If the function fis defined by f(x) = x ^ 5 - 1 then the inverse, f ^ - 1 is defined by f ^ - 1 * (x) =
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Given the function:
[tex]f(x)=x^5-1[/tex]We have that f(x) = y, so:
[tex]y=x^5-1[/tex]Substitute x with y:
[tex]x=y^5-1[/tex]And solve for y:
[tex]\begin{gathered} x+1=y^5-1+1 \\ x+1=y^5 \\ y=\sqrt[5]{x+1} \end{gathered}[/tex]Answer:
[tex]\sqrt[5]{x+1}[/tex]Related Questions
Aiden ipens a savings account with a deposit of 4500. The account pays 3% simple interest.3. If Aiden does not make any more deposits or withdrawals, how much will he have in the account at the end of two years?A 4527B 4635C 4680D 4774E 4905
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Answer: $4, 770
Aiden deposit $4500 into her account with an interest rate of 3%
Time = 2 years
Using the Simple Interest
[tex]\begin{gathered} I\text{ = }\frac{P\text{ x R x T}}{100} \\ P\text{ = \$4500} \\ R\text{ = 3\%} \\ T\text{ = 2} \\ I\text{ = }\frac{4500\text{ x 3 x 2}}{100} \\ I\text{ = }\frac{4500\text{ x 6}}{100} \\ I\text{ = }\frac{27000}{100} \\ I\text{ = \$270} \\ \text{The total amount in her account is } \\ \text{Balance = Principal + Interest} \\ \text{Balance = \$4500 + \$270} \\ \text{Balance = \$4, 770} \end{gathered}[/tex]i would like help understanding this form of math please.
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Question:
Solution:
If we have the formula:
[tex]\text{Height = }\frac{Cons\tan t}{\text{Width}}[/tex]The length of the rectangle below is 7 than it’s width. Given that the total distance around the rim of the shape is 46 units, what is the value of x?
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Identify the key features for the following equation: y=4sin(x)−5What kind of cyclic model is the equation?
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Given,
The equation of the function is:
[tex]y=4sinx-5[/tex]The standard equation of wave is,
[tex]y=Asin\text{ \lparen Bx+C\rparen+D}[/tex]Here, A is the amplitude
B is the period.
C is the phase shift.
D is vertical shift.
As the given function have the sine function so, the cyclic model of the wave is sine.
Amplitude = 4.
Midline = -5
Minimum = -9
Hence, the key feature of the cyclic model is identified.
(c) In the diagram below:ARga nainP.50°B65%DNot drawn to scale(i) Calculate the angle BDC (ii) Calculate angle ABD (iii) Find angle BAD(iv) What type of triangle is triangle ABD ?CS
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Given: Parallel lines PQ and RS. Triangle ABD and BDC are such that
[tex]\begin{gathered} BD=CD \\ m\angle ABR=50\degree \\ m\angle ADB=65\degree \end{gathered}[/tex]Required: To determine the triangle ABD type and calculate the angle BDC, ABD, and angle BAD.
Explanation: Since line PQ is parallel to line RS,
[tex]\angle ADB=\angle DBC=65\degree[/tex]Now since BD=CD, triangle BCD is an isosceles triangle. Hence,
[tex]\angle DBC=\angle DCB=65\degree[/tex]Now, in triangle BCD, we have
[tex]\begin{gathered} \angle B+\angle C+\angle D=180\degree\text{ \lparen Angle sum property\rparen} \\ 65\degree+65\degree+\angle D=180\degree \\ \angle D=50\degree \end{gathered}[/tex]Now RS is a straight line. Hence at point B, we have
[tex]\begin{gathered} 50\degree+\angle ABD+\angle DBC=180\degree\text{ \lparen Linear pair\rparen} \\ \angle ABD=65\degree \end{gathered}[/tex]Finally, in triangle ABD, we have
[tex]\begin{gathered} \angle A+\angle B+\angle D=180\degree \\ \angle A+65\degree+65\degree=180\degree \\ \angle A=50\degree \end{gathered}[/tex]Now since in triangle ABD, we have
[tex]\angle ABD=\angle ADB[/tex]The triangle ABD is isosceles.
Final Answer:
[tex]\begin{gathered} \angle BDC=50\degree \\ \angle ABD=65\degree \\ \angle BAD=50\degree \end{gathered}[/tex]The triangle ABD is isosceles.
An equilateral triangle and an isosceles triangle share a common side. What is the measure of /_ABC?
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The sum of all the angles in a triangle is equal to 180 degrees
For an equilateral triangle, all sides are equal
i.e 60 + 60 + 60= 180
For an isosceles triangle, two sides are equal
the first image is an isosceles triangle why the second image is an equilateral triangle
It is 6 miles in a kayak to the Fish Islands from my house. The trip to the island takes 2 hourstraveling against the current and 1¼ hours for the return trip (with the current). How fast can Ipaddle the Kayak if there was no current? The answer can be rounded to the nearest tenth.Solve Algebraically using linear systems
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It is given that the distance is 6 miles and the time is 2 hours upstream and one and a quarter hour downstream.
The time downstream is given by:
[tex]1\frac{1}{4}=\frac{4+1}{4}=\frac{5}{4}\text{ hours}[/tex]Since the distance is constant, it follows:
[tex]\begin{gathered} \text{ Speed=}\frac{\text{ Distance}}{\text{ Time}} \\ \text{ Distance=SpeedxTime} \end{gathered}[/tex]So the distance is constant hence:
[tex]\text{ Distance upstream=Distance Downstream}[/tex]Let the speed of kayak be x and speed of current be y so the speed downstream is x+y and speed upstream is x-y so it follows:
[tex]\begin{gathered} \frac{5}{4}(x+y)=2(x-y) \\ 4\times\frac{5}{4}(x+y)=4\times2(x-y) \\ 5x+5y=8(x-y) \\ 5x+5y=8x-8y \\ 5y+8y=8x-5x \\ 13y=3x \\ x=\frac{13}{3}y \end{gathered}[/tex]Use the individual equation to find x and y as follows:
[tex]\begin{gathered} 6=2(x-y) \\ 6=2(\frac{13}{3}y-y) \\ 3=\frac{13-3}{3}y \\ \frac{9}{10}=y \end{gathered}[/tex]Hence the speed of the water current is 9/10 miles per hour.
The speed of the kayak is given by:
[tex]\begin{gathered} x=\frac{13}{3}y \\ x=\frac{13}{3}\times\frac{9}{10} \\ x=\frac{39}{10}=3.9\text{ miles per hour} \end{gathered}[/tex]Hence the speed of the kayak without the water current is 3.9 miles per hour.
The time required without water current is:
[tex]\begin{gathered} \text{Time}=\frac{Dis\tan ce}{Speed} \\ t=\frac{6}{3.9}\approx1.5\text{ hours} \end{gathered}[/tex]Hence it will take approximately 1.5 hours without the current.
14. In your rectangular backyard, you knowthe width of the yard is three lessthan four times the length. If the perimeterof your yard is 24 yards, what isthe width?18 3/5yards3 yards9 yards15 yards
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ANSWER:
3rd option: 9 yards
STEP-BY-STEP EXPLANATION:
Given that:
Length = L
Width = W = 4L - 3
The perimeter is the sum of all the sides, therefore:
[tex]\begin{gathered} p=L+L+W+W \\ \\ \text{ We replacing:} \\ \\ 24=L+L+4L-3+4L-3 \\ \\ \text{ We solve for L} \\ \\ 24+3+3=10L \\ \\ L=\frac{30}{10} \\ \\ L=3\text{ yd} \\ \\ \text{ Therefore:} \\ \\ W=4L-3=4(3)-3=9\text{ yd} \end{gathered}[/tex]So the correct answer is 3rd option: 9 yards
I don't understand if this equation is a linear equation or not. Can you please help me?
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we have the equation
[tex]\frac{x}{4}-\frac{y}{3}=1[/tex]To remove the fractions, multiply both sides by (4*3=12)
[tex]\begin{gathered} \frac{12x}{4}-\frac{12y}{3}=12 \\ 3x-4y=12 \\ 4y=3x-12 \\ y=\frac{3}{4}x-3 \end{gathered}[/tex]this is the equation of a line
that means
is a linear equation
Which equation is an identity?O 3(x - 1) = x + 2(x + 1) + 1Ox-4(x + 1) = -3(x + 1) + 1O 2x + 3 = 1 (4x + 2) + 2(6x - 3) = 3(x + 1) – x-2
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Identity equations are always true, no matter the values that the variables take.
We have to calculate for each one, and if the result gives a true statement, then the equation is an identity:
1) 3(x - 1) = x + 2(x + 1) + 1
[tex]\begin{gathered} 3\left(x-1\right)=x+2\left(x+1\right)+1 \\ 3x-3=x+2x+2+1 \\ 3x-3=3x+3 \\ 3x-3x=3+3 \\ 0=6 \end{gathered}[/tex]This is FALSE (for any value of x), so the equation is not an identity.
2) x-4(x + 1) = -3(x + 1) + 1
[tex]\begin{gathered} x-4\left(x+1\right)=-3\left(x+1\right)+1 \\ x-4x-4=-3x-3+1 \\ x(1-4+3)=-2+4 \\ 0=2 \end{gathered}[/tex]This is FALSE, so the equation is not an identity.
3) 2x + 3 = 1 (4x + 2) + 2
[tex]\begin{gathered} 2x+3=14x+2+2 \\ 3-2-2=14x-2x \\ -1=12x \\ x=\frac{-1}{12} \end{gathered}[/tex]This equation holds true only for x=-1/12, so it is not an identity.
4) (6x - 3) = 3(x + 1) – x-2
[tex]\begin{gathered} \left(6x-3\right)=3\left(x+1\right)-x-2 \\ 6x-3=3x+3-x-2 \\ 6x-3=2x+1 \\ 6x-2x=1+3 \\ 4x=4 \\ x=1 \end{gathered}[/tex]This equation holds true only for x=1, so it is not an identity.
Neither of the options is an identity.
drawing a sketch, giving an example, or providing a written description, please indicate themeaning of each of the following shapes.
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For the given shapes, we will draw a sketch
a) A cone
the sketch of the cone will be as follows:
The cone has a circular base of radius = r, and a height of (h) and has a flat surface and curved surface as shown.
b) The diameter of the circle:
The diameter is a line segment (d) that connects two points lying on the circle through the center of the circle
c) The radius of the circle:
The radius of the circle (r) is a line segment that connects the center of the circle and any point lying on the circle
Devonte creates a scatter plot of the relationship between his hourly pay in dollars, y, and the number of customers he serves at a coffee shop, X. He calculates the equation of the trend line to be y = 2.52 +7. Part A What does the y-intercept represent? Enter the correct answers in the boxes. per hour when he serves customers. The y-intercept represents that Devonte earns $
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Given equation of line is,
Error Analysis Denzel identified (3, 2) as a point on the line y - 2 = 2/3 (x + 3). What is the error that Denzel made?
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Slope point formula:
y-y1= m (x-x1)
Where:
m= slope
(y1,x1) = point of the line
For:
y - 2 = 2/3 (x + 3)
m= 2/3
y1= 2
x1= -3
The error is that the point is not (3,2) is (-3,2)
y-2 = 2/3 (x-(-3))
y-2 = 2/3 (x+3)
4. Jill wants to buy $70,000 worth of insurance for her new house. If therate is $8.00 per $1000 of value, what will her insurance premium be?a. $590b. $560C. $530
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Let's calculate the insurance premium Jill will have to pay for her insurance of her new home:
Insurance premium = 70,000 / 1,000 * 8
Insurance premium = 70 * 8
Now you can calculate easily the payment Jill will have to afford.
4+(6x2²)-9 use pemdas
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Given:
[tex]4+(6\times2^2)-9[/tex]Required:
To solve the given expression.
Explanation:
Consider
[tex]\begin{gathered} =4+(6\times2^2)-9 \\ \\ =4+(6\times4)-9 \\ \\ =4+24-9 \\ \\ =28-9 \\ \\ =19 \end{gathered}[/tex]Final Answer:
[tex]4+(6\times2^2)-9=19[/tex]I don't understand how to add and subtract Intregers
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Explanation
First of, you should know that integers are whole numbers.
There are positive integers (positive whole numbers, that is, normal whole numbers greater than 0, for example, 7, 98, 14 etc.) and there are negative integers (negative whole numbers, that is, whole numbers less than 0, for example, -3, -37, -101 etc.)
So, the first tip about adding and subtracting these numbers is to look at them in monetary terms.
Always look at positive numbers like money you have in your pockets (cash at hand).
And look at negative numbers like money you're owing someone.
So, we can then go through the different types of addition and subtraction of integers now.
- Addition of two positive numbers
** 2 + 2
You can interprete this simple addition as having $2 and another $2 is given to you, this means you've got $4 now.
2 + 2 = 4
** 17 + 7
You can interprete this simple addition as having $17 and another $7 is given to you, this means you've got $24 now.
17 + 7 = 24
- Subtraction of two positive integers
** 7 - 3
Look at this like having $7, then -3 means $3 is taken away from it, then you've got only $4 left.
7 - 3 = 4
** -15 + 10
This means you're owing $15, and you've got only $10, after paying the $10, there's still a debt of $5 left. So,
-15 + 10 = -5
Before the next two further types of adding/subtracting integers, weneed to also note the following
(+) × (+) = (+)
(+) × (-) = (-)
(-) × (+) = (-)
(-) × (-) = (+)
These helps us to simplify these additions and subtractions that involve a mix of positve numbers and negative numbers or just strictly working with negative numbers.
Addition of two negative numbers
** -5 + (-3)
Normally, with the former approach, this just means a debt of $5 is added to a debt of $3, these come together to give a bigger debt of $8.
But we can simplify the given equation further because we know that
(+) × (-) = (-), So,
-5 + (-3) = -5 - 3 (The plus sign before the -3 and the minus sign in the bracket multiply to give a negative/minus sign).
So,
-5 + (-3) = -5 -3 = -8
** -7 + (-4)
-7 + (-4) = -7 - 4 = -11
Subtraction of two negative integers
** -5 - (-5)
Recall that
(-) × (-) = (+), So,
-5 - (-5) = -5 + 5
Which then translates to owing $
Function A Function B Tell whether each function is linear or nonlinear. х y 4 0 1 3 5 24 8 2 3 13 0 1 2 3 4 5 Function A is a function. Function B is a function.
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Function A is NOT LINEAR
Function B is LINEAR
The slope (change in y over change in x) does not follow a linear pattern in function A. That is the increase/decrease in the y coordinates is not at the same rate as that of the x coordinate. Whereas, for the other function, function B, the slope follows a linear pattern, that is the rate of change in y over the rate of change in x is the same rate, that is why function B has a straight line graph
Christian and Lea are in charge of planning the school prom. They will spend $250 on decorations. Dinner will cost $12 per person (p) that attends theprom. Which equation represents the total cost (t) of the prom for any number of students attending?p = 250t + 12p = 12 + 250t=12p - 250t = 250p + 12
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If one object costs $x then p objects will cost $px.
Given data:
It is given that they spend $250 on decorations and $12 per person for dinner.
Now the cost $250 is fixed.
Now, if cost od dinner for one person is $12.
So the cost of dinner for p persons will be $12p
Therefore, total cost 't' will be
[tex]t=12p+250[/tex]30 points helps What are the coordinates of each vertex if the figure is rotated 180° clockwise about the origin?[G.CO.2, G.CO.4, G.C0.5]
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The coordinates of each of the vertices after the figure is rotated 180° clockwise about the origin: A' = (2, -2), B' = (-2, -5), C' = (-6, -3), D' = (-4, 3)
Explanation:
The first thing we do is to write the vertices of the original shape:
A = (-2, 2)
B = (2, 5)
C = (6, 3)
D = (4, -3)
A rotation of (x, y) 180° clockwise about the origin = (-x, -y)
We take the negative of the x and y coordinates of the original shape
180° clockwise about the origin becomes:
A' = (-(-2), -2) = (2, -2)
B' = (-2, -5)
C' = (-6, -3)
D' = (-4, -(-3)) = (-4, 3)
The coordinates of each of the vertices after the figure is rotated 180° clockwise about the origin: A' = (2, -2), B' = (-2, -5), C' = (-6, -3), D' = (-4, 3)
What is the center and the radius of the circle: ( x + 7 ) 2 + ( y - 1 ) 2 = 9 ?
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Given:
There is a equation of circle given in the question as below
[tex]\left(x+7\right)^2+(y-1)^2=9[/tex]Required:
We want to find the center and radius of given circle
Explanation:
The general equation of circle is
[tex](x-h)^2+(y-k)^2=r^2[/tex]where (h,k) is the center of circle and r be the radius of circle
Now by comparing we get
[tex]\begin{gathered} (h,k)=(-7,1) \\ r^2=9\Rightarrow r=3 \end{gathered}[/tex]Final answer:
C
All of the following are equivalent exceptO (4)(y)O 4+ y04.1O 4 yASK FOR HELPUNT QUESTION
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Triangle ACD is dilated about the origin.10D'987-854DC92СA-5-4-3-2-102- 1-2Which is most likely the scale factor?0 1 / 3OOo
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Step 1
Find the length of any two sides of both figures
[tex]\begin{gathered} In\text{ the original image} \\ AC=3\text{ units} \\ CD=2\text{ units} \\ In\text{ the dilated image} \\ A^{\prime}C^{\prime}=9\text{ units} \\ C^{\prime}D^{\prime}=6\text{ units} \end{gathered}[/tex]Step 2
Write the ratio that will be used to get the dilation factor.
[tex]\begin{gathered} \frac{C^{\prime}D^{\prime}}{CD}=\frac{A^{\prime}C^{\prime}}{AC} \\ \frac{6}{2}=\frac{9}{3} \\ 3=3 \\ \text{Therefore, the scale factor = 3} \end{gathered}[/tex]4. Angelo gave 3 of a bag of pretzels to Ben. Ben ate a portion (x) of the pretzels and then gave 4 of the remaining pretzels in the bag to Connor. The expression below represents Connor's portion of the bag of pretzels. 2/3 314 Which expression is equivalent to Connor's portion of the bag of pretzels?
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we have Connor's portion of the pretzels
[tex]\frac{2}{3}\times(\frac{3}{4}-x)[/tex]then simply the expression
[tex]\begin{gathered} \frac{2}{3}\times\frac{3}{4}-\frac{2}{3}x \\ \frac{2\times3}{3\times4}-\frac{2}{3}x \\ \frac{6}{12}-\frac{2}{3}x \\ \frac{1}{2}-\frac{2}{3}x \end{gathered}[/tex]answer: C
how do you find 18.84 20.91 19.5 on a number line 14-22
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In order to find the given numbers on a number line thats moves between 14 and 22, we shall illustrate with a number line.
The number line illustrated above shows the numbers arranged in order from 14 to 22.
The numbers indicated in the question are printed in blue.
The position of the numbers are also indicated with a black "stroke" in relation to the position of the numbers 14 to 22.
find the length of arc FH. Round to the nearest hundredth.(Degrees)
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Given the circle G
As shown, m∠FGH = 36
And the radius of the circle = r = FG = 10 units
we will find the length of the arc FH using the formula:
[tex]\text{Arc}=\theta\cdot r[/tex]The given angle measured in degree, we will convert it to radian
So,
[tex]\theta=36\cdot\frac{\pi}{180}=\frac{\pi}{5}[/tex]So, the length of the arc =
[tex]\frac{\pi}{5}\cdot10=2\pi\approx6.283185[/tex]Round to the nearest hundredth.
So, the answer will be the length of the arc FH = 6.28
A total of 5000 tickets were sold for a raffle. the prizes are $1000, $500, $200, and $100. what price should be charged so there is a 60% profit per ticket?
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Answer: $0.576
Step-by-step explanation:
The total amount in prizes is $1800.
For there to be 60% profit, the total cost of the tickets need to be [tex]1800(1.6)=\$ 2880[/tex].
Thus, each ticket must sell for [tex]\frac{2880}{5000}=\$ 0.576[/tex]
$0.576 should be charged so there is a 60% profit per ticket.
What is Unitary Method?The unitary technique involves first determining the value of a single unit, followed by the value of the necessary number of units.
For example, Let's say Ram spends 36 Rs. for a dozen (12) bananas.
12 bananas will set you back 36 Rs. 1 banana costs 36 x 12 = 3 Rupees.
As a result, one banana costs three rupees. Let's say we need to calculate the price of 15 bananas.
This may be done as follows: 15 bananas cost 3 rupees each; 15 units cost 45 rupees.
Given:
The prizes are $1000, $500, $200, and $100.
So, total prize = 1000+ 500+ 200+ 100 = $1800.
The, the price of ticket to break
= 1800 / 5000
= $0.36
Now, the price for 60% ticket = 0.36 (1 + 0.6)
= 0.36 x 1.6
= $0.576
Learn more about unitary method here:
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Will a truck that is 14 feet wide carrying a load that reaches 12 feet above the ground clear the semielliptical arch on the one-way road that passes under the bridge shown in the figure on the right?
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Given the equation of an elipse
[tex]\frac{x^2}{a^2}+\frac{y^2}{b^2}=1[/tex]from the question,
[tex]\begin{gathered} \text{major axis}\Rightarrow2a \\ \therefore2a=52 \\ a=\frac{52}{2}=26ft \\ b=13ft \end{gathered}[/tex]Given that
[tex]x=14ft[/tex]Substitute, for a,b, and x in the elipse formula to find y
[tex]\begin{gathered} \frac{14^2}{26^2}+\frac{y^2}{13^2}=1 \\ \frac{196}{676}+\frac{y^2}{169}=1 \end{gathered}[/tex]Multiply through by 169
[tex]\begin{gathered} 49+y^2=169 \\ y^2=169-49 \\ y^2=120 \\ y=\sqrt[]{120}=10.95ft \end{gathered}[/tex]Hence, it clear the arch because the height of the archway of the bridge 7 feet from the center is approximatelyfeet
In ΔTUV, t = 82 inches, v = 86 inches and ∠V=41°. Find all possible values of ∠T, to the nearest degree.
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The value of ∠T is 38.722° as the definition of angle is "An angle is created by joining two line segments at one point, or we can say that an angle is the combination of two line segments at a common endpoint".
What is angle?An angle is created by joining two line segments at one point, or we can say that an angle is the combination of two line segments at a common endpoint. When two straight lines or rays intersect at a single endpoint, an angle is created. The vertex of an angle is the location where two points come together. The Latin word "angulus," which means "corner," is where the word "angle" originates. Based on measurement, there are different kinds of angles in geometry. The names of fundamental angles include acute, obtuse, right, straight, reflex, and full rotation. A geometrical shape called an angle is created by joining two rays at their termini. In most cases, an angle is expressed in degrees.
Here,
Side t = 82
Side u = 128.98238
Side v = 86
Angle ∠T = 38.722°
Angle ∠U = 100.278°
Angle ∠V = 41°
∠T = sin⁻¹(t·sin(V)/v)
=38.722°
Since the definition of an angle is "An angle is created by joining two line segments at one point, or we can say that an angle is the combination of two line segments at a common endpoint," the value of ∠T is 38.722°.
To know more about angles,
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the Firgure shows two triangles on a coordinate grid: Which set of transformations have been performed on triangle ABC to form triangle A'B'C'? A) Dilation by a scale factor of 1/3 followed by reflection about the x-axisB) Dilation by a scale factor of 3 followed by reflection about the x-axis C) Dilation by a scale factor of 1/3 followed by reflection about the y-axis D) Dilation by a scale factor of 3 followed by reflection about the y-axis
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In transformations, the Pre-Image is the original figure and the Image is the figure transformated.
In this case you can identify that the Pre-Image is the triangle ABC and the Image is the triangle A'B'C'.
Notice that the vertices of ABC are:
[tex]A(-3,3);B(0,0);C(-6,-3)[/tex]By definition, when the scale factor used in the dilation is between 0 and 1, the Image obtained is a reduction and, therefore, it is smaller than the Pre-Image. Since A'B'C' is smaller than ABC, then you can determine that ABC was dilated by this scale factor:
[tex]sf=\frac{1}{3}[/tex]When a figure is reflected across the y-axis, the rule is:
[tex](x,y)\rightarrow\mleft(-x,y\mright)[/tex]If you dilate ABC by the scale factor shown above, and then you reflect it across the y-axis, the coordinates of the Image will be:
[tex]\begin{gathered} A\mleft(-3,3\mright)\rightarrow A^{\prime}(-(\frac{-3}{3}),\frac{3}{3})\rightarrow A^{\prime}(1,1) \\ \\ B\mleft(0,0\mright)\rightarrow B^{\prime}\mleft(0,0\mright) \\ \\ C\mleft(-6,-3\mright)\rightarrow C^{\prime}(-(\frac{-6}{3}),\frac{-3}{3})\rightarrow C^{\prime}(2,-1) \end{gathered}[/tex]Notice that the coordinates of A'B'C' shown in the picture match with the vertices found above.
Therefore, the answer is: Option C.
The following is a sample of 20 measurements.Answer b part
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b)
Given:
[tex]\begin{gathered} \bar{x}=10.2 \\ s=2.12 \end{gathered}[/tex]Hence,
[tex]\begin{gathered} \bar{x}\pm s=10.2\pm2.12 \\ \bar{x}+s=12.32 \\ \bar{x}-s=8.08 \end{gathered}[/tex]So, the measurements in the data between 8.08 and 12.32 are 11, 9, 12, 10 12, 12 , 12, 9, 9, 9, 11, 11, 12 and 11.
Therefore, the number of measurements in interval x±s is 14.
The percentage of the measurements that fall between the interval x±s is,
[tex]\text{Percent}=\frac{14}{20}\times100=70[/tex]Therefore, the percentage of the measurements that fall between the interval x±s is 70%.
Now,
[tex]\begin{gathered} \bar{x}\pm2s=10.2\pm2\times2.12 \\ \bar{x}\pm2s=10.2\pm4.24 \\ \bar{x}+2s=14.44 \\ \bar{x}-2s=5.96 \end{gathered}[/tex]So, all the measurements in the data are between 5.96 and 14.44.Therefore, the number of measurements in interval x±2s is 20.
Therefore, the percentage of the measurements that fall between the interval x±2s is 100%.
Now,
[tex]\begin{gathered} \bar{x}\pm3s=10.2\pm3\times2.12 \\ \bar{x}\pm3s=10.2\pm6.36 \\ \bar{x}+3s=16.56 \\ \bar{x}-3s=3.84 \end{gathered}[/tex]So, all the measurements in the data are between 3.84 and 16.56.Therefore, the number of measurements in interval x±3s is 20.
Therefore, the percentage of the measurements that fall between the interval x±3s is 100%.
Last part: compare the percentage .
According to empirical rule, approximately 68% of the measurements in a sample will fall within the interval x±s.
From part b, the obtained percentage of measurements that fall within the interval x±s is 70%.
Therefore, percentage of measurements that fall within the interval x±s is greater than the predicted percentage for x±s using the empirical rule.
Option C is correct.