Answers
let's bear in mind that complex roots never come alone, their conjugate sister is always with her, so if we have the complex root of "i" or namely "0 + i", her conjugate is also coming along, or "0 - i", so we really have four roots, so
[tex]\begin{cases} x = 0+i &\implies x -i=0\\ x = 0-i &\implies x +i=0\\ x = \sqrt{2} &\implies x -\sqrt{2}=0\\ x = 3 &\implies x -3=0\\ \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{original~polynomial}{a ( x -i )( x +i )( x -\sqrt{2} )( x -3 ) = \stackrel{0}{y}}\hspace{5em}\stackrel{\textit{we are assuming that}}{a=1} \\\\\\ 1( x -i )( x +i )( x -\sqrt{2} )( x -3 ) = y\implies ( x -i )( x +i )( x -\sqrt{2} )( x -3 ) = y \\\\[-0.35em] ~\dotfill[/tex]
[tex]\stackrel{ \textit{difference of squares} }{( x -i )( x +i )}\implies x^2 - i^2\implies x^2-(-1)\implies x^2+1 \\\\[-0.35em] ~\dotfill\\\\ (x^2+1)( x -\sqrt{2} )( x -3 )\implies (x^2+1)(x^2-3x-x\sqrt{2}+3\sqrt{2}) \\\\\\ (x^2+1)[x^2-x(3+\sqrt{2})+3\sqrt{2}] \\\\\\ x^4-x^3(3+\sqrt{2})+3x^2\sqrt{2}+x^2-x(3+\sqrt{2})+3\sqrt{2} \\\\\\ \boxed{x^4-x^3(3+\sqrt{2})+x^2(3\sqrt{2}+1)-x(3+\sqrt{2})+3\sqrt{2}~~ = ~~y}[/tex]
Related Questions
92920625 rounded to the nearest million
Answers
Answer:
93 million
Step-by-step explanation:
93 Million
93 millllll
Please help meee don’t understand
Answers
so the total number of cds sold=91×3= 273.
273 is the answer
Jane made $143 for 11 hours of work. At the same rate, how many hours would she have to work to make $169 ?
Answers
13
First, we divide to figure out how much Jane makes an hour
143 ÷ 11 = 13
Second, we multiply multiply by 13 until we get 169
13 × 13 = 169
So Jane will make 169 dollars in 13 hours
Answer:
13 Hours
Step-by-step explanation:
$143 ÷ 11hrs = $13/hr that Jane is paid, so to find how many hours she will need to work to make $169
$169 ÷ $13 = 13 hours
The quadrilateral on the graph below is rotated about the point (0, 0). What are the new coordinates of Point B and Point C after a 90 degree clockwise rotation?
Answers
A 90 degree clockwise rotation about the point (0,0), the new coordinates of Point B are (3, -1) and the new coordinates of Point C are (1, -2).
How do you check if a quadrilateral is a rectangle on a graph?A quadrilateral can be proven to be a rectangle in a number different ways. Here are the three simplest methods: 1. Establish that all angles are 90 degrees; 2. Establish that two opposed angles are 90 degrees; and 3. Establish that the diagonals are equally long and intersect one another.
The following transformation matrix can be used to rotate a point (x,y) 90 degrees clockwise with respect to the origin (0,0):
|0 1|\s|-1 0|
This transformation matrix can be applied to each point to determine its new coordinates upon rotation.
Point B is the first point, and its initial coordinates are (-1, 3). The transformation matrix in use:
|0 1| |-1| |3|
|-1 0| x |3| = |-1|
After rotating Point B 90 degrees clockwise, the new coordinates are: (3, -1).
The transformation matrix is then applied to Point C, whose initial coordinates are (2, 1):
|0 1| |2| |1|
|-1 0| x |1| = |-2|
After rotating in a clockwise direction by 90 degrees, Point C's new coordinates are (1, -2).
As a result, after rotating 90 degrees in a clockwise direction around Point 0, Point B's new coordinates are (3, -1), while Point C's new coordinates are (1, -2).
To know more about Quadrilateral Graph visit:
https://brainly.com/question/12518144
#SPJ1
Sierra left $4.50 as a tip for a waiter. This was 18% of the bill before the tip. How much was her total bill before the tip?
$
Answers
Answer$81
Step-by-step explanation:
In the diagram, point B is a point of tangency. Find
the radius r of OC.
Answers
The radius r for the circle C is equal to 39 using the Pythagoras rule for the right triangle ABC
How to evaluate for the radius using Pythagoras ruleSince the line AB is tangent to the circle at point B, then the triangle ABC is a right triangle and the Pythagoras rule can be applied as follows:
(50 + r)² = r² + 80²
r² = (50 + r)² - 80²
r² = (50 + r - 80)(50 + r + 80) {difference of two square}
r² = (r - 30)(r + 130)
r² = r² + 130r - 30r - 3900 {expansion of brackets}
r² - r² + 130r - 30r = 3900 {collect like terms}
100r = 3900
r = 3900/100 {divide through by 100}
r = 39
Therefore, the radius r for the circle C is equal to 39 using the Pythagoras rule for the right triangle ABC.
Know more about Pythagoras here:https://brainly.com/question/343682
#SPJ1
2. Suppose a consumer has $30 available to be divided between commodities A and B and the unit price of B is fixed at $3. What will be his demand equation for A if his utility function is U = 4XgXb?
Answers
The demand equation for commodity A is Xa = (4/10)Xb.
What is demand equation?A demand equation is a mathematical representation of the relationship between the quantity of a good or service that consumers are willing to buy and the various factors that influence that demand, such as price, income, and preferences.
In the given question,
To find the consumer's demand equation for A, we need to use the utility maximization rule, which states that a consumer will allocate their budget in such a way as to maximize their total utility subject to their budget constraint.
Let Xa be the quantity of commodity A and Xb be the quantity of commodity B. We know that the consumer has $30 to spend, so the budget constraint is:
3Xb + pXa = 30
where p is the price of commodity A. We also know the utility function:
U = 4XgXb
To maximize U subject to the budget constraint, we can use the Lagrangian method:
L = 4XgXb + λ(30 - 3Xb - pXa)
where λ is the Lagrange multiplier.
To find the demand equation for A, we need to take the partial derivative of L with respect to Xa and set it equal to zero:
∂L/∂Xa = -λp = 0
This gives us λ = 0, which we can substitute back into the Lagrangian equation to get:
L = 4XgXb + 0(30 - 3Xb - pXa)
L = 4XgXb
To find the demand equation for A, we need to take the partial derivative of L with respect to p and solve for Xa:
∂L/∂p = -4XgXb/Xa = -30/3
Xa = (4/10)Xb
So the demand equation for commodity A is Xa = (4/10)Xb.
To know more about demand equation, visit:
https://brainly.com/question/13245362
#SPJ1
The demand equation for commodity A is Xa = (4/10)Xb.
What is demand equation?
A demand equation is a mathematical formula that expresses the relationship between the quantity of a good or service that consumers are willing and able to purchase and various factors that affect that quantity, such as price, income, and the prices of other goods.
According to given information:To find the consumer's demand equation for A, we need to use the utility maximization rule, which states that a consumer will allocate their budget in such a way as to maximize their total utility subject to their budget constraint.
Let Xa be the quantity of commodity A and Xb be the quantity of commodity B. We know that the consumer has $30 to spend, so the budget constraint is:
3Xb + pXa = 30
where p is the price of commodity A. We also know the utility function:
U = 4XgXb
To maximize U subject to the budget constraint, we can use the Lagrangian method:
L = 4XgXb + λ(30 - 3Xb - pXa)
where λ is the Lagrange multiplier.
To find the demand equation for A, we need to take the partial derivative of L with respect to Xa and set it equal to zero:
∂L/∂Xa = -λp = 0
This gives us λ = 0, which we can substitute back into the Lagrangian equation to get:
L = 4XgXb + 0(30 - 3Xb - pXa)
L = 4XgXb
To find the demand equation for A, we need to take the partial derivative of L with respect to p and solve for Xa:
∂L/∂p = -4XgXb/Xa = -30/3
Xa = (4/10)Xb
So the demand equation for commodity A is Xa = (4/10)Xb.
To know more about demand equation, visit:
brainly.com/question/13245362
#SPJ1
Select the correct answer. Which function has an average rate of change of -4 over the interval [-2,2]?
A. x | -2 | -1 | 0 | 1 | 2
m(x) | -12 | -5 | -4 | -3 | 4
B.
C.
D.
Answers
Answer:
The correct answer is option B.
To find the function with an average rate of change of -4 over the interval [-2,2], we need to calculate the slope of the function between the two points -2 and 2.
Average rate of change = (f(2) - f(-2))/(2 - (-2)) = (-4)
Option B has the function qx with values {-4, 0, 0, -4, -12} at x values {-2, -1, 0, 1, 2}. The average rate of change of this function over the interval [-2,2] is indeed -4.
[tex]\sqrt{2x} + 3 = 8[/tex]2 x + 3 = 8
Answers
Answer: x= 6241/2
Step-by-step explanation:
You deposit $5,000.00 in an account earning 8% interest compounded annually. How much will you have in the account in 5 years?
Answers
Answer:
Answer:
50313.28
Step-by-step explanation:
A = total amount
P = principal or amount of money deposited,
r = annual interest rate
n = number of times compounded per year
t = time in years
For the formula:
A=P(1+r/n)n⋅t
P=$5000 , r=8% , n=1 and t=30 years
Solution:
A= 5000(1+0.08/1) to the power 1.30
= 5000*1.08 to the power 30
= 5000*10.062657
= 50313.28
20 points!! please help!!
Answers
To find the area of the total figure, we need to first find the areas of the rectangle and triangle, and then add them together.Therefore, the area of the total figure is 200 square feet.
What is area?Area is the measurement of the size of a two-dimensional surface enclosed by a closed figure
Area of rectangle = length x width
= 20 ft x 8 ft
= 160 sq. ft
Area of triangle = 1/2 xbase xheight
= 1/2 x 8 ft x 10 ft
= 40 sq. ft
To find the base of the triangle, we can use the Pythagorean theorem, which states that the square of the hypotenuse (slope) of a right triangle is equal to the sum of the squares of its two sides. In this case, the hypotenuse is 12 ft, one of the other sides is the height of the triangle (10 ft), and the other side is the base of the triangle (b).
Using the Pythagorean theorem, we have:
12² = 10² + b²
144 = 100 + b²
44 = b²
b = √44
b ≈ 6.63 ft
Now that we know the base of the triangle, we can find the area of the total figure by adding the area of the rectangle and the area of the triangle:
Area of total figure = area of rectangle + area of triangle
= 160 sq. ft + 40 sq. ft
= 200 sq. ft
To know more about Pythagorean theorem visit:
https://brainly.com/question/15169173
#SPJ1
Solve 6x+14x+5=5(4x+1) and write a word problem to the equation or any relevant forms of it represents.
Answers
After solving the given expression, the value of x is 5.
What exactly are expressions?
An expression in mathematics is a set of numbers, variables, and mathematical operations (such as addition, subtraction, multiplication, and division) that may be evaluated to generate a value. Expressions can be simple or complicated, with one or more variables involved.
Now,
To solve the equation 6x+14x+5=5(4x+1), we first need to simplify both sides of the equation using the distributive property of multiplication:
6x + 14x + 5 = 20x + 5
Now we can simplify further by subtracting 20x and 5 from both sides of the equation:
6x + 14x - 20x = 0 - 5
Simplifying again:
x = -5
Finally, we can solve for x by multiplying both sides by -1:
x = 5
Therefore, the solution to the equation 6x+14x+5=5(4x+1) is x=5.
Word problem:
A clothing store sells two types of shirts: T-shirts and polo shirts. The store makes a profit of $6 on each T-shirt sold and a profit of $14 on each polo shirt sold. Last week, the store sold a total of 5 shirts and made a total profit of $25. If x represents the number of T-shirts sold, write an equation to represent the situation.
Solution:
Let x be the number of T-shirts sold, then the number of polo shirts sold is 5 - x (since a total of 5 shirts were sold). The total profit from selling x T-shirts and (5-x) polo shirts can be calculated as:
Profit = (profit per T-shirt x number of T-shirts) + (profit per polo shirt x number of polo shirts)
Profit = (6x) + (14(5-x))
Profit = 6x + 70 - 14x
Profit = -8x + 70
Since the total profit is given as $25, we can write the equation:
-8x + 70 = 25
Simplifying:
-8x = -45
x = 5.625
Since we can't sell a fraction of a shirt, we need to round down to the nearest integer. Therefore, the store sold 5 T-shirts.
To know more about expressions visit the link
brainly.com/question/13947055
#SPJ1
pls help me with this
Answers
Therefore , the solution of the given problem of unitary method comes out to be rectangle's size is 7/12 square inches.
An unitary method is what ?The objective can be accomplished by using what was variable previously clearly discovered, by utilizing this universal convenience, or by incorporating all essential components from previous flexible study that used a specific strategy. If the anticipated claim outcome actually occurs, it will be feasible to get in touch with the entity once more; if it isn't, both crucial systems will undoubtedly miss the statement.
Here,
=> A = L x W,
where A is the area, L is the length, and W is the breadth, is the formula for calculating the area of a rectangle.
Inputting the numbers provided yields:
=> A = (7/4) x (1/3)
These fractions can be made simpler by eliminating any shared variables in the numerator and denominator before being multiplied. Since 7 and 3 are both prime integers in this instance, there are no shared factors to cancel.
The new numerator and denominator can then be obtained by multiplying the numerators and denominators, respectively. Thus, we get:
=> A = (7 x 1) / (4 x 3)
When we multiply the numerator by the remainder, we obtain:
=> A = 7/12
The rectangle's size is 7/12 square inches as a result.
To know more about unitary method visit:
https://brainly.com/question/28276953
#SPJ1
b. Rewrite 4 x 63 as the product of a unit fraction and a whole number.
Solve.
Answers
Rewriting 4 x 3/6 as the product of a unit fraction and a whole number is: 12 * 1/6
How to multiply fractions?The parameters are given as:
Number - 4
Fraction - 3/6
The following steps can be used to determine the product as the product of a whole number and a unit fraction:
Step 1 - Remember the whole number are those numbers that involve all positive integers and zero.
Step 2 - Also remember that the unit fraction is nothing but a fraction whose numerator is 1.
Step 3 - Write the given expression.
4 * 3/6
Step 4 - Convert the given fraction into a unit fraction by multiplying 4 by 3 in the above expression.
4 * 3 * 1/6
Step 5 - Simplify the above expression.
12 * 1/6
Read more about Fraction Multiplication at: https://brainly.com/question/415490
#SPJ1
how far is sam from the top of a temple?
Answers
The distance between Sam from the top of the temple is 56. 6 feet
How to determine the distanceTo determine the distance, we need to know that trigonometric identities are mathematical identities that is mostly used to prove that all the values of the functions of trigonometry are true.
The types of trigonometric identities are;
tangentsinecosinecotangentcosecantsecantFrom the information given, we can deduce that;
The angle of elevation, θ = 62 degrees
The opposite side of the angle that is the height of the temple is 50 feet
The distance is the hypotenuse side
Then, using sine identity, we have;
sin 62 = 50/d
d = 50/0. 8829
d = 56. 6 feet
Learn about trigonometric identities at: https://brainly.com/question/22591162
#SPJ1
Can someone help me please
$4000 are invested in a bank account at an interest rate of 10 percent per year.
Find the amount in the bank after 7 years if interest is compounded annually.
--------------
Find the amount in the bank after 7 years if interest is compounded quarterly.
---------------
Find the amount in the bank after 7 years if interest is compounded monthly.
---------------
Finally, find the amount in the bank after 7 years if interest is compounded continuously.
---------------
Answers
Answer:
To find the amount in the bank after 7 years, we can use the formula:
A = P(1 + r/n)^(nt)
where:
A = the amount in the bank after 7 years
P = the principal amount (initial investment)
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the number of years
For the given problem:
P = $4000
r = 10% = 0.1
t = 7 years
a) Compounded Annually:
n = 1
A = 4000(1 + 0.1/1)^(1*7) = $7449.36
b) Compounded Quarterly:
n = 4
A = 4000(1 + 0.1/4)^(4*7) = $7650.13
c) Compounded Monthly:
n = 12
A = 4000(1 + 0.1/12)^(12*7) = $7727.27
d) Compounded Continuously:
n → ∞ (as n approaches infinity)
A = Pe^(rt) = 4000e^(0.1*7) = $8193.85
Therefore, the amount in the bank after 7 years increases as the compounding frequency increases. If interest is compounded continuously, the amount in the bank will be the highest.
At Spirit Night slices of pizza cost $2 and pretzels cost $1. The school store sold 150 items and made a total of $250. Write a systems of
equations to represent the situation where x represents the number of slices of pizza sold and y represnts the number o pretzels
Answers
Answer:
[tex]x + y = 150[/tex]
[tex]2x + y = 250[/tex]
A rectangular box has a length that is 4 feet longer than its width, w.
Write an algebraic expression, in simpliest form, to find the perimeter of the box.
Answers
Step-by-step explanation:
The length of the rectangle is 4 feet longer than its width w, which means the length is w + 4
The perimeter of a rectangle is the sum of the lengths of all four sides which can be expressed as:
Perimeter = 2(length + width)
Substituting w + 4 for length and w for width, we get:
Permiter = 2(w + 4 + w)
Simplifying this expression, we get:
Perimeter = 2(2w + 4)
Perimeter = 4w + 8
Therefore, the algebraic expression to find the perimeter of the rectangular box is 4w + 8
Florida Pick 3 In the Florida Pick 3 lottery, you can place a “straight” bet of $1 by selecting the exact order of three digits between 0 and 9 inclusive (with repetition allowed), so the probability of winning is 1/1000. If the same three numbers are drawn in the same order, you collect $500, so your net profit is $499.
Answers
The probability of losing is [tex]\frac{999}{1000}[/tex] (since there are 999 ways to lose and 1 way to win), so the odds against winning are [tex](\frac{\frac{999}{1000} }{\frac{1}{1000} })[/tex] = 999:1.
To understand the concept of odds against winning, we can use the analogy of flipping a coin. There are only two outcomes that can occur when we flip a fair coin: heads or tails. The probability of getting heads is 1/2 and the probability of getting tails is also [tex]\frac{1}{2}[/tex] . The odds against getting heads are the ratio of the probability of getting tails to the probability of getting heads, which is 1:1 or even odds. In the case of the Florida Pick 3 lottery, there are 1000 possible outcomes, but only one is a winning outcome. Therefore, the actual odds against winning the Florida Pick 3 lottery with a straight bet are 999 to 1.
To learn more about probability follow the link:
https://brainly.com/question/30034780
#SPJ1
The complete question is:
In the Florida Pick 3 lottery, you can place a “straight” bet of $1 by selecting the exact order of three digits between 0 and 9 inclusive (with repetition allowed), so the probability of winning is 1/1000. If the same three numbers are drawn in the same order, you collect $500, so your net profit is $499.
Find the actual odds against winning
the diameter of a spherical balloon is 21.6 centimeters
Answers
The parameter for determining the diameter of an object depends on the specific object being measured. Here are some examples of parameters that can be used to determine diameter the answer is 5276.7 cm3. Thus, option D is correct.
What are the parameter for determining the diameter?The formula for the volume of a sphere is [tex]V = (4/3)πr^3[/tex] , where r is the radius of the sphere.
Since we are given the diameter of the sphere, we can find the radius by dividing the diameter by 2:
[tex]r = 21.6 cm / 2 = 10.8 cm[/tex]
Substituting this value into the formula, we get:
[tex]V = (4/3)\pi(10.8)^3[/tex]
[tex]= 4.18879 \times (10.8)^3[/tex]
[tex]= 5276.794 cm^3[/tex]
Rounding to the nearest tenth, we get:
[tex]V \approx 5276.8 cm^3[/tex]
Therefore, the answer is D) 5276.7 cm3.
Learn more about diameter here:
https://brainly.com/question/5501950
#SPJ9
The given question is incomplete. the complete question is given below.
The diameter of a sphere is 21.6 cm. What is the sphere's volume? Round to the nearest tenth, if necessary. A) 693.2 cm3 B) 1453.8 cm3 C) 1868.5 cm3 D) 5276.7 cm3
Watch help video
Given circle E with diameter CD and radius EA. AB is tangent to E at A. If
EC = 3 and EA = 3, solve for AC. Round your answer to the nearest tenth if
necessary. If the answer cannot be determined, click "Cannot be determined."
C
A
B
Answers
The circle E with diameter CD and radius EA having the length of AC is approximately 4.2 units.
What is Pythagoras' Theorem?
In a right-angled triangle, the square of the hypotenuse side equals the sum of the squares of the other two sides.
Since EA is a radius of circle E, and AB is tangent to E at A, we know that AB is perpendicular to EA. Thus, triangle EAB is a right triangle.
Let x be the length of AC. Then, by the Pythagorean Theorem in triangle EAC, we have:
[tex]AC^{2} = EA^{2} +EC^{2}[/tex]
[tex]AC^{2} = 3^{2} + 3^{2}[/tex]
[tex]AC^{2} = 18[/tex]
AC ≈ 4.2 (rounded to the nearest tenth)
Therefore, the length of AC is approximately 4.2 units.
Learn more about the Pythagorean theorem here:
https://brainly.com/question/14930619
#SPJ1
Identify the nonlinear equation.
Responses
A y = 3x - 7y = 3 x - 7
B y = xy = x
C y = 3y = 3
D y = x2
PLS HELP
Answers
Answer:y=0.5
Step-by-step explanation:Comme tu peux le voir, y est égal à au tiers de 3, se qui équivaut à 1. Si 2x=y, cela signifie que x=y/2, soit 0,5
[tex]65y - 147y[/tex]
Math problem.
I need help.
Answers
Answer: 82y
Step-by-step explanation:
147y - 65y = 82y
Just perform simple subtraction
Write the next three terms of the geometric sequence where a_1 = - 8 and r = -2
a_1 = -8
a_2 =
a_3 =
a_4 =
Answers
Answer:
a_2 = 16
a_3 = -32
a_4 = 64
Step-by-step explanation:
Multiply each term by r to get the next term.
a_1 = -8
a_2 = -8 × (-2) = 16
a_3 = 16 × (-2) = -32
a_4 = -32 × (-2) = 64
The graph shows the velocity versus time for 4 different cars on a race track. If all four cars have the same mass, which one experiences the largest net force?
Answers
Answer:
Step-by-step explanation: 1
What is the fourth term of the sequence:
Write the number in the blank only.
a_1 = 5
a_n = 2a_n-1 + 3
Answers
The fourth term of the sequence with the definition of functions a₁ = 5 and aₙ = 2aₙ₋₁ + 3 is 61.
Calculating the fourth term of the sequenceGiven the following definition of functions
a₁ = 5
aₙ = 2aₙ₋₁ + 3
To find the fourth term of the sequence defined by a₁ = 5aₙ = 2aₙ₋₁ + 3, we can use the recursive formula to generate each term one by one:
a₂ = 2a₁ + 3 = 2(5) + 3 = 13
a₃ = 2a₂ + 3 = 2(13) + 3 = 29
a₄ = 2a₃ + 3 = 2(29) + 3 = 61
Therefore, the fourth term of the sequence is 61.
Read more about sequence at
https://brainly.com/question/29431864
#SPJ1
En un canal se necesitan diariamente 36 kg de maiz para alimentar a 480 gallinas ¿Cuantos kg se necesitan ahora si se vendieron 120 gallinas?
Resolver con reglla de 3
Answers
Answer:
Step-by-step explanation: it is -2x + 430298= -9n79000.
Scientists are making an aerial study of a volcano. Their helicopter is circling at a 4 km radius around the volcano's crater, and one of the scientists notices a new vent that is 45° east of due south from the crater. What is the position of the new vent relative to the crater?
Answers
Answer:
2√2 km south and 2√2 km west of the volcano's crater.
Step-by-step explanation:
If the scientist is at the center of the circle with the volcano's crater, then the new vent is located 45° east of due south, or 135° counterclockwise from due north.
To describe the position of the new vent relative to the crater, we can use the bearing or direction angle, which is the angle between the north direction and the line connecting the crater and the new vent, measured counterclockwise.
To find the bearing, we can draw a right triangle with the hypotenuse equal to the distance from the center of the circle to the new vent, which is also the radius of the circle, or 4 km. The opposite side of the triangle is the north-south component of the line connecting the crater and the new vent, which is equal to the radius times the sine of the angle between the line and due south. The adjacent side is the east-west component of the line, which is equal to the radius times the cosine of the angle.
Using trigonometric functions, we can calculate:
Opposite side = 4 km x sin(135°) = 4 km x (-√2/2) = -2√2 km (southward direction)
Adjacent side = 4 km x cos(135°) = 4 km x (-√2/2) = -2√2 km (westward direction)
Therefore, the new vent is located 2√2 km south and 2√2 km west of the volcano's crater. Its position relative to the crater can be described as "southwest by south."
I need to find f(g) f(x) please
Answers
The answers f(g(x)) = x and g(f(x)) = x tell us that the two functions f(x) and g(x) are inverses of each other.
What is inverse?The inverse of a function is a second function that "undoes" the effect of the first function. More specifically, if f is a function that maps elements from a set A to a set B, then its inverse function, denoted as f^(-1), maps elements from B back to A.
According to question:(a) To find f(g(x)), we need to substitute the expression for g(x) into f(x):
f(g(x)) = g(x) / (6 + g(x))
Substituting the expression for g(x) yields:
f(g(x)) = (6x / (1 - x)) / (6 + (6x / (1 - x)))
This equation can be made simpler by first locating a common denominator:
f(g(x)) = (6x / (1 - x)) / ((6(1 - x) / (1 - x)) + (6x / (1 - x)))
f(g(x)) = (6x / (1 - x)) / ((6 - 6x + 6x) / (1 - x))
f(g(x)) = (6x / (1 - x)) / (6 / (1 - x))
f(g(x)) = 6x / 6
f(g(x)) = x
To find g(f(x)), we need to substitute the expression for f(x) into g(x):
g(f(x)) = 6f(x) / (1 - f(x))
Substituting the expression for f(x) yields:
g(f(x)) = 6(x / (6 + x)) / (1 - (x / (6 + x)))
To simplify this expression, we can first find a common denominator:
g(f(x)) = 6(x / (6 + x)) / (((6 + x) / (6 + x)) - (x / (6 + x)))
g(f(x)) = 6(x / (6 + x)) / ((6 + x - x) / (6 + x))
g(f(x)) = 6(x / (6 + x)) / (6 / (6 + x))
g(f(x)) = x
(b) The answers f(g(x)) = x and g(f(x)) = x tell us that the two functions f(x) and g(x) are inverses of each other. This means that when we apply one function and then the other, we get back to the original input value. Specifically, if we apply f(x) to x and then apply g(x) to the result, we get x back, and if we apply g(x) to x and then apply f(x) to the result, we also get x back. This is a useful property when analyzing functions and their relationships.
Learn more about inverse visit:
https://brainly.com/question/24584973
#SPJ1
In ΔLMN, n = 27 inches, l = 70 inches and ∠M=149°. Find ∠N, to the nearest degree.
Answers
Using trigonometric functions, we can find that the value of the angle N is 3°.
What are trigonometric functions?The six fundamental trigonometric operations make up trigonometry. Trigonometric ratios are useful for describing these methods. The sine, cosine, secant, co-secant, tangent, and co-tangent functions are the six fundamental trigonometric functions. On the ratio of a right-angled triangle's sides, trigonometric identities and functions are founded. Trigonometric formulas are used to determine the sine, cosine, tangent, secant, and cotangent values for the perpendicular side, hypotenuse, and base of a right triangle.
Here, using the cosine theorem:
CosM = n² + l² - m²/2nl
⇒ Cos 149° = 27² + 70² - m²/2 × 27 × 70
⇒ -0.981 = 729 + 4900 - m²/3780
⇒ 5629 - m² = -3708
⇒ m² = 9337.
Now Cos N = m² + l² - n²/2ml
= (9337 + 4900 - 729) / (2 × √9337 × 70)
= 0.9985
Cos N = 0.9985
Putting [tex]Cos^{-1}[/tex] on both sides:
[tex]Cos^{-1}[/tex] Cos N = [tex]Cos^{-1}[/tex] 0.9985
⇒ N ≈ 3°
To know more about trigonometric functions, visit:
https://brainly.com/question/6904750
#SPJ1
The complete question is:
In ΔLMN, n = 27 inches, l = 70 inches and ∠M=149°. Find ∠N, to the nearest degree.