An approximate solution to an equation is found using this iterative process.
(x₂)³-1
and x₁ = -1
4
Xn+1 =
4/34
a) (i) Work out the value of x₂
(ii) Work out the value of x3
b) Work out the solution to 6 decimal places.
Answer:
a) i) -0.5
ii) -0.28125
b) -0.254102 (6 d.p.)
Step-by-step explanation:
Given iteration formula:
[tex]x_{n+1}=\dfrac{\left(x_n\right)^3-1}{4} \quad \textsf{and} \quad x_1=-1[/tex]
Part (a)(i)
Substitute the value of x₁ into the formula and solve for x₂ :
[tex]\begin{aligned}\implies x_2 & =\dfrac{\left(x_1\right)^3-1}{4}\\\\& =\dfrac{\left(-1\right)^3-1}{4}\\\\ & = \dfrac{-1-1}{4}\\\\ & = \dfrac{-2}{4}\\\\ & = -0.5\end{aligned}[/tex]
Part (a)(ii)
Substitute the value of x₂ into the formula and solve for x₃ :
[tex]\begin{aligned}\implies x_3 & =\dfrac{\left(x_2\right)^3-1}{4}\\\\& =\dfrac{\left(-0.5\right)^3-1}{4}\\\\ & = \dfrac{-0.125-1}{4}\\\\ & = \dfrac{-1.125}{4}\\\\ & = -0.28125\end{aligned}[/tex]
Part (b)
To find the solution to 6 decimal places, keep substituting each new value into the iteration formula until the answers are the same when rounded to the required level of accuracy.
[tex]\implies x_4=-0.2555618286...[/tex]
[tex]\implies x_5=-0.2541728038...[/tex]
[tex]\implies x_6=-0.2541051331...[/tex]
[tex]\implies x_7=-0.2541018552...=-0.254102\:\: \sf (6 \:d.p.)[/tex]
[tex]\implies x_8=-0.2541016964...=-0.254102\:\: \sf (6 \:d.p.)[/tex]
[tex]\implies x_9=-0.2541016888...=-0.254102\:\: \sf (6 \:d.p.)[/tex]
Therefore, the solution is -0.254102 (6 d.p.).
Let g(x) = f(x+3) for some function f(x). Is g(x) a function? From the previous problem, let the domain of f(x) be [1, 4). What is the domain of g(x)?
The domain of g(x) is [4, 7)
How to determine the domain of g(x)?The function is given as:
g(x) = f(x + 3)
Since f(x) is a function, then g(x) is also a function
The domain of f(x) is given as:
[1, 4)
The equivalent of these values in g(x) are
x = 1 + 3 = 4
x = 4 + 3 = 7
Hence, the domain of g(x) is [4, 7)
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An American roulette wheel has 38 slots: two slots are numbered 0 and 00, and the remaining slots are numbered from 1 to 36. Find the probability that the ball lands in an odd-numbered slot. Write your answer as a whole number or reduced fraction.
Answer:
9/19
Step-by-step explanation:
To write the probability as a fraction, put the total number on the bottom and put the number of ways the event can happen, put that on top.
They said there are 38 slots. That's the total, it goes on the bottom.
Landing on an Odd Number is the event we're looking at. There are 18 ways that can happen: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35
That number 18 (bc 18 ways to get an odd number) goes on top.
P(Odd) = 18/38
Always simplify when possible.
18/38
= 9/19
The numbering system we commonly use is called the decimal numbering system because it uses ____ symbols to represent all possible numbers.
The numbering system we commonly use is called the decimal numbering system because it uses 10 symbols to represent all possible numbers.
A way of writing numbers is known as a number system.
There are the following number systems:
Decimal number system: "Deci" meaning 10, implies that the number system consists of 10 digits or symbols, namely 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.Binary number system: "Bi" meaning 2, implies that the number system consists of 2digits or symbols, namely 0, and 1.Octal number system: "Oct" meaning 8, implies that the number system consists of 8 digits or symbols, namely 0, 1, 2, 3, 4, 5, 6, 7, and 8.Hexa-Decimal number system: "Hexa-Deci" meaning 16, implies that the number system consists of 16 symbols, namely 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F.Hence, we can say that the numbering system we commonly use is called the decimal numbering system because it uses 10 symbols to represent all possible numbers.
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What is solve -x+4=x+6
please help me with this problem
Answer:
Answer is 544
Hope it helps you
Evaluate the function f(x) = − x² + 10x – 6 Find f( -5)
Answer:
f (-5) = -81
Step-by-step explanation:
f (-5) = - (-5)² + 10.(-5) - 6 = -81
if z varies jointly with x and y when x=3, y=-1, and z=12, what is z when x=-5 and y=6
Step-by-step explanation:
z = 12 = k × 3 × -1
12 = k×-3
-4 = k
z = k× -5×6 = k×-30 = -4 × -30 = 120
Consider Mary's experiment regarding whether learning of 6th graders on a math lesson is affected by background noise level. Mary has collected her data. What is the null hypothesis for her study
a) If μ₁ is the average learning rate of the 6th graders without background noise level and μ₂ is the average learning rate of the 6th graders with the background noise level
The Null Hypothesis is that the learning rate of the 6th graders is not affected by the background noise level.
The null hypothesis, H₀: μ₁ = μ₂
b) The Alternative Hypothesis is that the learning rate of the 6th graders is affected by the background noise level.
An alternative hypothesis, Hа: μ₁ ≠ μ₂
c) Assumptions that must be met about the data before she can correctly use an independent t-test
There must be a random selection of the 6th graders
That the two groups are normally similar in their learning abilities
The division of students into the two groups should be at random
d) She has to make these assumptions to prevent bias and inaccuracy of results. If these assumptions are not made, the outcome of the experiment may not reflect the true effect of background noise on the learning of the 6th graders.
She can still get accurate results if she includes some bias in the selection to prove a particular result.
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The difference of two numbers is 8, and the sum of the squares of these two numbers are 320. what are the numbers
Answer:
There are two sets of numbers that meet the criteria:
1) 8 and 16, or
2) -8 and -16
Step-by-step explanation:
Let's call the two numbers A and B.
We are told:
1) A-B=8, and
2) A^2 + B^2 = 320
Rearrange 1) to solve for A:
A=8+B
Use this value of A in the second equation:
A^2 + B^2 = 320
(8+B)^2 + B^2 = 320
(64+ 16B + B^2) + B^2 = 320
2B^2 + 16B + 64 = 320
B^2 + 8B + 32 = 160
B^2 + 8B + 32 = 160
B^2 + 8B - 128 = 0
(B+16)(B-8)=0
B is either -16 or +8
Assuming B = 8
Then from A-B=8 we find that A = B+8 and A = 16
Does A^2 + B^2 = 320?
(16)^2 + (8)^2 = 320?
256 + 64 = 320? YES
--------------------------------
Assuming B=-16
If B=-16, then from A-B=8 we find that A = 8+(-16) and A = -8
Does A-B=8?
-8 - (-16) = 8? YES
Does A^2 + B^2 = 320?
(-8)^2 + (-16)^2 = 320?
64 + 256 = 320? YES
Select all of the solutions to the equation.
The solution to the equations is; +1/2, -1/2, +√2, and -√2.
What is a system of equations?A system of equations is two or more equations that can be solved to get a unique solution. the power of the equation must be in one degree.
Given;
[tex]4x^4 + 7x^2 - 2 = 0\\\\[/tex]
The factored form of the polynomial regression
[tex](4x^2 -1)(x^{2} + 2)[/tex]
So,
[tex](4x^2 -1) = 0\\\\4x^2 = 1\\\\x = \pm \dfrac{1}{2}[/tex]
Also,
[tex](x^{2} + 2) = 0\\\\x^{2} = - 2\\\\x = \pm \sqrt{2}[/tex]
Hence, the solution to the equations are;
+1/2, -1/2, +√2, and -√2.
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Answer:
ABCF
Step-by-step explanation:
Pete drive 50 miles in 4 hours work out his average speed in miles per hour
Answer:
Pete's average speed is 12.5 miles per hour
Step-by-step explanation:
Miles per hour means how many miles were driven in 1 hour so :
50 miles in 4 hours = ? miles in 1 hour
4 ÷ x = 1
4/x = 1
x = 4
50÷4 = ?
50/4 = ?
? = 12.5
Pete's average speed is 12.5 miles per hour
Hope this helped and have a good day
how to find volume of a prism
Answer:
- HOW TO FIND VOLUME OF PRISM -
1st step - Write the formula for finding the volume of a regular pentagonal prism. The formula is V = [1/2 x 5 x side x apothem] x height of the prism.
2nd step - Find the area of the pentagonal base face. Let's say the length of a side is 6 cm and the length of the apothem is 7 cm.
3rd step - Find the height. Let's say the height of the shape is 10 cm.
4th step - Multiply the area of the pentagonal base face times the height. ...
and last step - State your answer in cubic units.
Step-by-step explanation:
:):):)
I don’t understand this, can somebody help?
[tex]not \: penny \: or \: dime = quarter \: or \: nickle[/tex]
[tex]p(x) = \frac{13 + 18}{86} = \frac{31}{86} = 0.36[/tex]
30)[tex]gallons \: for \: car \: 1 = \frac{450}{18} = 25 \: gal[/tex]
[tex]gallons \: for \: car \: 2 = \frac{450}{25} = 18 \: gal[/tex]
[tex]we \: need \: 25 + 18 = 43 \: gallons[/tex]
[tex]43 \times 2.49 = 107.07 \: dollars[/tex]
Victoria needs to order some new supplies for the restaurant where she works. The restaurant needs at least 673 glasses. There are currently 368 glasses. If each set on sale contains 10 glasses, write and solve an inequality which can be used to determine x, the number of sets of glasses Victoria could buy for the restaurant to have enough glasses.
Using a linear function, we have that:
The inequality is: [tex]368 + 10x \geq 673[/tex].The solution is: [tex]x \geq 31[/tex].What is a linear function?A linear function is modeled by:
y = mx + b
In which:
m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.b is the y-intercept, which is the value of y when x = 0, and can also be interpreted as the initial value of the function.Since there are currently 368 glasses(y-intercept), and each set on sale comes with 10 glasses(slope), the number of glasses after x sales is given by:
y = 368 + 10x.
She needs at least 673 glasses, hence the inequality is:
[tex]y \geq 673[/tex]
[tex]368 + 10x \geq 673[/tex].
Now we solve the inequality to find the number of sets needed:
[tex]10x \geq 305[/tex]
[tex]x \geq 30.5[/tex]
As the number of sets is an integer number, the solution is:
[tex]x \geq 31[/tex].
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Writing Exercises
587. When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?
Answer:
Therefore, we can say that the exponent will be negative if the number is less than 1 and vice versa.
Step-by-step explanation:
We are asked to explain that when we convert a decimal number to scientific notation how will we know whether the exponent is negative.
If the given number is greater than 1 then the exponent is positive and if less than 1 the exponent is negative.
Triangle G H J is rotated 90 degrees about point X to form triangle S T R.
Triangle GHJ is rotated 90° about point X, resulting in triangle STR. Which congruency statement is true?
TR ≅ GJ
∠S ≅ ∠H
TS ≅ HG
∠R ≅ ∠G
The congruency statement is true TS ≅ HG. so option C is correct.
What is the congruent triangle?Two triangles are said to be congruent if the length of the sides is equal, a measure of the angles are equal and they can be superimposed.
Given information;
Triangle G H J is rotated 90 degrees about point X to form triangle S T R.
Triangle GHJ is rotated 90° about point X, resulting in triangle STR.
Now, if any triangle is rotated 90 degrees about a point the two sides will be ≅ to each other.
So, ST = GH
RT= HJ
SR= GJ
<S= <G
<T = <H
<R=<J
The side TS ≅ HG.
Hence, option (c) is correct.
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Answer:
C
Step-by-step explanation:
Trust me bro
A gardener is planning to fill her garden with mulch. she plots it on a grid to plan how much she will need. the garden is in the shape of a rectangle with vertices at (3, 9) (5, 9) (3, 3) (5, 3). one bag of mulch covers 5ft^2 and costs $5.00. how much will it cost her to cover her garden?
The cost to cover the garden with mulch will be $12.00, as the area of the rectangular garden is 12 ft.².
The cost of one bag of mulch is given as $5.00.
The area covered by one bag of mulch is 5 ft.².
Therefore, the cost to cover 1 ft.² of the garden with mulch is $(5/5) = $1.00.
To find the area of the rectangular plot representing the garden, we do the follows:
The distance formula between two points (x₁, y₁) and (x₂, y₂) is,
d = √((x₂ - x₁)² + (y₂ - y₁)²).
The length of the rectangular plot between points (3, 9) and (5, 9):
Length = √((5 - 3)² + (9 - 9)²) = √(2² + 0²) = √(4) = 2.
The width of the rectangular plot between the points (3, 3) and (3, 9):
Width = √((3 - 3)² + (9 - 3)²) = √(0² + 6²) = √(36) = 6.
Therefore, the area of the rectangular plot = Length*Width = 2*6 = 12 ft.².
Therefore, the cost to cover the rectangular plot with mulch, at the cost of $1.00 per 1 ft.² = $1*12 = $12.00.
Therefore, the cost to cover the garden with mulch will be $12.00, as the area of the rectangular garden is 12 ft.².
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If an equilateral triangles sides are 8 inches long, what is the height?
I think it’s 8.48 but i’m not sure.
Answer:
6.93
Step-by-step explanation:
formula to find height of equilateral triangle is [tex]\frac{\sqrt{3} }{2} *A[/tex] A is the side of the equilateral triangle
Answer:
6.93 inches
(first option listed)
Step-by-step explanation:
we can find the height of this triangle by turning it into a right triangle, and then using the Pythagorean Theorem
(process shown in photo)
hope this helps!! have a lovely day :)
justification two common tangents
Two common tangents occur when two circles intersect each other at two points.
How to illustrate the information?It should be noted that two common tangents are also referred to as transverse common tangents.
In this case, two common tangents occur when two circles intersect each other at two points.
The diagram is attached.
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answer with steps please
Answer:
Step-by-step explanation:
Answer: [tex]\frac{-70a^3-195^2-20a-78}{60}[/tex] or [tex]-\frac{7}{6} a^3-\frac{13}{4} a^2-\frac{1}{3} a-\frac{13}{10}[/tex]
When we say subtract this from that, the value or word after from is the one which is ahead of the equation. For example, subtract 9 from 10, then the expression will be 10-9 . [Answer is 1]
The expression that can be formed from the picture is and the solution is shown in the steps below.
[tex](\frac{a^3}{3} -\frac{3a^2}{4} -\frac{5}{2}) - (\frac{5a^2}{2} +\frac{3a^3}{2} +\frac{a}{3} -\frac{6}{5} )[/tex] [Formed the expression]
[tex]\frac{a^3}{3} -\frac{3a^2}{4} -\frac{5}{2} - \frac{5a^2}{2} -\frac{3a^3}{2} -\frac{a}{3} +\frac{6}{5}[/tex] [Removed the brackets]
[tex]\frac{a^3-a}{3} +\frac{6}{5} -\frac{3a^2}{4} +\frac{-5a^2-3a^3-5}{2}[/tex] [Brought common denominators together]
[tex]\frac{20(a^3-a)}{60} +\frac{6*12}{60} -\frac{15(3a^2)}{60} +\frac{30(-5a^2-3a^3-5)}{60}[/tex] [LCM of 3,5,4 and 2]
[tex]\frac{20a^3-20a+72-45a^2-150a^2-90a^3-150}{60}[/tex] [Multiplied the numerators]
[tex]\frac{-70a^3-195^2-20a-78}{60}[/tex] [simplified]
[tex]-\frac{7}{6} a^3-\frac{13}{4} a^2-\frac{1}{3} a-\frac{13}{10}[/tex] [More simplified]
I tried my best to show my steps and solve it. Thank you.
∩_∩
(„• ֊ •„)♡
┏━∪∪━━━━┓
hope it helped
┗━━━━━━━┛
Can someone help me with question 3? it geometry, gotta need an answer fast
Answer:
Step-by-step explanation:
So for the 1st one you're going to use your protractor and line it up so that BC is at 0 degrees, that way you can look at the point A to determine the angle. Now that you have the angle you're going to take your protractor and draw one point which will be where the two lines meet, line this up in the middle and then draw a point at 0 degrees and a point at the angle you found. Now draw another point at half the angle (like in the previous answer I answered). Draw a line from the middle point and connect it to all these three points (4 in total but one of them is going connected to all of them, like the point B in the image you sent).
For the second question you're going to take the angle you found in the first answer and multiply it by 2. Now start off by drawing a middle point which will connect the two other lines. Line up your protractor so that point is at bottom but in the middle (Check my photo for clarification). Now go to the value that is twice the angle and draw a point there, and also draw a point at 0 degrees) and then connect the two points to the middle point.
For question 3 you don't really need the hint, all you need is to multiply the angle by 1.5 but I'll send another image explaining what it's trying to explain.
What word describes the slope of the line passing through points (5, 12) and
(5,-13)?
Answer: undefined
Step-by-step explanation:
[tex]\frac{y2-y1}{x2-x1}[/tex]
[tex]\frac{-13-12}{5-5}[/tex]
[tex]\frac{-25}{0}[/tex]
= undefined
1. How could you figure out the coordinates of a reflection over the x-axis? What would the coordinates of that reflected point be?
2. Then also tell how you could determine the coordinates when (-8,2) is reflected over the y-axis, and tell the coordinates of that new point.
The reflection of a point over the x axis is given by the equation (x, y) ⇒ (x, -y).
If the point (-8,2) is reflected over the y-axis, the new point would be (8, 2)
What is a transformation?Transformation is the movement of a point from its initial location to a new location. Types of transformation are reflection, translation, rotation and dilation.
The reflection of a point over the x axis is given by the equation (x, y) ⇒ (x, -y).
If the point (-8,2) is reflected over the y-axis, the new point would be (8, 2)
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Which statement about the function is true?
O The function is increasing for all real values of x
where
x<-4.
Answer:
A.) The function is increasing for all real values of x where x < -4
Step-by-step explanation:
On the interval x < -4, all of the x values are increasing. Before x = -4, the line assumes a positive slope because it is heading upwards. After this interval, the function is decreasing because the slope is heading downwards.
On the interval -6 < x < -2, all of the x values are positive because they lie above the x-axis. This does not necessarily mean that all of the values are increasing.
On the interval x < -6 and x > -2, all of the x values are negative because they lie below the x-axis.
a property sells for one and a half million dollars. the five owners each receive a fifth of the money.
how much money does each owner receive?
Answer: Each owner gets $300000
Step-by-step explanation:
one and a half million is 1500000
then divide by 5
you get $300000
Please help!!! What is the slope of a line perpendicular to the following line:
Answer:
b.) 2
Explanation:
Parallel slopes have same slope.
Perpendicular slopes have negatively inverse slope.
[tex]\sf Here \ given \ equation: y = -\dfrac{1}{2} x - 6[/tex]
Comparing it with slope intercept form, "y = mx + b" where m is slope.
Here parallel slope: -1/2
Perpendicular slope: -(-1/2)⁻¹ = 2
Prism A and Prism B have the same volume. Which could be the dimensions of prism B. Please help!
Question : Prism A and Prism B have the same volume. Which could be the dimensions of prism B?
Prism A :
Length = 8 feet
Width = 4 feet
Height = 5 feet
Prism B:
Length = ? feet
Width = ? feet
Height = ? feet
Option 1:
Length = 6 feet
Width = 10 feet
Height = 2 feet
Option 2:
Length = 4 feet
Width = 4 feet
Height = 10 feet
Option 3:
Length = 5 feet
Width = 4 feet
Height = 7 feet
Option 4:
Length = 8 feet
Width = 4 feet
Height = 6 feet
Here, the dimensions of the prism B will be length = 4 feet, width = 4 feet, and height = 10 feet.
We know that if the a and b be the length and width of the base respectively and c be the height of a prism, then the volume of the prism is abc.
Given that
length of 1st prism = 8 feet
width of 1st prism = 4 feet
height of 1st prism = 5 feet
So volume of 1st prism = 8 * 4 * 5 = 160 feet³
Now the volume of 1st and 2nd prisms are the same.
So, we have to check the options for length, width, and height of the 2nd prism.
1st option:
Height = 6 feet
Width = 10 feet
Length = 2 feet
So, volume = 6 * 10 * 2 = 120 feet³
Then the 1st option is incorrect.
2nd option:
Height = 4 feet
Width = 4 feet
Length = 10 feet
So, volume = 4 * 4 * 10 = 160 feet³
Then the 2nd option is correct.
Therefore, the dimensions of the prism B will be length = 4 feet, width = 4 feet, and height = 10 feet. The 2nd option is correct.
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Determine the equation, in vertex form, for the following parabolas.
A. The graph below
B. has a vertex at (−2, 5) and passes (b) through the point (2, 0)
5x²+20x+16y-15=0 is the equation, in vertex form, for the given parabolas.
We need to determine the equation, in vertex form, for the following parabolas.
How to find the equation, in vertex form?We can find the parabola's equation in vertex form following two steps:
Step 1: Use the (known) coordinates of the vertex, (h,k), to write the parabola's equation in the form: y=a(x−h)²+k.
Step 2: Find the value of the coefficient a by substituting the coordinates of the point into the equation written in step 1 and solving for a.
Now, substitute (−2, 5) in y=a(x−h)²+k
⇒y=a(x−(-2))²+5
Now, substitute (2, 0) in y=a(x−(-2))²+5.
0=a(2+2)²+5
⇒a=-5/16
So, y=-5/16(x+2)²+5
⇒16y=-5(x²+4x+4)+5
⇒16y=-5x²-20x-20+5
⇒5x²+20x+16y-15=0.
Therefore, 5x²+20x+16y-15=0 is the equation, in vertex form, for the given parabolas.
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Line I is parallel to line M. If the measure of 6 is 75 degrees, what is the measure of 3?
(please look at photo attached)
In the given diagram, the measure of ∠3 will be 105°.
In the given diagram, ∠3 and ∠6 are consecutive interior angles.
How to form supplementary angles by transversal?If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles formed are supplementary.
That is,
∠3 + ∠6 = 180°
From the given information,
∠6 = 75°
Then,
∠3 + 75° = 180°
∠3 = 180° - 75°
∠3 = 105°
Hence, the measure of ∠3 will be 105°.
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