For given[tex]$\mathrm{n}$[/tex] and [tex]$\mathrm{X}_{(1)}$[/tex], we can obtain the confidence interval [tex]$(\mathrm{a}, \mathrm{b})$[/tex].
Let[tex]$\mathrm{X}_1, \mathrm{X}_2, \ldots, \mathrm{X}_{\mathrm{n}}$[/tex] be a sequence of size [tex]$\mathrm{n}$[/tex] from the exponential distribution with rate[tex]$\lambda$.[/tex] Then the probability density function of the random is a given [tex]$\mathrm{f}\left(\mathrm{x}_{\mathrm{i}}, \lambda\right)=\lambda \exp (-\lambda \mathrm{x}) ; \quad 0 \leq \mathrm{x} < \infty$[/tex]
Let us consider a ordered random sample of size [tex]$\mathrm{n}$[/tex] as [tex]$\mathrm{X}_{(1)}, \mathrm{X}_{(2)}, \ldots, \mathrm{X}_{(\mathrm{n})}$[/tex]. Then [tex]$\mathrm{X}_{(1)}$[/tex] is the smallest of the sample. The density function of the smallest sample [tex]$\mathrm{X}_{(1)}$[/tex] is give by
[tex]$$\mathrm{f}\left(\mathrm{X}_{(1)}, \lambda\right)=\mathrm{n}\left[1-\mathrm{F}_{\mathrm{x}}(\mathrm{x}, \lambda)\right]^{\mathrm{n}-1} \times \mathrm{f}(\mathrm{x}, \lambda)$$[/tex]
As the cumulative density is given by
[tex]=\int_0^{\mathrm{x}} \mathrm{f}(\mathrm{x}, \lambda) \mathrm{dx} \\&=\int_0^{\mathrm{x}} \lambda \times \exp (-\lambda \mathrm{x}) \mathrm{dx} \\&=1-\exp (-\lambda \mathrm{x})\end{aligned}$$[/tex]
The density of the minimum of the sample is given by
[tex]=\mathrm{n} \times[1-(1-\exp (-\lambda \mathrm{x}))]^{\mathrm{n}-1} \times \lambda \times \exp (-\lambda \mathrm{x})$[/tex]
[tex]$$\begin{aligned}&=\mathrm{n} \times[\exp (-\lambda \mathrm{x})]^{\mathrm{n}-1} \times \lambda \times \exp (-\lambda \mathrm{x}) . \\&=\mathrm{n} \lambda \times \exp (-\mathrm{n} \lambda \mathrm{x})\end{aligned}$$[/tex]
Thus, here [tex]$\mathrm{X}_{(1)}$[/tex] follows gamma distribution with parameter [tex]$\alpha=1 \quad$[/tex] and [tex]$\beta=\frac{1}{n \lambda}$[/tex].
We make use of the condition that if Y follows Gamma distribution with parameter with parameter [tex]$\alpha$[/tex]and [tex]$\beta$[/tex]. Then, we have
[tex]$\frac{2 \mathrm{Y}}{\beta} \sim \chi^2(2 \alpha, \mathrm{n}-1)$[/tex]
Thus we have that
[tex]\mathrm{T}=\frac{2 \mathrm{X}_{(1)}}{\frac{1}{\mathrm{n} \lambda}} \sim \chi^2(2, \mathrm{n}-1)$$[/tex]
[tex]$$\begin{aligned}\mathrm{b} &=\frac{\chi^2\left(1-\frac{\alpha}{2}, \mathrm{n}-1\right)}{2 \mathrm{X}_{(1)} \times \mathrm{n}} \\&=\frac{\chi^2\left(1-\frac{0.03}{2}, \mathrm{n}-1\right)}{2 \mathrm{X}_{(1)} \times \mathrm{n}} \\&=\frac{\chi^2(0.985, \mathrm{n}-1)}{2 \mathrm{X}_{(1)} \times \mathrm{n}} \\\mathrm{a} &=\frac{\chi^2\left(\frac{0.03}{2}, \mathrm{n}-1\right)}{2 \mathrm{X}_{(1)} \times \mathrm{n}} \\&=\frac{\chi^2(0.015, \mathrm{n}-1)}{2 \mathrm{X}_{(1)} \times \mathrm{n}}\end{aligned}$$[/tex]
Therefore for given[tex]$\mathrm{n}$[/tex] and [tex]$\mathrm{X}_{(1)}$[/tex], we can obtain the confidence interval [tex]$(\mathrm{a}, \mathrm{b})$[/tex].
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Question
POSSIBLE POINTS: 1.25
Rewrite the following equation in Slope-Intercept Form (y=mx+b), then use your equation to complete the table and find the slope, the y-intercept, and the x-intercept.
6x−2y=12
Equation:
Table:
x y
-2
-1
0
1
2
y-intercept: (
,
)
x-intercept: (
,
)
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The slope–intercept form of the equation, 6•x - 2•y = 12, is y = 3•x - 6, the completed table is therefore;
x; -2, -1, 0, 1, 2
y; -12, -9, -6, -3, 0
The slope of the equation is 3
The y–intercept of the graph of the equation is (0, -6)
The x–intercept is (2, 0)
What is the slope–intercept form of a straight line equation?The slope–intercept form of a straight line equation is the form; y = m•x + c, which shows (indicates) the slope, m and the y–intercept, c
The given equation is; 6•x - 2•y = 12
Required;
To rewrite the equation in the slope–intercept form of a straight line equation, y = m•x + c
Solution;
6•x - 2•y = 12
Rearranging the above equation gives;
6•x - 12 = 2•y
From the symmetric property of equality, we have;
2•y = 6•x - 12
Dividing both sides of the equation by 2 gives;
y = (6•x - 12) ÷ 2 = 3•x - 6
Therefore;
y = 3•x - 6
The above equation is in the slope–intercept form, y = m•x + c
Where;
The slope, m = 3The y–intercept, c = -6Which gives the y–intercept as the point (0, -6)
The table of values can therefore be completed as follows;
x; -2, -1, 0, 1, 2
When x = -2, y = 3×(-2) - 6 = -12
When x = -1, y = 3×(-1) - 6 = -9
At the point, x = 0, y = 3×(0) - 6 = -6
The point where x = 1, gives; y = 3×(1) - 6 = -3
At the point where x = 2, y = 3×(2) - 6 = 0
Which gives the completed table as follows;
x; -2, -1, 0, 1, 2
y; -12, -9, -6, -3, 0
The x–intercept is the point where the y–value is 0, which from the completed table gives the x–intercept as the point (2, 0)
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Answer and if you can put an explanation that would ben nice.
Explanation not necessary
The answer is b
In this image is the explanation, just click on it
Help Please
Which of the following is equivalent to −4(2x + 3y) − 6(3q + 7p)?
A. −8x + 12y − 18q + 42p
B. −8x − 12y − 18q − 42p
C. −8x + 3y − 18q + 7p
D. −24(2x + 3y − 3q + 7p)
Answer: B. −8x − 12y − 18q − 42p
I need to know the answer please…
i need help with this
In the expression in order to least to greatest is,
[tex](\frac{1}{6} )^{6}[/tex] < [tex]6^{-3}[/tex] < [tex]\frac{6^{6} }{6^{2} }[/tex] < [tex]\frac{6^{12} }{6^{6} }[/tex] by using [tex]\frac{x^{a} }{x^{b} } = x^{a-b}[/tex] formula.
Based on the given condition,
⇒ [tex]\frac{6^{12} }{6^{6} }[/tex] [tex]\frac{6^{6} }{6^{2} }[/tex] [tex]6^{-3}[/tex] [tex](\frac{1}{6})^{6}[/tex]
Negative numbers grow as they get closer to 0 and positive numbers grow as they distance from 0
First classify numbers as positive and negative
We can use formula,
[tex]\frac{x^{a} }{x^{b} } = x^{a-b}[/tex]
We can write,
⇒ [tex]\frac{6^{12} }{6^{6} }[/tex] = [tex]6^{12-6}[/tex]
[tex]\frac{6^{12} }{6^{6} }[/tex] = [tex]6^{6}[/tex]
⇒ [tex]\frac{6^{6} }{6^{2} }[/tex]= [tex]6^{6-2}[/tex]
[tex]\frac{6^{6} }{6^{2} }[/tex] = [tex]6^{4}[/tex]
⇒ [tex]6^{-3}[/tex] = [tex]6^{-3}[/tex]
⇒ [tex](\frac{1}{6} )^{6}[/tex] = [tex]\frac{1}{6^{6} }[/tex] = [tex]6^{-6}[/tex]
So,
[tex]\frac{6^{12} }{6^{6} }[/tex] = [tex]6^{6}[/tex]
[tex]\frac{6^{6} }{6^{2} }[/tex] = [tex]6^{4}[/tex]
[tex]6^{-3}[/tex] = [tex]6^{-3}[/tex]
[tex](\frac{1}{6} )^{6}[/tex] = [tex]6^{-6}[/tex]
We can write,
[tex]6^{-6}[/tex] < [tex]6^{-3}[/tex] < [tex]6^{4}[/tex] < [tex]6^{6}[/tex]
Then,
[tex](\frac{1}{6} )^{6}[/tex] < [tex]6^{-3}[/tex] < [tex]\frac{6^{6} }{6^{2} }[/tex] < [tex]\frac{6^{12} }{6^{6} }[/tex]
Therefore,
In the expression in order to least to greatest is,
[tex](\frac{1}{6} )^{6}[/tex] < [tex]6^{-3}[/tex] < [tex]\frac{6^{6} }{6^{2} }[/tex] < [tex]\frac{6^{12} }{6^{6} }[/tex] by using [tex]\frac{x^{a} }{x^{b} } = x^{a-b}[/tex] formula.
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Derek is 70 and his granddaughter
Vanessa is 5.
In how many years will Vanessa be ¹/6 Derek's
age?
Two opposite numbers are 10 units away from each other. What are the numbers?
Answer:
-5 and 5
Step-by-step explanation:
5 + 5 = 10 and 5 and -5 are 10 spaces away from each other, hope this helps! :)
Answer:
5, -5
Step-by-step explanation:
5 = 5 units away from 0
-5 = 5 units away from zero
5 units + 5 units = 10 units
-5 to 5 = 10 units apart 5 and -5 are opposites of each other because positive is the opposite of negative and vice versa
hope this helped :) ^^
Gabriel is deigning a rectangular planter box for their garden.It needs to cover an area of 15 1/2 m2 . Gabriel wants it to be 7 3/4 m long. How wide does the planter box need to be?
The width of the planter box will be 2 meter.
What is mean by Rectangle?
A quadrilateral with four right angles is called a Rectangle.
The area of a rectangle is defined as length times width.
Given that;
The area of a rectangular planter box = [tex]15\frac{1}{2}[/tex] m²
The length of cover to cover a rectangular garden = [tex]7 \frac{3}{4}[/tex] meter
Since, The area of a rectangle = Length x Width
Let the width of planter box = x meter.
Then, We can substitute all the values we get;
The area of a rectangular planter box = Length x Width
⇒ [tex]15 \frac{1}{2} = 7 \frac{3}{4} * x[/tex]
Change into improper fraction as;
⇒ 31/2 = 31/4 × x
Solve for x as;
⇒ 31/2 = 31/4 × x
Divide by 31/4;
⇒ x = 31/2 × 4/31
⇒ x = 4/2
⇒ x = 2 meter
Thus, The width of the planter box will be 2 meter.
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Tell whether each statement about 20 and 30 is true or false the greatest common factor is 5 true or false 10 is a common multiple true or false the least common multiple is 60. 2 is a common factor true or false
The true statements are:
The greatest common factor of 20 and 30 is 10.
The least common multiple of 20 and 30 is 60.
The common factor is 2.
Consider the numbers 20 and 30,
The factors of 20 are 1, 2, 3, 4, 5, 10, and 20.
The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.
The common factors are 1, 2, 3, 5, 6, 10 and hence the greatest common factors are 10.
And, 2 is a common factor.
The multiples of 20 are 20, 40, 60, 80, 100, ....
The multiples of 30 are 30, 60, 90, 120, ...
Therefore, the least common multiple is 60.
The GCF and LCM of 20 and 30 are 10 and 60 respectively. The common factor is 2.
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Lisa is saving for college. The account is modeled by the function:
, when x represents how many years she has saved.
Xavier is also saving college. His account is modeled by this table:
x 0 1 2 3
g(x) 200 270 364.5 492.08
Answer the following questions:
The amount Lisa has in the account after 3 years is; 488.28.
The amount Xavier has in the account after 3 years is; 492.08.
The positive difference in their account after 3 years is; 4.20.
What is the positive difference in their account after 3 years?It follows from the task content that Lisa savings as described is modelled by the function;
F (x) = 250(1.25)^x
1. It therefore follows that the amount Lisa would have in the account after 3 years would be;
F(3) = 250 (1.25)³
F(3) = 250 × 1.953125
F(3) = 488.28125.
Hence, when rounded to the nearest hundredth; we have; F(3) = 488.28.
2. Also, the amount Xavier has after 3 years can be determined from the table as; 492.08.
3. The positive difference can therefore be determined as; | 492.08 - 488.28 | = 4.20.
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Solve this system of linear equations using substitution.
{(3x+2y=10),(2x+y=1):}
A (-8,17)
B(-4,17)
C(-8,6)
D (8,17)
[tex]y = 1 - 2x.....(3) \\ \\ 3x + 2(1 - 2x) = 10 \\ 3x + 2 - 4x = 10 \\ 3x - 4x = 10 - 2 \\ - x = 8 \\ \frac{ - x}{ - 1} = \frac{8}{ - 1} \\ x = - 8 \\ \\ \\ y = 1 - 2( - 8) \\ y = 1 + 16 \\ y = 17[/tex]
NOTE I FIRST DERIVED THE THIRD EQUATION FROM THE SECOND EQUATION BY MAKING Y THE SUBJECT OF THE EQUATION AND SUBSTITUTED THE DERIVED EQUATION TO THE FIRST EQUATION BY PLUGGING IN 1-2x IN THE PLACE OF Y AND SOLVED SO ON.
(-8,17)
OPTION A IS THE ANSWER.
GOODLUCK
Answer:
A
Step-by-step explanation:
3x + 2y = 10 → (1)
2x + y = 1 ( subtract 2x from both sides )
y = 1 - 2x → (2)
substitute y = 1 - 2x into (1)
3x + 2(1 - 2x) = 10
3x + 2 - 4x = 10
- x + 2 = 10 ( subtract 2 from both sides )
- x = 8 ( multiply both sides by - 1 )
x = - 8
substitute y = - 8 into (2)
y = 1 - 2(- 8) = 1 + 16 = 17
solution is (- 8, 17 )
jose bought 2 hammers and “y” number of tarps. The total cost was $87. How many tarps did he buy
Answer: He bought y tarps
Step-by-step explanation: this confusing
7% turned into a fraction?
Percentages can be expressed as decimals.
7% = 1 x 0.07
Decimals can be expressed as fractions.
7% = 1 x 7/100
7% = 7/100
Answer:
7/100
Step-by-step explanation:
Percentage is out of 100 - Denominator
7% of 100 7-Numerator
7/100
the average score of 100 students taking a statistics final was 72 with a stanndard deviation of 7. assuming a normal distribution, what is the probability that a student scored greater than 52
The probability of a student scoring higher than 52 is 0.0021.
When the data are normally distributed and the mean and standard deviation are known. Calculate the percentage of data that is greater than or less than a certain number. The standard normal distribution and the Z-score or the standard normal distribution for this.
In the question, the mean and standard deviation are provided as μ = 72 and σ = 7
It has also been said that the data is typically dispersed.
Determine the proportion or percentage of data that is greater than 52. The Z-Score, also known as the standard normal variate, is defined as:
Z = (x - μ) ÷ σ
Calculate the requisite Z-score by substituting the values of μ, σ, and x:
Z = (72 - 52) ÷ 7
Z = 2.857
The area under the curve is approximately 0.0021 on the right side of the Z value.
As a result, the probability of a student scoring more than 52 is 0.0021.
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Abigail works as a salesperson at an electronics store and sells phones and phone accessories. abigail earns a $14 commission for every phone she sells and a $3.50 commission for every accessory she sells. on a given day, abigail made a total of $129.50 in commission and sold 7 more accessories than phones. write a system of equations that could be used to determine the number of phones sold and the number of accessories sold. define the variables that you use to write the system.
The number of phones sold and the number of accessories sold will be 6 and 13 respectively.
What is the equation?An equation is a statement that two expressions, which include variables and/or numbers, are equal. In essence, equations are questions, and efforts to systematically find solutions to these questions have been the driving forces behind the creation of mathematics.
It is given that Abigail earns a $14 commission for every phone she sells and a $3.50 commission for every accessory she sells. on a given day, Abigail made a total of $129.50 in commission and sold 7 more accessories than phones
Suppose the number of accessories and phone be a p respectively.
If he sold 7 more accessories than phones the obtained equation is,
a = p+7
If Abigail receives a commission of $14 for each phone sold and a commission of $3.50 for each accessory sold. Abigail earned a total of $129.50 in commission on a certain day; the resulting calculation is
⇒3.5a + 14p = 129.50
⇒3.5(p+7) + 14p = 129.5
⇒3.5p + 24.50 + 14p = 129.5
⇒17.5p = 105.0
⇒p = 6
Substituting the value of p in the equation we get, a = 13
Thus, the number of phones sold and the number of accessories sold will be 6 and 13 respectively.
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Need some help can anyone offer a hand?
∠SQU ≅ ∠VQT by the Vertical angles theorem.
From the figure, lines UV and WZ are parallel.
We need to arrive at the conclusion that angle VQT is congruent to angle WRS. i.e., ∠WRS ≅ ∠VQT.
So first we have the angles ∠SQU and ∠VQT. Both are vertical angles or opposite angles at Q.
The Vertical Angles Theorem states that two vertical angles are congruent to each other.
Thus, ∠SQU ≅ ∠VQT by the Vertical angles theorem.
Then we have the angles ∠SQU and ∠WRS. They are corresponding angles from the figure.
The Corresponding Angles Theorem states that two corresponding angles are congruent to each other.
Thus, ∠SQU ≅ ∠WRS by The Corresponding Angles Theorem.
Finally, ∠VQT ≅ ∠WRS by the Transitive property of Equality .
The transitivity property can be defined as if a ≅ b and b ≅ c, then a ≅ c.
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PLS HELP I ONLY HAVE 1 MIN 20 PTS!!! NO FAKE ANSWERS (MULTIPLE ANSWER)
Answer:
Step-by-step explanation:
3/4 first is correct
third one and 4th one
4. Which value for x makes the sentence true?
3x - 8 = 16
x=
Answer: x = 8
Step-by-step explanation:
This is a very, very, very simple algebra problem. I assume you know the basics of algebra.
First, move 8 to the other side:
3x = 24
divide by 3 on both sides
x = 8
There! Would it kill you to give me a quartic question to spice my life up?
2.83 in expanded form
Answer:
2+0.8+0.03
Step-by-step explanation:
Which number is prime?
77?
57?
63?
59?
What is the constant of variation for the quadratic variation?
7y = 0.28x²
A. 0.04
B. 0.28
C. 0.98
D. 7
Answer:
A.004
Step-by-step explanation:
I been Doing this for some Years And I knw it cuz I go on it EvenyDay
M measures 12°
(a) Find the supplement of M.
(b) Find the complement of M.
168° is the supplementary angle of 12°.
The complement of the initial angle, which is 12 degrees, is 90 – 12 degrees, or 78 degrees.
What is an example of a supplementary angle?
Angles that add up to 180 degrees are referred to as supplementary angles. For instance, angle 130° and angle 50° are complementary angles since the sum of these two angles is 180°.
Two angles are said to be supplementary angles if they add up to 180 degrees. For example, if ∠A + ∠B = 180°, then ∠A and ∠B are called supplementary angles. Supplementary angles always form a straight angle (180 degrees) when they are put together.
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If the original angle is 12 degrees, then 90-12 = 78 degrees is the complement of that angle. Therefore, the angles of 12 degrees and 78 degrees are complimentary.
What is supplemetary and complementary angles?Two angles are said to be supplementary angles because they combine to generate a linear angle when their sum is 180 degrees. When two angles add up to 90 degrees, however, they are said to be complimentary angles and together they make a right angle.
complementary angles have a sum of 180 degrees. Supplementary and complementary angles can share a vertex or side but are not need to be near to one another.
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1 to the power of 2 + 2 to the power of 3 + 3 to the power of 4
Answer: 90
Step-by-step explanation:
Which expressions are equivalent to the one below? Check ALL that apply. log2 - log8
using properties of log (a/b) = log (a) - log (b)
in this case log 2 - log 8 = log (2/8) = log (1/4)
first option
in a recent survey, % of the community favored building a police substation in their neighborhood. if 14 citizens are chosen, find the probability that exactly of them favor the building of the police substation. round the answer to the nearest thousandth.
The probability that exactly 5 of them favor the building of the police substation is 0.0258.
What is probability?
Probability theory is the mathematical underpinning that makes it possible to analyze random events in a way that makes sense.
The probability of an occurrence is a quantitative measure of how probable it is that a given event will transpire.
This number is always between 0 and 1, where 0 indicates impossible and 1 indicates confidence.
Given that,
n=14, p=0.63
P(x=5) = [tex]^{14}C_5(0.63)^5(0.37)^9[/tex]
=0.0258
The probability that exactly 5 of them favor the building of the police substation is 0.0258.
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Someone please help me!!!
answer the question in the photo!
Part a: 4 = 4
Each real number is equal to itself, based on the reflexive property of equality. We can convey it mathematically as x = x for any arbitrary real number x.Thus, 4 = 4 is a Reflexive property of equality.
Part b: -3 = x and x = -3
Whenever a real number x is equal to a real number y, we can say that y is equal to x, according to the symmetric property of equality. This property can be stated as follows: if x = y, then y = x.Thus, -3 = x and x = -3 is the symmetric property of equality.
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solve this equation
9+7n=-2+2(n-7)
Estimate Jupiter’s mass using a one-digit whole number times a power of 10. Be sure to include units.
The most appropriate choice for exponent will be given by-
Mass of jupiter = 1.898 [tex]\times[/tex] [tex]10^{27}[/tex] kg
What are exponent?
Exponent tells us how many times a number is multiplied by itself.
For example : In [tex]2^4 = 2 \times 2 \times 2 \times 2[/tex]
Here, 2 is multiplied by itself 4 times.
If [tex]a^m = a\times a \times a\times....\times a[/tex] (m times), a is the base and m is the index.
The laws of index are
[tex]a^m \times a^n = a^{m+n}\\\\\frac{a^m}{a^n} = a^{m - n}\\\\a^0=1\\\\(a^m)^n = a^{mn}\\\\(\frac{a}{b})^m =\frac{a^m}{b^m}\\\\a^{-m} = \frac{1}{a^m}[/tex]
Here,
Mass of jupiter = 1.898 [tex]\times[/tex] [tex]10^{27}[/tex] kg
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desribe a sequence of trransformations that take isoceles trapezoid 7 degrees to its image
From T to T', there is a 90 degree counterclockwise transformation.
Transformation entails shifting a shape's location.
A figure is transformed when it is moved from its original location to another.
Angle of Rotation: The extent of the angle at which a figure is rotated. The common rotational angles are 45°, 90°, and 180°. A transformation that does not alter the size of a figure is called an isometric transformation.
There is a right angle between the isosceles trapezoid T and its image T' (i.e. 90 degrees)
To the right of T is Figure T'.
This implies:
Trapezoid T will reach T' if a 90 degree counterclockwise turn is made through point D.
Therefore, a 90 degree counterclockwise transformation through point D is necessary.
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Arianys broke a cell sample into 99 batches, each weighing 6.6\times 10^{-6}6.6×10 −6 grams. How much did the original sample weigh? Use scientific notation to express your answer.
Arianys broke a cell sample into 99 batches each weighing 6.6 x 10⁻⁶, the weight of the original cell sample is 653.4 x 10⁻⁶ grams which when expressed in scientific notation is 6.534 x 10⁻⁴ grams
Arianys broke a cell sample into 99 batches each weighing 6.6 x 10⁻⁶
If the weight of each batch of the original cell sample is 6.6 x 10⁻⁶
The weight of 99 batches = 6.6 x 10⁻⁶ x 99 grams
The weight of 99 batches = 653.4 x 10⁻⁶ grams
The weight of the original cell sample is 653.4 x 10⁻⁶ grams
Scientific notation is a way of expressing numbers in decimal form that are too large or too small. The absolute value of the coefficient of any number is greater than or equal to 1 but it should be less than 10
As per scientific notation it can be written as 6.534 x 10⁻⁴ grams
Therefore, if Arianys broke a cell sample into 99 batches each weighing 6.6 x 10⁻⁶, the weight of the original cell sample is 653.4 x 10⁻⁶ grams which when expressed in scientific notation is 6.534 x 10⁻⁴ grams
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