallons of frozen yogurt and ice cream should not exceed the available freezer space of 115 gallons.
Space Constraint: Y + I ≤ 115
3. Ice cream sales constraint: The weekly ice cream sales should be at least twice the frozen yogurt sales.
Ice Cream Sales Constraint: I ≥ 2Y
4. Non-negativity constraint: The number of gallons for both products cannot be negative.
Non-negativity Constraint: Y ≥ 0, I ≥ 0
The model using graphical analysis, we can plot the constraints on a graph and find the feasible region. Then, we can evaluate the objective function at each corner point of the feasible region to determine the optimal solution.
Since the constraints involve three variables (Y, I, and Z), it is not feasible to graphically represent the entire feasible region in three dimensions.
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Question one
Let F be a Boolean function defined by the following Boolean expression:
F = (¬A ∧ ¬B ∧ ¬C ∧ D) ∨ (¬A ∧ B ∧ C ∧ D) ∨ (A ∧ B ∧ ¬C ∧ D) ∨ (A ∧ ¬B ∧ C ∧ D) ∨ (A ∧ ¬B ∧ ¬C ∧ ¬D)
Where A, B, C, and D are Boolean variables.
a) Construct a truth table for F.
b) Find the minimal sum-of-products form of F using Quine-McCluskey method.
c) Using Boolean algebra, simplify the Boolean expression of F and state the simplified Boolean expression in terms of the three variables A, B, and C.
Answer:
a) The truth table for F is as follows:
A B C D F
0 0 0 0 1
0 0 0 1 0
0 0 1 0 0
0 0 1 1 1
0 1 0 0 0
0 1 0 1 1
0 1 1 0 1
0 1 1 1 0
1 0 0 0 0
1 0 0 1 1
1 0 1 0 1
1 0 1 1 0
1 1 0 0 1
1 1 0 1 0
1 1 1 0 0
1 1 1 1 1
b) Using the Quine-McCluskey method, we can find the minimal sum-of-products form of F:
F = ¬A ∧ ¬B ∧ D ∨ ¬A ∧ C ∧ D ∨ A ∧ B ∧ ¬C ∧ D ∨ A ∧ ¬B ∧ C ∧ D
c) Using Boolean algebra, we can simplify the Boolean expression of F as follows:
F = (¬A ∧ ¬B ∧ D) ∨ (¬A ∧ C ∧ D) ∨ (A ∧ B ∧ ¬C ∧ D) ∨ (A ∧ ¬B ∧ C ∧ D)
= (¬A ∧ (¬B ∧ D ∨ C ∧ D)) ∨ (A ∧ (B ∧ ¬C ∧ D ∨ ¬B ∧ C ∧ D))
= (¬A ∧ (¬B ∨ C) ∧ D) ∨ (A ∧ ((B ∧ ¬C) ∨ (¬B ∧ C)) ∧ D)
= (¬A ∧ ¬B ∧ C ∧ D) ∨ (¬A ∧ B ∧ ¬C ∧ D) ∨ (A ∧ B ∧ C ∧ D) ∨ (A ∧ ¬B ∧ ¬C ∧ D)
Therefore, the simplified Boolean expression of F in terms of the three variables A, B, and C is:
F = (¬A ∧ ¬B ∧ C ∧ D) ∨ (¬A ∧ B ∧ ¬C ∧ D) ∨ (A ∧ B ∧ C ∧ D) ∨ (A ∧ ¬B ∧ ¬C ∧ D)
Step-by-step explanation:
Sarah is setting up a garden plot in the shape of a right triangle. One leg of the triangle will be 7 feet longer than the other and the hypotenuse will be 1 foot longer than the longest leg. Find the length of each side of the garden plot. As a part of your work, include a sketch with the sides labeled. Show your work algebraically and be sure to use proper units in your answer.
value of the sides of the triangle are given as follows;
shorter leg = 5
longer leg = 12
hypotenuse = 13
What is Pythagoras theorem?Pythagoras Theorem states that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse.
If the short leg = x
the longer length = x+7
the hypotenuse = x+7+1 = x+8
Using Pythagorean theorem
(x+8)² = x² + (x+7)²
x² +16x +64 = x²+ x² + 14x +49
collecting like terms
x²-2x²+16x -14x+64 -49= 0
-x² + 22x +15 = 0
x² -2x -15 = 0
x² -5x +3x -15 = 0
(x² -5x) (3x -15) = 0
x(x-5) +3(x-5)
(x+3)(x-5) = 0
x= -3 or +5
Since length is discreet we choose the positive answer
therefore x = 5
shorter leg = 5
longer leg = 5+7 = 12
hypotenuse = 5+7+1 = 13
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NO LINKS!!! URGENT HELP PLEASE!!!
Solve ΔABC using the Law of Sines
1. A = 29°, C = 63°, c = 24
2. A = 72°, B= 35°, c = 21
Answer:
1) B = 88°, a = 13.1, b = 26.9
2) C = 73°, a = 20.9, b = 12.6
Step-by-step explanation:
To solve for the remaining sides and angles of the triangle, given two sides and an adjacent angle, use the Law of Sines formula:
[tex]\boxed{\begin{minipage}{7.6 cm}\underline{Law of Sines} \\\\$\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$\\\\\\where:\\ \phantom{ww}$\bullet$ $A, B$ and $C$ are the angles. \\ \phantom{ww}$\bullet$ $a, b$ and $c$ are the sides opposite the angles.\\\end{minipage}}[/tex]
Question 1Given values:
A = 29°C = 63°c = 24As the interior angles of a triangle sum to 180°:
[tex]\implies A+B+C=180^{\circ}[/tex]
[tex]\implies B=180^{\circ}-A-C[/tex]
[tex]\implies B=180^{\circ}-29^{\circ}-63^{\circ}[/tex]
[tex]\implies B=88^{\circ}[/tex]
Substitute the values of A, B, C and c into the Law of Sines formula and solve for sides a and b:
[tex]\implies \dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}[/tex]
[tex]\implies \dfrac{a}{\sin 29^{\circ}}=\dfrac{b}{\sin 88^{\circ}}=\dfrac{24}{\sin 63^{\circ}}[/tex]
Solve for a:
[tex]\implies \dfrac{a}{\sin 29^{\circ}}=\dfrac{24}{\sin 63^{\circ}}[/tex]
[tex]\implies a=\dfrac{24\sin 29^{\circ}}{\sin 63^{\circ}}[/tex]
[tex]\implies a=13.0876493...[/tex]
[tex]\implies a=13.1[/tex]
Solve for b:
[tex]\implies \dfrac{b}{\sin 88^{\circ}}=\dfrac{24}{\sin 63^{\circ}}[/tex]
[tex]\implies b=\dfrac{24\sin 88^{\circ}}{\sin 63^{\circ}}[/tex]
[tex]\implies b=26.9194211...[/tex]
[tex]\implies b=26.9[/tex]
[tex]\hrulefill[/tex]
Question 2Given values:
A = 72°B = 35°c = 21As the interior angles of a triangle sum to 180°:
[tex]\implies A+B+C=180^{\circ}[/tex]
[tex]\implies C=180^{\circ}-A-B[/tex]
[tex]\implies C=180^{\circ}-72^{\circ}-35^{\circ}[/tex]
[tex]\implies C=73^{\circ}[/tex]
Substitute the values of A, B, C and c into the Law of Sines formula and solve for sides a and b:
[tex]\implies \dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}[/tex]
[tex]\implies \dfrac{a}{\sin 72^{\circ}}=\dfrac{b}{\sin 35^{\circ}}=\dfrac{21}{\sin 73^{\circ}}[/tex]
Solve for a:
[tex]\implies \dfrac{a}{\sin 72^{\circ}}=\dfrac{21}{\sin 73^{\circ}}[/tex]
[tex]\implies a=\dfrac{21\sin 72^{\circ}}{\sin 73^{\circ}}[/tex]
[tex]\implies a=20.8847511...[/tex]
[tex]\implies a=20.9[/tex]
Solve for b:
[tex]\implies \dfrac{b}{\sin 35^{\circ}}=\dfrac{21}{\sin 73^{\circ}}[/tex]
[tex]\implies b=\dfrac{21\sin 35^{\circ}}{\sin 73^{\circ}}[/tex]
[tex]\implies b=12.5954671...[/tex]
[tex]\implies b=12.6[/tex]
PLEASE HELP!!!!!!
it’s a division equation
Answer:
x+1
DOMAIN: x [tex]\neq[/tex] -5
Step-by-step explanation:
We can factor [tex]x^{2}[/tex]+6x+5 as (x+1)(x+5) since this expands to [tex]x^{2}[/tex]+5*x+1*x+5*1, which is [tex]x^{2}[/tex]+6x+5.
Now, we have:
[tex]\frac{(x+1)(x+5)}{(x+5)}[/tex]
We can cancel out x+5, because this expression is the same as (x+1)*[tex]\frac{x+5}{x+5}[/tex], which is (x+1).
HOWEVER, the denominator cannot be equal to zero because that will make the expression undefined (you can't divide by zero). So, since x+5[tex]\neq[/tex]0, x[tex]\neq[/tex]-5.
In Algebra, an expression could have all of the four arithmetic operations, which would include multiplying, dividing, adding and subtracting.
(a) True
(b) False
Answer:
It is true that in algebraic expressions there could have all four arithmetic operations
lin's family needs to travel 325 miles to reach her grandmother's house at 377 miles what percentage have they completed
Lin's family has completed approximately 86.15% of the total Distance to her grandmother's house.
The percentage of the distance Lin's family has completed, the ratio of the distance they have traveled to the total distance.
The distance traveled is 325 miles, and the total distance to Lin's grandmother's house is 377 miles.
To calculate the percentage, we can use the following formula:
Percentage = (Distance Traveled / Total Distance) * 100
Plugging in the values, we have:
Percentage = (325 / 377) * 100
= 0.8615 * 100
= 86.15
Therefore, Lin's family has completed approximately 86.15% of the total distance to her grandmother's house.
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Use the conversion factor 1 gallon=3.785 to convert 4 liters into gallons
Using the conversion factor, we can convert the liters into the number of gallons of 1. 057 gallons.
How to convert the liters ?A conversion factor is a mathematical ratio that is used to convert one unit of measurement to another. It is a multiplier or divisor that relates two different units of measurement for the same quantity.
To convert 4 liters into gallons using the conversion factor 1 gallon = 3.785 liters, you divide the given value (in liters) by the conversion factor.
This means that the number of gallons in 4 liters would be:
= 4 liters ÷ 3. 785 liters/gallon
= 1. 057 gallons
Therefore, 4 liters is approximately equal to 1. 057 gallons.
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2. Six months after John William becomes a shareholder, Peixotto Media announces
its first shareholder meeting. Because the co-presidents will be making some very
important announcements about decisions that will be voted upon by the
shareholders, John William attends. He finds that of the 20,000 shares that have
been offered so far, the shareholders who attend the meeting own 6,000 of them.
a. The co-presidents propose adding a new member to the company's board of
directors. The shareholders are allowed to vote on this matter. What percentage of
the total vote will John William control? (2 points)
b. The company's co-presidents announce that Peixotto Media will be issuing
another offering of stock, this time in the amount of 50,000 shares. Given his
preemptive rights as an existing shareholder, how many shares of this stock is John
William entitled to purchase before the offering is made to the public? (3 points)
John William is entitled to purchase 15,000 shares before the offering is made to the public.
a. To calculate the percentage of the total vote John William will control, we need to find the proportion of shares he owns out of the total shares represented at the meeting.
John William owns 6,000 shares out of the 20,000 shares offered so far. Therefore, the proportion of shares he owns is 6,000/20,000, which simplifies to 3/10 or 0.3.
To convert this proportion to a percentage, we multiply by 100:
Percentage of the total vote = 0.3 * 100 = 30%
Thus, John William will control 30% of the total vote at the. shareholder meeting.
b. Preemptive rights allow existing shareholders to purchase new shares before they are offered to the public. To determine the number of shares John William is entitled to purchase, we need to calculate the proportion of his current ownership.
John William owns 6,000 shares out of the total shares offered so far, which is 20,000. Therefore, his ownership proportion is 6,000/20,000, or 3/10.
If Peixotto Media plans to issue 50,000 new shares, John William's entitlement would be:
Entitlement = (3/10) * 50,000 = 15,000 shares.
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pleaseeeeee helppppppp
The statements that complete the function transformations are left, up, stretch and stretch
Completing the function transformationsFrom the question, we have the following parameters that can be used in our computation:
f(x) = (x + 94)²
Also, we have
f(x) = x²
This implies that f(x) = (x + 94)² is obtained by shifting f(x) = x² left by 94 units
Next, we have
f(x) = x² + 94
This implies that f(x) = x² + 94² is obtained by shifting f(x) = x² up by 94 units
Next, we have
f(x) = 94√x from f(x) = √x
This implies that f(x) = 94√x can be obtained from vertically stretching f(x) = √x by a factor of 94
Also, the graph of f(x) = √94x can be obtained from horizontally stretching f(x) = √x by a factor of 1/94
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(HELP ME BRAINLIEST OF THE BRAINLESTS!!!!!!)
A 1. Some square tiles measure 3 1/2 inches on each side. Seven Tiles are placed in a row. How long is the row of tiles?
The row would be _____ inches long
B 2. Suppose that 10 tiles like those problem 1 were placed in a row. How long would that row of tiles be
It would be ______ inches long
The row of tiles would be 24 1/2 inches long.
B:The row of 10 tiles would be 35 inches long.
What is the length of the row?A 1. To be able to know the length of the row of tiles, one need to multiply the length of one tile by the number of tiles in the said row.
So, all tile measures 3 1/2 inches on each side, Hence the length of one tile is 3 1/2 inches.
Based on the fact that there are 7 tiles in a row, we have to multiply the length of one tile by 7:
Length of the row = 3 1/2 inches/tile x 7 tiles
= 24 1/2 inches
So, the row of tiles is 24 1/2 inches long.
B 2. Also, to find the length of a row of 10 tiles, we need to multiply the length of one tile by 10.
Note that:
Length of one tile = 3 1/2 inches
Number of tiles in the row = 10
Hence:
Length of the row = 3 1/2 inches/tile x 10 tiles
= 35 inches
So, the row of 10 tiles is 35 inches long.
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Find the current balance for Jeff’s savings account if he had a balance of $396.80, made three $15 deposits, withdrew $125, and earned $1.04 interest.
The current balance for Jeff's savings account is $317.84.
To find the current balance for Jeff's savings account, we need to calculate the net effect of the deposits, withdrawals, and interest on his initial balance.
Initial balance: $396.80
Deposits: 3 × $15 = $45
Withdrawals: $125
Interest: $1.04
Adding the deposits and interest to the initial balance:
$396.80 + $45 + $1.04 = $442.84
Then, subtracting the withdrawal:
$442.84 - $125 = $317.84
Therefore, the current balance for Jeff's savings account is $317.84.
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In Algebra, a minus symbol [-] will represent two things, an operation of subtraction and a negative number, depending on where it's placed.
(a) True
(b) False
Answer:
It is true that in algebra minus symbol represent two things
Step-by-step explanation:
And it is true that it depend on where it placed such as
8-2 this minus indicate the substraction of two terms
-3 this show that the number has negative quantity
100 Points! Algebra question. Photo attached. Please show as much work as possible. Thank you!
The equation is verified, and we have shown that E_0 = cos⁴θ is equivalent to E_0 = E_0 (1/2 + cos2θ/2)².
We have,
To verify the equation
E_θ = E_0 cos^4θ = E_0 (1/2 + cos2θ/2)², we can expand the right-hand side and simplify it.
Starting with the right-hand side of the equation:
E_0 (1/2 + cos2θ/2)²
Expanding the square using the binomial expansion:
E_0 [(1/2)² + 2(1/2)(cos2θ/2) + (cos2θ/2)²]
Simplifying:
E_0 [1/4 + cos2θ + (cos2θ)²/4]
Combining the terms:
E_0 [1/4 + 4cos²θ/4 + (cos²θ)²/4]
Simplifying further:
E_0 [1/4 + 4cos²θ/4 + cos⁴θ/4]
Combining the terms again:
E_0 [1/4 + 4cos²θ/4 + cos⁴θ/4] = E_0 [1/4 + 4cos²θ/4 + cos⁴θ/4]
Simplifying the expression yields:
E_0 cos⁴θ = E_0 cos⁴θ
Therefore,
The equation is verified, and we have shown that E_0 = cos⁴θ is equivalent to E_0 = E_0 (1/2 + cos2θ/2)².
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Answer this and I'll get you Brainlest!
The total cost of the meal if the meal costs $50 is $60
How to calculate the total cost of the mealFrom the question, we have the following parameters that can be used in our computation:
Tip percent = 20%
From the question, we can use any value as the cost of the meal
So, we can use the following
Meal cost = $50
So, we have
Tip amount = Tip percent * Meal cost
This gives
Tip amount = 20% * 50
Tip amount = 10
Next, we have
Total meal cost = Tip amount + Meal cost
This gives
Total meal cost = 10 + 50
Evaluate
Total meal cost = 60
Hence, the total cost of the meal is $60
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A student is establishing the A.A criterion for the similarity of triangles [MN and [QR. The student writes LMLN ~ ZQLR What other information can the student use to establish the AA criterion?
The other information can the student use to establish the AA criterion is Angle LMN congruent angle LQR or angle LMN congruent angle LRQ
The student can use the following information to establish the AA criterion:
Angle MLN congruent angle QLR (already given)Angle LMN congruent angle LQR or angle LMN congruent angle LRQ (either one will work)These two angles correspond to the two angles in the other triangle (LQR or LRQ) that are not congruent to the angle already known to be congruent (angle QLR).
Therefore, the AA congruent for similarity can be congruent .
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Find the area of the given circle. Round your
answer to two decimal places. (Use pi = 3.14)
In this figure, we have been given a circle. The area of the circle is 314 sq ft.
The number of square units required to fill a circle is known as its area. The region occupied inside the boundaries of a round item or 2d figure is defined as the area in general. The measurement is done in square units, with square metres (m2) being the usual unit.
There are predefined formulas for calculating area for squares, rectangles, circles, triangles, and so on.
Area of a circle = πr²
We have been given r as 10 ft
and the value of π as 3.14, so we will just put it in the formula
Area of circle = 3.14 × 10²
= 3.14 × 100
314 sq ft.
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50 Points! Multiple choice geometry question. Photo attached. Thank you!
The shape of the fence is a rhombus.
Option C is the correct answer.
We have,
To determine the shape of the fence with the given coordinates, we can analyze the properties of different quadrilaterals.
Square: All four sides are equal in length, and all angles are right angles.
Rectangle: Opposite sides are equal in length, and all angles are right angles.
Rhombus: All four sides are equal in length, but the angles are not necessarily right angles.
Trapezoid: At least one pair of opposite sides are parallel.
Let's analyze the given coordinates to determine the shape of the fence:
Side 1: Distance between (-16, 1) and (-6, 5) = √(((-6) - (-16))² + (5 - 1)²) = √(10² + 4²) = √(116)
Side 2: Distance between (-6, 5) and (4, 1) = √((4 - (-6))² + (1 - 5)²) = √(10² + (-4)²) = √(116)
Side 3: Distance between (4, 1) and (-6, -3) = √((-6 - 4)² + (-3 - 1)²) = √((-10)² + (-4)²) = √(116)
Side 4: Distance between (-6, -3) and (-16, 1) = √((-16 - (-6))² + (1 - (-3))²) = √((-10)² + 4²) = √(116)
We can see that all four sides of the fence have equal lengths, which means it is either a square, rectangle, or rhombus.
To further differentiate between these shapes, we need to consider the angles.
The angle between side 1 and side 2:
tan(angle) = (5 - 1)/(-6 - (-16)) = 4/10 = 0.4
angle ≈ 21.8°
The angle between side 2 and side 3:
tan(angle) = (1 - 5)/(4 - (-6)) = (-4)/10 = -0.4
angle ≈ -21.8°
The angle between side 3 and side 4:
tan(angle) = (-3 - 1)/(-6 - (-16)) = (-4)/10 = -0.4
angle ≈ -21.8°
The angle between side 4 and side 1:
tan(angle) = (1 - (-3))/(-16 - (-6)) = 4/10 = 0.4
angle ≈ 21.8°
All four angles are approximately 21.8° or -21.8°. This indicates that the fence is a rhombus since its sides are equal in length but its angles are not right angles.
Therefore,
The shape of the fence is a rhombus.
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2. Evaluate (5+5√3i)^7 using DeMoivre’s theorem.
Write your answer in rectangular form.
Using DeMoivre’s theorem, the answer in regular form would be (5 + 5√3i)⁷ = -5000000 + 8660254.03i
How do we Evaluate (5+5√3i)⁷ using DeMoivre’s theorem?The De Moivre's Theorem is used to simplify the computation of powers and roots of complex numbers and is used in together with polar form.
Convert the complex number to polar form. The polar form of a complex number is z = r(cos θ + isin θ),
r = |z| magnitude of z
it becomes
r = √((5)² + (5√3)²) = 10
θ = arg(z) is the argument of z.
θ = atan2(b, a) = atan2(5√3, 5) = π/3
(5 + 5√3i) = 10 × (cos π/3 + i sin π/3)
De Moivre's theorem to raise the complex number to the 7th power
(5 + 5√3i)⁷
= 10⁷× (cos 7π/3 + i sin 7π/3)
= 10⁷ × (cos 2π/3 + i sin 2π/3)
Convert this back to rectangular form:
Real part = r cos θ = 10⁷× cos (2π/3) = -5000000
Imaginary part = r sin θ = 10⁷ × sin (2π/3) = 5000000√3 = 8660254.03i
∴ (5 + 5√3i)⁷ = -5000000 + 8660254.03i
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Answer:10^7 (1/2 - √3/2 i)
Step-by-step explanation:
To use DeMoivre's theorem, we first need to write the number in polar form. Let's find the magnitude and argument of the number:
Magnitude:
|5 + 5√(3i)| = √(5^2 + (5√3)^2) = √(25 + 75) = √100 = 10
Argument:
arg(5 + 5√(3i)) = tan^(-1)(√3) = π/3
So the number can be written in polar form as:
5 + 5√(3i) = 10(cos(π/3) + i sin(π/3))
Now we can use DeMoivre's theorem:
(5 + 5√(3i))^7 = 10^7 (cos(7π/3) + i sin(7π/3))
To simplify, we need to find the cosine and sine of 7π/3:
cos(7π/3) = cos(π/3) = 1/2
sin(7π/3) = -sin(π/3) = -√3/2
Explanation:
So the final answer in rectangular form is:
10^7 (1/2 - √3/2 i)
find the center and radius by completing the square x2+6x+y2-16y-8=0
Which type of retirement account is an investment option for ANY young person?
a. traditional IRA
b. pension
c. 401(k)
d. social security
The type of retirement account is an investment option for ANY young person is 401(k). option C
What is a 401(k) retirement account?A 401(k) retirement account is described as an option for investing money, created and arrangement for youths with access to any lucrative job that pays a reasonable amount of money.
This retirement account creates room for individuals to contribute a portion of their tax income towards retirement savings, and also contributes to grow tax-deferred until withdrawal.
However, pensions are employer-funded retirement plans and may be unavailable to all young individuals
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Giselle works as a carpenter and as a blacksmith.
She earns
$
20
$20dollar sign, 20 per hour as a carpenter and
$
25
$25dollar sign, 25 per hour as a blacksmith. Last week, Giselle worked both jobs for a total of
30
3030 hours, and earned a total of
$
690
$690dollar sign, 690.
How long did Giselle work as a carpenter last week, and how long did she work as a blacksmith?
Giselle worked as a carpenter for
hours and as a blacksmith for
hours last week.
Answer:
Giselle worked as a carpenter for 12 hours and as a blacksmith for 18 hours last week
Step-by-step explanation:
Let's assume that Giselle worked as a carpenter for x hours and as a blacksmith for y hours last week.
We know that she earns $20 per hour as a carpenter and $25 per hour as a blacksmith, and that she worked a total of 30 hours, so we can write two equations based on this information:
x + y = 30 (total hours worked)
20x + 25y = 690 (total earnings)
To solve for x and y, we can use substitution or elimination. Let's use substitution.
From the first equation, we can solve for x in terms of y:
x = 30 - y
Substitute this expression for x in the second equation:
20(30 - y) + 25y = 690
Simplify and solve for y:
600 - 20y + 25y = 690
5y = 90
y = 18
So Giselle worked as a blacksmith for 18 hours last week.
To find how long she worked as a carpenter, we can substitute y = 18 into the first equation and solve for x:
x + 18 = 30
x = 12
Therefore, Giselle worked as a carpenter for 12 hours last week.
evaluate 6a +4b -3c when a=4 , b=7 , and c=-2 (a) 26 (b) 16 (c) 46) (d) 58
Answer:
(d) 58
Step-by-step explanation:
Let's start with the equation:
6a+4b-3c
Now, since we got the values of a, b, and c, we can just plug them into the equation:
6(4)+4(7)-3(-2)
24+28+6=58
So the answer is d) 58
Which data set has a variation, or mean absolute deviation, similar to the data
set in the given dot plot?
50 51 52 53 54 55 56 57 58 59 60 61
Ο Α.
OB.
O C.
D.
4612
15 17 19
0123
8 9 1011
55 57 59 61 63 65
Option C (15 17 19) has a similar mean absolute deviation to the data set in the given dot plot.
The informational collection in the given speck plot has a mean of roughly 55 and a moderately little scope of values, with most of the information bunched around the mean. An informational index with a comparable mean outright deviation would have a comparative degree of variety in the information values.
Out of the given choices, the informational index that has a variety like the given informational collection is choice C: 15 17 19. This informational collection has a mean of roughly 17 and a little scope of values, with most of the information bunched around the mean.
The mean outright deviation of this informational collection is around 1.63, which is like the mean outright deviation of the given informational collection. Choice A (50 51 52 53 54 55 56 57 58 59 60 61) has a bigger scope of values and a bigger mean outright deviation contrasted with the given informational collection.
Choice B (4612) and Choice D (8 9 1011 and 55 57 59 61 63 65) have altogether unique mean qualities and scopes of values, and consequently their mean outright deviations are not quite the same as the given informational collection.
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Graph by completing the square x2+8x+y2-10y-32=0
The circle equation x² + 8x + y² -10y - 32 = 0 can be graphed using (x + 4)² + (y - 5)² = 73
Graphing the circle equation by completing the squareFrom the question, we have the following parameters that can be used in our computation:
x² + 8x + y² -10y - 32 = 0
Add 32 to both sides of the equation
This gives
x² + 8x + y² -10y = 32
Group the terms in two's
So, we have
(x² + 8x) + (y² -10y) = 32
When we complete the square on each group, we have
(x + 4)² + (y - 5)² = 16 + 25 + 32
Evaluate the like terms
(x + 4)² + (y - 5)² = 73
Hence, the circle equation can be graphed using (x + 4)² + (y - 5)² = 73
See attachment for the graph
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Consider the experiment of selecting a playing card from a deck of 52 playing cards. Each card corresponds to a sample point with a
1
52
probability.
(a)
List the sample points in the event an ace is selected.
S = {x | x is a card from the deck that is not a club a spade or a diamond}
S = {x | x is a card from the deck but not king, jack, or queen}
S = {1 of clubs, 1 of diamonds, 1 of hearts, 1 of spades}
S = {ace of clubs, ace of diamonds, ace of hearts, ace of spades}
S = {king of clubs, king of diamonds, king of hearts, king of spades}
If Each card corresponds to a sample point with a 1/52 probability then the sample points in the event an ace is selected is S = {ace of clubs, ace of diamonds, ace of hearts, ace of spades}
In a standard deck of 52 playing cards, there are four aces.
Each ace corresponds to a different suit: clubs, diamonds, hearts, and spades.
So, when we talk about the event of selecting an ace, we are referring to the act of choosing one of these four specific cards from the deck.
The sample points in the event "an ace is selected" are the individual cards that match this criteria.
Therefore, the sample points in this event are:
S = {ace of clubs, ace of diamonds, ace of hearts, ace of spades}
These four cards are the possible outcomes when we randomly select a single card from the deck, focusing only on the event of selecting an ace.
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The bottom part says how many student tickets where brought? Can anyone pls help me PLSS
255 number of adults and 355 number of students bought ticket.
Here, we have,
Let the number of adults bought tickets are x and the number of students that bought tickets is (x + 100).
Since it is given that 100 more students brought tickets than adults.
Now, each adult ticket costs $5 and each student's ticket costs $3.5 and the total collected value of tickets is $2517.5.
So, 5x + 3.5(x + 100) = 2517.5
⇒ 8.5x + 350 = 2517.5
⇒ 8.5x = 2167.5
⇒ x = 255
So, 255 number of adults and (255 + 100) = 355 number of students bought ticket. (Answer)
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Help me with this and expiation pls
The radius of the cylinder, considering it's lateral surface area, is given as follows:
r = 31.5 cm.
How to obtain the lateral surface area of a cylinder?
The lateral surface area of a cylinder of radius r and height h is given by the equation presented as follows:
Sl = 2πrh.
The parameters for this problem are given as follows:
Sl = 693π, h = 11.
Hence the radius of the cylinder, considering it's lateral surface area, is given as follows:
r = Sl/2πh
r = 693/22 (simplifying π in the numerator and denominator)
r = 31.5 cm.
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BTS-2 has coordinates (-8,6) and the edge connecting vertices P and Q has the equation y = 4.
(b) Write down the coordinates of BTS-4.
a. Jason will receive the strongest signal from BTS-4 because he is located in the Voronoi cell of BTS-4.
b. The coordinates of BTS-4 are (−8,4).
How to explain the informationa. Jason will receive the strongest signal from BTS-4 because he is located in the Voronoi cell of BTS-4. A Voronoi cell is a region of space that is closer to a given point than any other point. In this case, the given point is BTS-4. The Voronoi diagram is a partitioning of the plane into Voronoi cells, one for each point.
b. The edge connecting vertices P and Q has the equation y=4, which means that it is a horizontal line that intersects the y-axis at 4. The coordinates of BTS-2 are (−8,6), which means that it is located 8 units to the left of the origin and 6 units above the origin. Therefore, BTS-4 must be located 8 units to the left of the origin and 4 units above the origin, which gives it the coordinates (−8,4).
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