At a point on the ground 15ft from the base of a tree, the distance to the top of the tree is 1 ft more than 2 times the height of the tree. Find the height of the tree.
The height of the tree is 8ft.
Given that,
A right triangle has two legs, which are height and (15 feet) from the base.
The hypotenuse is 15 feet from the base to the top of the tree.
or 2(height) + 1ft.
Let us take a as height of the tree; b=distance from base=15 ft; c=hypotenuse=2a+1ft
The area of the square created on the hypotenuse of any right triangle is equal to the sum of the areas of the squares formed on the triangle's legs:
[tex]a^{2} +b^{2} =c^{2}[/tex]
[tex]a^{2} +(15ft)^{2}=(2a+ 1ft) ^{2}[/tex]
[tex]a^{2} +225ft^{2}=4a^{2} +4a+1\\ 0=3a^{2}+4a-224\\ 0=(a-8)\times(3a+28)\\[/tex]
Now,
a-8=0 (or) 3a+28=0
a=8 (or) a=[tex]\frac{-28}{3}[/tex]
Therefore, The height of the tree is 8ft.
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A water molecule has a mass
of 3 x 10-29 kg. A bottle
contains 1.7x 1028 molecules of water. Calculate the mass
of water in the bottle.
Answer: 5.1 x 10⁻¹ kg
To find the mass of water in the bottle, we can multiply the mass of one molecule by the number of molecules.
Dimensional AnalysisBy using dimensional analysis, or by cancelling out units, we can figure out what to multiply or divide. Dimensional analysis uses fractions written with units. Dividing the same unit cancels out units we don't need.
To solve:
Figure out what units we need in the answer.Write the fractions using given information from the question.Put the fractions in an equation.Cancel out units to solve.Units we need in the answerWe are looking for the mass of H₂O in the bottle.
The units we need in the answer are: [tex]\displaystyle{ \frac{kg}{bottle} }[/tex]
Write fractions with given informationThe question tells us:
the mass of one H₂O molecule is 3 x 10⁻²⁹ kg: [tex]\displaystyle{ \frac{(3 * 10^{-29}\ kg)}{1\ H_{2}O}}[/tex]the bottle has 1.7 x 10²⁸ molecules of H₂O: [tex]\displaystyle{ \frac{(1.7*10^{28}\ H_{2}O)}{1\ bottle}}[/tex]Write an equationOur equation puts the fractions in an equation, making them multiply to get the units we need.
[tex]\displaystyle{ \frac{(3 * 10^{-29}\ kg)}{1\ H_{2}O} \ *\ \frac{(1.7*10^{28}\ H_{2}O)}{1\ bottle}}\ =\ \frac{kg}{bottle}[/tex]
Why is this equation correct?
Remember that dividing the same unit will cancel them out. In the equation, we have H₂O in a numerator and H₂O in a denominator. So, H₂O will cancel out. In the end, we will only have kg in the numerator and bottle in the denominator.
Cancel out units to solveTake the equation we wrote. Now that we know our equation gives us the right units, we only need the front part.
[tex]\displaystyle{ \frac{(3 * 10^{-29}\ kg)}{1\ H_{2}O} \ *\ \frac{(1.7*10^{28}\ H_{2}O)}{1\ bottle}}[/tex]
Cancel out H₂O:
[tex]=\displaystyle{ \frac{(3 * 10^{-29}\ kg)}{1} \ *\ \frac{(1.7*10^{28})}{1\ bottle}}[/tex]
Simplify the equation:
[tex]=\displaystyle{ \frac{(3 * 10^{-29})*(1.7*10^{28})\ kg}{1\ bottle}[/tex]
[tex]=\displaystyle{ \frac{3 * 10^{-29}*1.7*10^{28}\ kg}{1\ bottle}[/tex]
Use exponent rules to multiply 10⁻²⁹ with 10²⁸:
[tex]=\displaystyle{ \frac{3 *1.7*10^{28-29}\ kg}{1\ bottle}[/tex]
[tex]=\displaystyle{ \frac{3*1.7*10^{-1}\ kg}{1\ bottle}[/tex]
Multiply within the numerator to simplify:
[tex]=\displaystyle{ \frac{5.1*10^{-1}\ kg}{1\ bottle}[/tex]
∴ The mass of water in the bottle is 5.1 x 10⁻¹ kg.
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If 6 is added to the square of a positive number, the sum is 31. Find the number.
Answer:
The number is 5.
Step-by-step explanation:
Given: If 6 is added to the square of a positive number, the sum is 31. Find the number.
Step-by-step solution:
Step 1: Formula
We know that the square of a positive number plus 6 is equal to 31. In a formula, it would look like [tex]\sqrt{x}[/tex] + 6 = 31 where x replaces the positive number.
Step 2: Subtract 6
We need to get x by itself to find the number. Subtract 6 on both sides to cancel it out and move it to the left
so, [tex]\sqrt{x}[/tex] + 6 -6 = 31 -6 = [tex]\sqrt{x}[/tex] = 25
Step 3: Square Root of 25
The Square Root of 25 is 5. We know this because 5*5 = 25.
Therefore, the number is 5.
I have a few math problems I need help with so they're all in picture form (and in separate questions). I'd appreciate it if they were explained to me, but it is not a requirement.
Answer:
In those questions you asked, we can use equation to find the answer but the 1st question's x did not get an exact value so you can do no solution. Thank you
Two in-line skaters, Nell and Kristin start from the same point and skate in the same direction.Nell skates at 8 mile per hour and Kristin skates at 10 miles per hour. if they can keep up the pace for 4.5 hours, how far apart will they be at the end of the time?
At the end of 4.5 hours, Nell and Kristin will be at a distance of 9 miles apart.
Here, we are given that Nell and Kristin start from the same point and skate in the same direction.
Speed of Nell = 8 miles per hour
Time for which he skates = 4.5 hours
Thus, the distance travelled by Nell will be-
Distance = Speed × time
= 8 × 4.5
= 36 miles
Similarly, Speed of Kristin = 10 miles per hour
Time for which he skates = 4.5 hours
Thus, the distance travelled by Kristin will be-
= 10 × 4.5
= 45 miles
Thus, the distance between them will be-
45 - 36 = 9 miles.
Thus, at the end of 4.5 hours, Nell and Kristin will be at a distance of 9 miles apart.
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Nell and Kristin, two inline skaters, begin at the same location and move in the same direction. While Kristin skates at 10 miles per hour, Nell skates at 8 miles per hour. 5 miles apart will they be at the conclusion of the 4.5 hours if they maintain their current rate.
Distance =Rate x time ....... d=rt
The difference in the distances traveled will be how far apart they are 12(2.5)-10(2.5)=2(2.5)=5 miles apart
Note that you can find their individual distances and subtract those to get the same answer:
Nell: distance =10(2.5)=25 miles
Kristin: distance =12(2.5)=30$ miles
30-25=5 miles apart
Distance is the sum of an object's movements, regardless of direction. Distance may be defined as the amount of space an item has covered, regardless of its beginning or finishing position.
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Circle all rational numbers from the list below.
EXPLAIN how you know if they are rational or irrational.
2.0
1 7/8
π
√16
6.39
√5
3.45_
Answer:√16 and 6.39 are rational numbers because they can be expressed as a ratio and a fraction.
Match the solution to its expression.
3/4*1/4 = 3/16
3/7*1/2 = 6/7
the base is of this pyramid is a square. how do you find the area of a square.
Answer:
If you're just looking for the area of the square, then it's 49m. If you want the surface area of the whole pyramid, it's 189.
Step-by-step explanation:
For just the square, it's easy: Multiply the side lengths. 7x7 is 49.
For the surface area, find the area of the square (49) and the four triangles. To find the areas of the triangles, multiply the base (bottom, here it's the side length of the square) by the height (in this case 10m) and divide by 2. 7x10=70 and 70/2=35. Now since there are 4 triangles, multiply that by 4. 35x4=140, and now add the area of the square; 140+49=189. I don't think you wanted the surface area though, but I hope this helps!
The container that holds the water for the football team is 2/5 full. After pouring in 6 gallons of water, it’s 7/10 full. How many gallons can the container hold
Answer:
20
Step-by-step explanation:
We need to determine what fraction of the container 6 gallons is. We know the difference between 2/5 and 7/10 is 6 gallons, so all we have to do is solve for 7/10 - 2/5. The common denominator is 10, as both 5 and 10 are factors of 10. 10/5 is 2, so we multiply the numerator of 2/5 by 2.
7/10 - 4/10 = 3/10
We now know that 6 gallons is 3/10 of the container. Now we solve.
10/3 = 3 1/3
So:
6 x 3 1/3 = 20
Answer:
20 gallons.-
Step-by-step explanation:
Let x be the number of gallons that the container holds.
The we have the equation
2/5 x + 6 = 7/10 x
Multiply through be 10
4x + 60 = 7x
3x = 60
x = 20.
12. Consider the following hypothetical basketball records for Spelman and Morehouse
Colleges.
Home games
Away games
a.
Spelman College
10 wins, 19 losses
12 wins, 4 losses
C.
Morehouse College
9 wins, 19 losses
56 wins, 20 losses
Give numerical evidence to support the claim that Spelman College has a better team
than Morehouse College.
b. Give numerical evidence to support the claim that Morehouse College has a better team
than Spelman College.
Which claim makes more sense and why?
The claim made by Morehouse college make more sense as their ratio of winning a game than losing it is higher.
a) In home games, the Spelman college won 10 wins whereas Morehouse College won only 9 games.
So, Spelman College has a better team than Morehouse College in home games.
b ) In Away games, Morehouse College won 56 games whereas Spelman College only won 12 games. So, Morehouse College has a better team than Spelman College in Away games.
c ) In total, Spelman College has won 22 games and lost 23 games. The ratio of win is to lose is 22 : 23 that is it won 0.96 games against it lost 1 game. In total, Morehouse College won 65 games and lost 39 games. The ratio of win it to lose is 65 : 39 that is it won 1.67 games against the 1 game it lost.
So, the claim of Morehouse College makes more sense.
Therefore, we get that the claim made by Morehouse college make more sense as their ratio of winning a game than losing it is higher.
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Solve the system using the substitution method.
x - 7y = 23
8x - y = 74
Answer:
y = -2
x = 9
Step-by-step explanation:
x - 7y = 23
8x - y = 74
-----------------
Extra:
(x - 7y = 23) -8
-8x + 56y = -184
-----------------------
-8x + 56y = -184
8x - y = 74
--------------------------
55y = -110
÷55 ÷55
-------------------
y = -2
x - 7y = 23
x - 7(-2) = 23
x + 14 = 23
-14 -14
-------------------
x = 9
I hope this helps!
Determine whether the sequence converges or diverges. How do you know if it converges or diverges? If it converges, give the limit. (7)
The sequence converges as the degree of the numerator and denominator are the same.
How to find the limit of a rational equation
In this question we have the case of a rational equation representing a sequence, of which we must determine if the sequences converges and diverges. An useful form to prove the convergence of rational equations is using limit and algebra properties:
(5 · n - 1) / (n + 1)
[(5 · n - 1) / (n + 1)] · 1
[(5 · n - 1) / (n + 1)] · (n / n)
[(5 · n - 1) · n] / [(n + 1) · n]
[(5 · n - 1) / n] / [(n + 1) / n]
(5 - 1 / n) / (1 + 1 / n)
As n → + ∞, then 1 / n → 0 and the limit of (5 · n - 1) / (n + 1) is 5. Thus, the sequence converges as the degree of the numerator and denominator are the same.
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eight more than the quotient of a number and 4 is equal to 2
The algebraic expression from the given statement is 8+x/4=x
Given the statement is the "eight more than the quotient of a number and 4 is equal to 2."
In mathematics, an expression is defined as a set of numbers, variables, and operations that are formed according to context-dependent rules.
Lat us assume the number be x.
In the given statement the quotient of a number and 4 means x/4.
And eight more than the quotient of a number and 4 means 8 is the sum of quotient of a number and 4 and it can be written as
8+x/4 ......(1)
As given in the statement eight more than the quotient of a number and 4 is equal to 2 means we want to equate equation (1) with 2 and written as
8+x/4=2
Hence, the algebraic expression from the given statement "eight more than the quotient of a number and 4 is equal to 2" is 8+x/4=2.
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Explain how negative powers of 10 can be
helpful when writing and comparing small
numbers.
Negative powers of 10 helpful when writing and comparing small numbers.
Small numbers (numbers between 0 and 1) are written with a negative power of ten. The specific power of 10 indicates just how big or how small the number is. You can compare two numbers written in scientific notation by looking at their powers of 10. The number with the greater power of 10 will be the greater number.
When comparing small numbers, write these numbers in scientific notation (only one non-xero digit must be before point) and then
if the powers of 10 are the same in compared numbers are the same, just compare the numbers which are multiplied by these powers of 10. For example, 3.6× 10∧-6 is greater than 2.6× 10∧-6 because powers are the same (-6) and 3.6 is greater than 2.6if the powers are different, then the smaller is power, the smaller is number (number with the smaller negative power has more places after decimal point). For example,3.6× 10∧-6 is greater than 2.6× 10∧-7 because -6 > -7.10 to the power of negative 7 is equal to 0.0000001.For example- 0.00003450000
3.45 x 10 -5
3.450000 x 10 -5
0.345 x 10 -4
0.3450000 x 10 -4
Hence it is prooved that negative powers of 10 helpful when writing and comparing small numbers.
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Find the product of 2(5) and (2) in simplest form. Also, determine whether the result is irrational or rational and explain
The product of 2√5 and √2 is 2√10.
And, The result is an irrational number.
Here,
We have to find the product of 2√5 and √2.
And, also determine the result is a rational or irrational number.
What is Irrational number?
A number which cannot be written in the form of fraction p/q, is called an Irrational number.
Now,
The product of 2√5 and √2 is;
⇒ 2√5 * √2 = 2√5*2 = 2√10
Hence, The product of 2√5 and √2 = 2√10.
Since, √10 is an irrational number because it cannot be written in the form of fraction.
Hence, 2√10 is an irrational number.
Therefore, The product of 2√5 and √2 is 2√10.
And, The result is an irrational number.
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The correct question is this;
Find the product of 2√5 and √2 in simplest form. Also, determine whether the result is irrational or rational and explain
Carton has a length of 2 2/3, a weight of one and 1/8, and a height of 1 1/5. What is the volume of the carton
Rawr!~ Allow me to help you today!
Answer:
[tex]V=\frac{2}{5}[/tex] cubic units!
Step-by-step explanation:
Step 1: Draw the carton (Carton is in the image)
Step 2: Calculate the volume V
[tex]V=LWH[/tex]
[tex]V=\frac{8}{3}[/tex] × [tex]\frac{1}{8}[/tex] ×[tex]\frac{6}{5}[/tex]
[tex]V=\frac{2}{5}[/tex] cubic units
I hope this helps you! <3
Answered by:AshTheNeko
wich numbers are a distance of 2.2 units from -3 on a number line
So, The numbers on a distance of 2.2 units from −3 on a number line are
-3 - 2.2 and -3+2.2
= - (3+2.2) and - (3-2.2)
= -5.2 and - 0.8
In an effort to keep low-wage workers' salaries commensurate with the cost of living, a number of states have amended their constitutions to allow the minimum wage to be adjusted with inflation.
You are the accountant for a company that owns a chain of 16 fast food restaurants in a state which adjusts the minimum wage for inflation. Each restaurant employs 55 workers, each averaging 20 hours per week at the current federal minimum wage, $7.25 per hour.
(d)
How much in "additional wages" will the company have to pay out next year at the adjusted rate (in $)? (Round your answer to the nearest dollar.)
The "additional wages" that the company will have to pay out next year at the inflation-adjusted rate of 7.5% (in $) is $494,208.
What is the inflation-adjusted pay rate?The inflation-adjusted pay rate is the rate after adjusting for inflation.
Data and Calculations:The number of fast-food restaurants = 16
The number of workers per restaurant = 55
Average weekly hours per worker = 20 hours
The number of weeks per year = 52
The current federal minimum wage per hour = $7.25
The total hours worked by the workers = 915,200 hours (16 x 55 x 20 x 52)
Total current wages = $6,635,200 (915,200 x $7.25)
Inflation rate = 7.5%
Inflation-adjusted wage rate = $7.79 ($7.25 x 1.075)
Total wages for next year = $7,129,408 ($7.79 x 915,200)
Additional wages next year = $494,208 ($7,129,408 - $6,635,200)
Thus, the "additional wages" that the company will have to pay out next year at the inflation-adjusted rate of 7.5% (in $) is $494,208.
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A travel agent is booking trips for tourists who travel from New York to Chicago. Tourists have three choices for how to travel from New York to Chicago. They can take an airplane for $350, a bus for $150, or a train for $225. Once they arrive in Chicago, they can travel by van to their hotel for $60 or take a cab for $40. If each option is equally likely to occur, what is the probability that a tourist will spend more than $250 on these 2 legs of the trip?
From the calculation below, the probability that a tourist will spend more than $250 on the 2 legs of the trip is 2/3.
How do we calculate the amount spent using probability?From the question, the number of possible options available and their total amount is as follows:
Airplane and Van = $350 + $60 = $410
Airplane and Cab = $350 + $40 = $390
Bus and Van = $150 + $60 = $210
Bus and Cab = $150 + $40 = $190
Train and Van = $225 + $60 = $285
Train and Cab = $225 + $40 = $265
From the above, it can be observed that we have a total number of 6 different options available and 4 of the options are more than $250.
Therefore, we have:
Probability of spending more than $250 = Number of options that are more than $250 / Total number of different options = 4/6 = 2/3
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f=2+4/5f-3 solve for f
Answer:
f=-5
Step-by-step explanation:
PLS COULD YALL HELP !!?!??
Answer:
x = 11
m<H = 39°
Step-by-step explanation:
m<G = 6x - 15
m<H = 3x + 6
complementary angles = angles that add up to 90
6x - 15 + 3x + 6 = 90
9x = 99
x = 11
m<H = 3x + 6, plug in 11 as x
3 * 11 + 6
= 33 + 6
m<H = 39°
No
5) Determine if the list of numbers is an arithmetic sequence by choosing Yes or No. If Yes
then choose the option with the correct common difference "d."
18, 16, 14, 12, 10,...
Yes, d= -2
Yes, d = -3
Yes, d = 2
Helppp!!
The option yes, d = -2 is correct
The arithmetic sequence or progression is a sequence of numbers such that the difference between the consecutive terms is constant.
The formula of an arithmetic sequence is
[tex]a_{n}[/tex] = [tex]a_{1}[/tex] + (n-1)d
Here the sequence is 18, 16, 14, 12, 10...... which is in arithmetic sequence as there is a common difference between them.
Now we have to find the common difference between the two consecutive terms of the series,
common difference (d) = 16- 18 = 14 - 16 = -2
Therefore the series is an arithmetic sequence and the common difference between them is -2.
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8. Mrs. Shin has a piece of lumber that is 11- inches wide. She plans to split the width of lumber into 3 equal pieces. How wide will each piece be?
the answer is 8
Step-by-step explanation:
just minus 11by 3
Han's younger sister is riding her tricycle in a straight line. After 3 s, her speed is 0.5 m/s. After 5 s, her speed is 1.5 m/s. what is her acceleration during this time?
The acceleration during this time will be 0.5 m/s².
What is Algebra?The analysis of mathematical representations is algebra, and the handling of those symbols is logic.
Han's younger sister is moving straight ahead on her tricycle. Her speed is 0.5 m/s after 3 seconds. Her speed is 1.5 m/s after 5 seconds.
We know that the first equation of motion is given as,
v = u + at
The information is given below.
v = 1.5 m/s, u = 0.5 m/s, and t = 2 s. Then the value of the acceleration will be
v = u + at
1.5 = 0.5 + 2a
2a = 1
a = 1/2
a = 0.5 m/s²
The acceleration during this time will be 0.5 m/s².
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HELP PLEASE!! I WILL GIVE BRAINLYEST
find the x and y intercept
6.8x+8.8y+9.8=0
From the given equation 6.8x+8.8y+9.8=0 the value of x-intercept is -98/68 and value of y-intercept is -98 /88.
As given in the question,
Given equation :
6.8 x +8.8 y +9.8= 0
To get x -intercept substitute y =0 in the equation :
6.8 x+ 8.8 (0) + 9.8=0
⇒6.8x= -9.8
⇒x= 98 /68
To get y -intercept substitute x =0 in the equation :
6.8 (0)+ 8.8y + 9.8 =0
⇒8.8y= -9.8
⇒ y= 98 /68
Therefore, from the given equation 6.8x+8.8y+9.8=0 the value of
x-intercept is -98/68 and value of y-intercept is -98 /88.
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The difference of the squares of two positive consecutive even integers is 84. Find the integers. Use the fact that, if x represents an even integer, then x + 2 represents the next consecutive even integer.
Answer:
The integers are 20 and 22
Step-by-step explanation:
Let the first even integer be x . Then the next consecutive even integer
= x + 2
The relationship between these two integers is the difference of the squares = 84
This means
(x + 2)² - x² = 84
Use the relationship a² - b² = (a + b) (a -b)
Here a = x + 2 and b = x
So
a + b = (x + 2 ) + x = x + 2 + x = x + x + 2 = 2x + 2
a - b = (x + 2) - x = x + 2 - x = x -x + 2 = 2
Therefore
(a + b)(a - b) in terms of x is
(2x + 2) x 2 = 4x + 4
We know this is 84
So 4x + 4 = 84
Subtract 4 on both sides:
4x = 84 -4
4x = 80
x = 20
So the next consecutive even integer = 22
The integers are 20 and 22
Distribute and simplify these radicals.
2√3 (√2+√3)
O 2√5 +6
O2√15
O 30
O6√2+6
probability from a two way table
Answer:
82% or 23/28
Step-by-step explanation:
2+3+18=23
2+3+18+6=28
23/28=
write a polynomial function of least degree with integral coefficients that has the given zeros. -1, 3+3i
The least degree polynomial function with integral coefficients and the given zeros, -1, 3+3i is f(x) = x³ + x².
What do we mean by polynomial function?A polynomial function is one that involves only non-negative integer powers or positive integer exponents of a variable in an equation such as the quadratic equation, cubic equation, and so on. For example, 2x+5 is a polynomial with an exponent of one.So, given zeroes are -1, 3+3i.
We got: x = -1Then, x + 1 = 0.Hence, one factor of the polynomial function is (x+1).Similarly,
We have: 3+3iRewrite as: x = 3 + √-9Square both the sides: x² = 9 - 9So, the other factor is x².So, the polynomial has the function: (x+1)x²
(x+1)x²x³ + x²f(x) = x³ + x²Therefore, the least degree polynomial function with integral coefficients and the given zeros, -1, 3+3i is f(x) = x³ + x².
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Which of these is NOT a rational number?
0.54225422...
0.55555555...
0.3434343...
0.61166117...
0.61166117... is not an rational number as it has no regularly repeating digit or group of digits.
What is rational number?A rational number is one that can be stated mathematically as the ratio or fraction p/q of two integers, p and q, which must both be non-zero. Every integer, including 3/7 (3 = 5/1) and 7 is a rational number.
Boldface Q is typically used to represent the field of rationals, also known as the field of rational numbers, the set of all rational numbers, or "the rationals."
Real numbers are those that make sense. The rational real numbers are those whose decimal expansion either ends after a finite number of digits (for example, 3/4 = 0.75), or eventually starts to repeatedly repeat the same finite sequence of digits (for instance, 9/44 = 0.20454545).
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