Answer:
3x^2 - x - 10
Step-by-step explanation:
Foil (First, outside, inside, last)
3x^2 + 5x + -6x - 10
combine like terms
3x^2 - x - 10
Rosa has $300. Suppose she spends 1/4 of the money on art supplies and 2/5 of the remainder on a pair of boots. How much did she spend on the arts supplies?
How much on the boots? How much money does she have left after makina both purchases?
WILL GIVE BRAINLIST
PLS HELP ASAPPPPP
Let's call the amount of money Rosa spent on art supplies "x". Since Rosa spent 1/4 of her money on art supplies, we can write the equation:
x = (1/4) * 300
x = 75
So Rosa spent $75 on art supplies.
Now we can find out how much money she had left after spending the $75 on art supplies.
remaining = 300 - 75
remaining = 225
And we can find out how much she spent on the boots.
boots = (2/5) * remaining
boots = (2/5) * 225
boots = 90
So Rosa spent $90 on the boots.
Let's call the amount of money Rosa spent on art supplies "A". Since Rosa spent 1/4 of her original $300 on art supplies, we can write the equation:
A = 1/4 * $300
A = $75
So, Rosa spent $75 on art supplies.
Next, let's find out how much money she had left after buying the art supplies. We can subtract the amount she spent on art supplies from her original $300:
$300 - $75 = $225
Now, let's find out how much she spent on the boots. We know that she spent 2/5 of the remainder on the boots, so we can write the equation:
B = 2/5 * $225
B = $90
So, Rosa spent $90 on the boots.
A Gallup poll report revealed that 72% of teens said they seldom or never argue with their friends. Yvonne wonders if this result holds true in her large high school. So she surveys a random sample of 150 students at her school and finds that 96 of them say they rarely or never argue with friends. She uses the data to perform a test of 0:=0.72H0:p=0.72 versus ≠0.72,Ha:p=0.72, where p is the true proportion of teens in Yvonne's school who rarely or never argue with their friends. The test yields a P-value of 0.0291. What conclusion would you make for each of the following significance levels? =0.01α=0.01
We reject the null hypothesis at a significance level of =0.01.
At a significance level of α=0.01, we reject the null hypothesis if the p-value is less than 0.01. In this case, the p-value of 0.0291 is greater than 0.01, so we fail to reject the null hypothesis. Therefore, we do not have sufficient evidence to conclude that the proportion of students in Yvonne's school who rarely or never argue with their friends is significantly different from the national average of 0.72 at a significance level of α=0.01.
In other words, we do not have enough evidence to say that Yvonne's school is significantly different from the national trend of 72% of teens who say they seldom or never argue with their friends.
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Question 1
A store owner want to develop a new snack mix by mixing chocolate and trail mix. How many pounds of
chocolate costing $8.20 per pound should be mixed with trail mix costing $3.40 per pound to create a 23
mixture worth $5.49. (round to the nearest pound)
The number of pounds of chocolate costing $8.20 per pound should be mixed with trail mix is; 18 pounds
How to solve Algebra Word Problems?Let c represent the pounds of chocolate needed.
The weight of the snack mix is c + 23.
Thus, the algebraic expression here is;
8.2c + 3.4(23) = 5.49(c + 23)
8.2c + 78.2 = 5.49c + 126.27
8.2c - 5.49c = 126.27 - 78.2
2.71c = 48.07
c = 48.07/2.71
c ≈ 18 pounds
This value represents the number of pounds of chocolate costing $8.20 per pound should be mixed with trail mix
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Click “show your work” and graph the absolute value function shown below.
f(x) =|x – 4| + 1
The ratio of Green Apple shampoo is 1 fl oz shampoo to 18 fl oz water. How much shampoo is needed to fill a one-gallon container?
Approximately 7.11 fluid ounces of shampoo to fill a one-gallon container.
How to find out how much shampoo is needed to fill a one-gallon container?One gallon is equal to 128 fluid ounces.
The ratio of shampoo to water is 1:18. This means for every 1 fl oz of shampoo, we need 18 fl oz of water.
To find out how much shampoo is needed to fill a one-gallon container, we need to set up a proportion:
1 fl oz shampoo / 18 fl oz water = X fl oz shampoo / 128 fl oz water
To solve for X, we can cross-multiply:
18 fl oz water * X fl oz shampoo = 1 fl oz shampoo * 128 fl oz water
18X = 128
X = 7.11 fl oz shampoo
So, we need approximately 7.11 fluid ounces of shampoo to fill a one-gallon container.
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On November 28, 2024, Shocker receives a $3,450 payment from a customer for services to be rendered evenly over the next three months. Deferred Revenue is credited. Record the adjusting entry for deferred revenue at its year-end of December 31
The adjusting entry for deferred revenue at its year-end of December 31 is $1,150
How to calculate simple interest amount?If the initial amount (also called as principal amount) is P, and the interest rate is R% annually, and it is left for T years for that simple interest, then the interest amount earned is given by:
[tex]I = \dfrac{P \times R \times T}{100}[/tex]
We are given that;
The amount of payment=$3,450
Now,
The adjusting entry for deferred revenue at its year-end of December 31
Debit: Deferred Revenue $1,150 (which is one-third of the total payment received)
Credit: Revenue $1,150 (which represents the portion of the service that has been provided in December)
Therefore, by the given interest answer will be $1,150
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Find the following probabilities based on a standard normal variable Z. Use Table 1. (Round your answers to 4 decimal places.)
a. P(Z > 0.74) b. P(Z ≤ −1.92) c. P(0 ≤ Z ≤ 1.62) d. P(−0.90 ≤ Z ≤ 2.94)
a. 0.7787; b. 0.0228; c. 0.7453; d. 0.9793. The probabilities are found by looking up the Z-value in Table 1 and finding the corresponding probability.
a. To find the probability that Z is greater than 0.74, we look up 0.74 in Table 1 and find the corresponding probability. The probability that Z is greater than 0.74 is 0.7787, which is the area to the right of the Z-value in the table.
b. To find the probability that Z is less than or equal to −1.92, we look up −1.92 in Table 1 and find the corresponding probability. The probability that Z is less than or equal to −1.92 is 0.0228, which is the area to the left of the Z-value in the table.
c. To find the probability that Z is between 0 and 1.62, we look up 0 and 1.62 in Table 1 and find the corresponding probabilities. The probability that Z is between 0 and 1.62 is 0.7453, which is the area between the two Z-values in the table.
d. To find the probability that Z is between −0.90 and 2.94, we look up −0.90 and 2.94 in Table 1 and find the corresponding probabilities. The probability that Z is between −0.90 and 2.
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assume a child is under observation by a researcher. he is given 6 blocks uniquely labeled a, b, c, d, e, and f. the child is equally likely to choose any of the blocks. find the probability the child arranges the blocks in such a way that block f is in either the 1st position or the last position.
The probability the child arranges the blocks in such a way that block f is in either the first position or the last position is found to be 1/3.
The child under observation of the researcher is given 6 unequally labelled block that are number as, a, b, c, d, e and f.
Now, the total number ways in which the child can arrange these block is,
Total possible ways = 6 x 5 x 4 x 3 x 2 x 1
Total possible ways = 720
Now, if the position of the block labelled f is fixed at either 1st or the last position then, total such ways are,
= 2 x 5 x 4 x 3 x 2 x 1
= 240
So, the probability that the child arranges the blocks in such a way that block f is in either the 1st position or the last position,
P = 240/720
P = 1/3
So, the probability is found to be 1/3.
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- A local college will construct a 2-floor building next
year. Each of the 5 classrooms on the 1st floor will
have 20 seats, and each of the 8 classrooms on the
2nd floor will have 35 seats. To comply with the 2010
ADA standard, what is the fewest total number of
wheelchair spaces needed in the 13 classrooms?
F. 4
G. 6
H. 11
J. 13
K. 21
Answer:
According to the 2010 ADA standard, the number of wheelchair spaces required in a classroom depends on the total number of seats in the classroom. Specifically, 1 wheelchair space is required for the first 25 seats, and an additional wheelchair space is required for each 25 seats thereafter.
For the 5 classrooms on the 1st floor, there are a total of 5 x 20 = 100 seats. Therefore, we need 1 wheelchair space for the first 25 seats, and 1 additional wheelchair space for the next 25 seats. So in total, we need 2 wheelchair spaces for the 1st floor classrooms.
For the 8 classrooms on the 2nd floor, there are a total of 8 x 35 = 280 seats. Therefore, we need 1 wheelchair space for the first 25 seats in each classroom, and an additional wheelchair space for the next 25 seats. This means that each classroom needs 2 wheelchair spaces. So in total, we need 8 x 2 = 16 wheelchair spaces for the 2nd floor classrooms.
Therefore, the total number of wheelchair spaces needed in all 13 classrooms is 2 + 16 = 18.
However, it's important to note that the question asks for the "fewest total number" of wheelchair spaces needed. Since we can't have a fraction of a wheelchair space, we need to round up to the nearest whole number. Therefore, the answer is 18 rounded up to the nearest whole number, which is 19.
So the correct answer is not one of the options given. The closest option is K. 21, but that is too high.
Mr. Phil T. Rich had a boat load of money. He decided to leave it to his family. But wanted to make sure his goldfish, Bubbles was taken care of too. He decided to leave 1/2 of his estate to his wife 30,000$ to his daughter, half of what was left over for his butler, Niles half of what was left to bubbles (his goldfish). He realized 8,000$ remaking, so he donated it to charity. What was the original value of his estate? GO STEP BY STEP. WORK BACKWARDS
If Mr. Phil T. Rich had a boat load of money. The original value of his estate is $152,000.
How to find the original value?Let's call the original value of Mr. Phil T. Rich's estate "x".
After he gave 1/2 of the estate to his wife, he was left with x/2.
After he gave 30,000 to his daughter, he was left with x/2 - 30,000.
After he gave half of what was left to his butler Niles, he was left with (x/2 - 30,000)/2.
After he gave half of what was left to Bubbles, he was left with (x/2 - 30,000)/4.
Finally, after he donated the 8,000 to charity, he was left with (x/2 - 30,000)/4 - 8,000.
We know that the final amount after all these transactions was $0, so we can set up an equation to solve for x:
0 = (x/2 - 30,000$)/4 - 8,000
Multiplying both sides by 4 to isolate x/2:
0 * 4 = (x/2 - 30,000) - 8,000 * 4
Expanding the right side:
0 = (x/2 - 30,000) - 32,000
Adding 30,000 and 32,000 to both sides:
30,000+ 32,000 = (x/2)
Dividing both sides by 1/2:
62,000 * 2 = x
Multiplying both sides by 2:
62,000* 2 * 2 = x * 2
Expanding the left side:
62,000* 4 = x
Dividing both sides by 4:
62,000 * 4 / 4 = x / 4
Expanding the left side:
$62,000 = x
Therefore the original value of Mr. Phil T. Rich's estate was $62,000.
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What is 32/34 in simplest form
Answer:
Step-by-step explanation:16/17
Answer:
16/17 is the simplest form
The students in the choir were lined up in rows of 9. The chorimaster rearranged them into equal number of rows and columns. There were more than 10 and fewer than 50 students in the choir. How many sutdents were there in the choir?
Answer: 36
Step-by-step explanation:
It must be a multiple of 9 for them to be in rows of 9 at the beginning therefore it can only be one of 18, 27, 36, 45 as it has to be bigger than 10 and smaller than 50 it cant be 9 or 54
It must also be a square number in order to have an equal amount of rows and column.
So it must be 36 as it is both a multiple of 9 and a square number.
Answer this one question
Answer:
3(m-7)
Step-by-step explanation:
because 3 times the quantity: m-7
Answer:
6. 3 times the quantity m minus 7
O 3(m – 7)Step-by-step explanation:
You're welcome.
if one of these 1 million people is selected randomly, find the following probabilities: (a) p(b1), (b) p(a1), (c) p(a1 | b2), and (d) p(b1 | a1). (e) in words, what do parts (c) and (d) say?
The probability of carrying the AIDS virus in B1 is 0.5%
Probability is a fundamental concept in understanding the ELISA test results for AIDS diagnosis.
The probability of carrying the AIDS virus (B1) can be calculated by dividing the number of people who carry the virus by the total number of people who were tested.
From the contingency table, we can see that the number of people who carry the virus is 5000. Therefore, the probability of carrying the virus can be expressed as:
=> P(B1) = Number of people who carry the virus / Total number of people tested
=> 5000 / 1000000
=> 0.005 or 0.5%
This means that out of 1 million people who were tested, only 0.5% were found to be carrying the AIDS virus.
In this context, probability helps us understand the likelihood of carrying the virus and the accuracy of the test results. It also allows us to estimate the prevalence of the virus in a population and make informed decisions about public health policies and interventions.
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Complete Question:
A common test for AIDS is called the ELISA(enzyme-linked immunosorbent assay) test. Among 1 million peoplewho are given the ELISA test. we can expect results similar tothose given in the following table:
B1: Carry AIDS Virus B2: Do not Carry AIDS Virus Totals
A1: Test 4885 73630 78515
Positive
A2: Test 115 921370 921485
Negative
Totals 5000 995000 1000000
If one of these 1 million people is selected randomly, find the value of P(B1).
wich valule of X makes the equation 3x+5=4x+3 true.
Answer:
x=2
Step-by-step explanation:
We will begin by subtracting both sides by 3
3x+2=4x
We will then subtract both sides by 3x since 3x and 4x are like terms
2=x OR x=2
Hope this helps!
A bookmark is shaped like a rectangle with a semicircle attached at both ends. The rectangle is 15 cm long and 4 cm wide. The diameter of each semicircle is the width of the rectangle. What is the area of bookmark ? Use 3.14 for pie
The area of the bookmark is 72.56 cm².
What is Area of Rectangle?The area of Rectangle is length times of width.
The width of the rectangle is 4 cm, so the diameter of each semicircle is also 4 cm.
Radius of each semicircle is 2 cm.
The area of the rectangle is length x width
= 15 cm x 4 cm
= 60 cm².
The area of each semicircle is (1/2) x π x radius²
= (1/2) x 3.14 x 2² = 6.28 cm².
The total area of the two semicircles is 2 x 6.28 = 12.56 cm².
The total area is 60 cm² + 12.56 cm² = 72.56 cm²
Therefore, the area of the bookmark is 72.56 cm².
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saif made 60 runs in 5 over and anus made 72 runs in 8 over whose performance was better
Saif's performance was better.
What is an expression?An expression contains one or more terms with addition, subtraction, multiplication, and division.
We always combine the like terms in an expression when we simplify.
We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.
Example:
1 + 3x + 4y = 7 is an expression.
3 + 4 is an expression.
2 x 4 + 6 x 7 – 9 is an expression.
33 + 77 – 88 is an expression.
We have,
We will compare the run rates of Saif and Anus.
The run rate is the average number of runs scored per over.
Now,
The run rate of Saif.
= 60 / 5
= 12 runs per over
And,
The run rate of Anus
= 72 / 8
= 9 runs per over
Therefore,
Saif's performance was better.
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Please help I dont understand this thank you
Answer:17x25 i think
Step-by-step explanation:
Find the negation for the following formula: Vx €Z[p(x) ^ q(x)] O Jxez[ p(x) v q(x)] O Jxez[~ p(x) ^ q(x)] O Vxez[ p(x) v q(x)] O for allxelement of straight integer numbers open square brackets tilde p left parenthesis x right parenthesis logical or tilde q left parenthesis X right parenthesis close square brackets
The negation for the formula is Jxez[~p(x) v ~q(x)] ^ Vxez[~p(x) ^ ~q(x)] ^ Vxez[p(x) v ~q(x)]
Negation is a concept used in mathematics to express the opposite or contradictory statement of a given formula.
To find the negation of this formula, we need to apply De Morgan's laws, which state that the negation of a logical conjunction (and) is a logical disjunction (or) with the negation of each of the operands, and vice versa.
First, let's apply De Morgan's laws to the first expression in the formula:
~[Vx €Z[p(x) ^ q(x)]]
The negation of "for all x in Z, p(x) and q(x)" is "there exists x in Z such that ~(p(x) and q(x))," which can be expressed as:
Jxez[~p(x) v ~q(x)]
Now, let's apply De Morgan's laws to the second expression in the formula:
=> ~[Jx €Z[p(x) v q(x)]]
The negation of "there exists x in Z such that p(x) or q(x)" is "for all x in Z, ~(p(x) or q(x))," which can be expressed as:
=> Vxez[~p(x) ^ ~q(x)]
Finally, let's apply De Morgan's laws to the third expression in the formula:
~[Jx €Z[~p(x) ^ q(x)]]
The negation of "there exists x in Z such that not p(x) and q(x)" is "for all x in Z, ~(not p(x) and q(x))," which can be expressed as:
Vxez[p(x) v ~q(x)]
Putting these three expressions together, we get the negation of the original formula as:
=> Jxez[~p(x) v ~q(x)] ^ Vxez[~p(x) ^ ~q(x)] ^ Vxez[p(x) v ~q(x)]
In this case, the original formula involved three functions (p(x), q(x), and logical operators) that were negated using De Morgan's laws to obtain the final expression.
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cans collected each day was
1. The eighth grade class at Robison Middle School is holding a canned food drive. The number of
45, 21, 3, 15, 32, 97, 68, 27, 29, and 52. Explain how to find the interquartile range of cans
collected each day.
The interquartile range, IQR=31.
What is interquartile range?
Interquartile range is a measure of statistical dispersion, which is the spread of the data. The interquartile range is calculated by the difference between the upper and lower quartile. The formula for the interquartile range is given below
[tex]IQR=Q_3-Q_1[/tex]
The given data : 45, 21, 3, 15, 32, 97, 68, 27, 29, 52.
Arrange in ascending order,
3, 15, 21, 27 , 29 ,32 , 45 , 52 , 68 , 97
By dividing in to 2 partition as lower half and upper half from let to right.
Lower half = 3, 15, 21, 27 , 29
Upper half = 32 , 45 , 52 , 68 , 97
Lower Quartile[tex]Q_{1}[/tex] = Median of Lower half of data
⇒[tex]Q_{1}[/tex]= 21 (Median is the middle most term)
Upper Quartile[tex]Q_{3}[/tex] = Median of Lower half of data.
⇒[tex]Q_{3}[/tex]=52 (Median is the middle most term)
Interquartile range = [tex]IQR=Q_3-Q_1[/tex]
= 52-21
= 31
Hence, the interquartile range IQ=31.
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b) Would this circle work for the pony's path? Why or why not? Explain your
answer by using the equation, the location of the stake, and the length of the
rope to support your reasoning. (2 points)
The equation for the pony's path is -
(x - 8)² + (y - 8)² = 128.
What is the equation of a circle?The general equation for a circle is -
(x - h)² + (y - k)² = r²
where (h, k) is the center and (r) is the radius.
Given is that the pony is staked at the coordinate point (8, 8).
The general equation for a circle is -
(x - h)² + (y - k)² = r²
We can write the equation for the pony's path as -
(x - 8)² + (y - 8)² = 128
Therefore, the equation for the pony's path is -
(x - 8)² + (y - 8)² = 128.
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Two buses leave Nashville at the same time traveling in opposite directions. One bus travels at 54mph and the other at 55mph. How soon will they be 490.5 miles apart?
This figure was created with three different rectangles. Find the total area in square centimeters.
The total area for the figure is given as follows:
288 cm².
How to obtain the area of a rectangle?The area of a rectangle is given by the multiplication of the width and the length of the triangle, as follows:
A = lw.
The dimensions of each rectangle are given as follows:
8 cm and 9 cm.9 cm and 12 cm.9 cm and 12 cm.Hence the total area is given as follows:
A = 8 x 9 + 2 x 9 x 12
A = 288 cm².
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The heights of the 228 men who filled out Survey 1 last semester follow the normal curve fairly closely with an average of about 70.5" and a SD of about 2.5 ".
What percent of the students are over 71.5 "?
Do the problem in steps
First convert 71.5 " to a z score.0.40 tries 0/5In the problem above you were given the height and asked to find the percentile. Now, you'll be going in the opposite direction-- you'll be given the percentile (placement on the normal curve) and asked to find the height. In both cases the first step is to find the z-score.
If someone is in the 92 th percentile in height (i.e., is taller than 92 % of the other males in the class), let's work on finding his height. First we'll find the z score.
(Do NOT look up the z score for Area= 92 % on the table b/c the Areas given on the table are MIDDLE areas. 92 th percentile means 92 % to the LEFT, and 8 % in the right tail.)
What is the Z-score corresponding to the middle area?(If the middle area falls between two lines on the table, you may use the closest one. Put Z-score in blank below.)
Now change the z score to a height. (Remember the z score tells how many SD's the value is from the average. So a z score of 2 means the person is 2 SDs above average which would be 2*(2.5") + 70.5".)
What is his height?
The male in the 92nd percentile in height would be 73.75 inches tall.
To find the z-score corresponding to the 92nd percentile, we need to find the value on the standard normal curve such that 92% of the area is to the left of that value.
Using the standard normal table, we can find the z-score closest to 0.92 which is 1.75.
Now that we have the z-score, we can use the following formula to find the height:
height = z-score * standard deviation + average height
So, using the given values of average height (70.5") and standard deviation (2.5"):
height = 1.75 * 2.5 + 70.5
height = 73.75"
Therefore, a male in the 92nd percentile in height would be 73.75 inches tall.
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given the following list of prices (in thousands of dollars) of randomly selected trucks at a car dealership, find the median. 20,46,19,14,42,26,33 provide your answer below: $$median
The median of the randomly selected list of prices of trucks will be 26 thousands dollars.
To find the median , first we need to arrange the list in either ascending form or descending form. Let us arrange the list in ascending form :
List before sorting : 20,46,19,14,42,26,33
List after sorting it in ascending form : 14,19,20,26,33,42,46
Here, the number of observations is 7 which is odd. So, The formula to find the median term from a given number of "Odd" observations in a set is :
⇒ {(n+1) / 2}th term (where n is the number of observations in a set)
⇒({7+1}/2)
⇒8/2
⇒4th term
Therefore, the 4th term in our list of prices of trucks is the median which is 26 thousand dollars
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PQ = 7x + 13, QR = 10x-2, PR = x + 27 what is the range of x-values
The range of x-values is 1 < x < 21.
What is triangle Inequality?According to triangle inequality theorem, for any given triangle, the sum of two sides of a triangle is always greater than the third side.
Given:
We have three sides of Triangle as
PQ = 7x + 13, QR = 10x-2, PR = x + 27.
So, PQ + QR > PR
(7x + 13) + (10x - 2) > x + 27, which simplifies to x > 1
and, PQ + PR > QR
(7x + 13) + (x + 27) > 10x - 2, which simplifies to x < 21
and, QR + PR > PQ
(10x - 2) + (x + 27) > 7x + 13, which simplifies to x > -3
So, the Range of sides is 1 < x < 21.
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Someone help me with these math problems
1. Two lines are MT and UW. 2. Two segments are SX and SY. 3. A pair of congruent angles is ∠QNT and ∠QNM. 4. Four right angles are ∠QNT and ∠QNM. ∠TNY and ∠MNY.
What are obtuse vertical angles?Angles that are both vertical and obtuse are known as obtuse vertical angles. In order to comprehend what obtuse vertical angles are, we must first learn the definitions of each of these kinds of angles.
Angles that cross one other at the junction of two lines are known as vertical angles. There is always an equal amount of vertical angles.
With these definitions, we get that obtuse vertical angles are angles that are in front of one another at the intersection between two lines, and both of which share a comparable measure that is between 90° and 180°.
From the given figure we observe that:
1. Two lines are MT and UW.
2. Two segments are SX and SY.
3. A pair of congruent angles is ∠QNT and ∠QNM.
4. Four right angles are ∠QNT and ∠QNM. ∠TNY and ∠MNY.
5. Two pairs of obtuse vertical angles ∠MSW and ∠UST. ∠VNT and ∠PNM.
6. Two pairs of adjacent supplementary angles are ∠MSW and ∠WST. ∠WNM and WNT.
7. Two pair of complementary angles are ∠TNP and ∠PNY. ∠QNW and ∠WNM.
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Two circles are internally tangent at a point T and have radii of 1 and 3. The maximum possible area for a triangle with one vertex at T, another vertex on the small circle, and the third on the large circle can be expressed in the form a √(b)/c, where a,b, and c are positive integers, b is not divisible by the square of any prime, and a and c are relatively prime. Find a+b+c.
The maximum possible area of such a triangle is $\frac{1}{2} \cdot \sqrt{4 + 2\sqrt{3}} = \frac{\sqrt{12 + 6\sqrt{3}}}{2} = 3\sqrt{3} + 3$, and $a + b + c = 3 + 3 + 3 = \boxed{9}$.
Let O and O' be the centers of the circles with radii 1 and 3, respectively, and let P and Q be points on the small and large circles, respectively, such that TPQ is a triangle. ]
Since the radius of the small circle is 1, we have TP = 1. Let R be the midpoint of PQ, so that TR is the altitude of triangle TPQ from T. Let x = TP = 1, y = TQ, and z = TR. [asy]
unit size(0.6 cm);
pair O, OO, P, Q, R, T;
O = (0,0);
OO = (4,0);
T = (0,3);
P = intersection points(Circle(O,1),Circle(T,2))[1];
Q = intersection points(Circle(OO,3),Circle(T,2))[0];
R = (P + Q)/2;
draw(Circle(O,1));
draw(Circle(OO,3));
draw(T--P--Q--cycle);
draw(T--R);
label("$O$", O, SW);
label("$O'$", OO, SE);
label("$P$", P, NW);
label("$Q$", Q, NE);
label("$R$", R, S);
label("$T$", T, N);
label("$x$", (T + P)/2, W);
label("$y$", (T + Q)/2, E);
label("$z$", (T + R)/2, W);
[/asy]
Then we have TR = z, and by the Pythagorean Theorem in right triangle TPQ we have
\begin{align*}
TQ^2 - 1 &= TR^2 = z^2, \
TQ^2 + 9 &= (TQ + TR)^2 = (TQ + z)^2.
\end{align*}Solving for TQ in the first equation and substituting into the second, we obtain
[(1 + z)^2 + 9 = (1 + z)^2 + 4z^2,]which simplifies to $z^2 - 2z - 2 = 0$. The positive root of this quadratic equation is $z = 1 + \sqrt{3}$, so $TQ = \sqrt{4 + 2\sqrt{3}}$. Then the area of triangle TPQ is
[\frac{1}{2} \cdot TP \cdot TQ = \frac{1}{2} \cdot \sqrt{4 + 2\sqrt{3}}.]Thus the maximum possible area of such a triangle is $\frac{1}{2} \cdot \sqrt{4 + 2\sqrt{3}} = \frac{\sqrt{12 + 6\sqrt{3}}}{2} = 3\sqrt{3} + 3$, and $a + b + c = 3 + 3 + 3 = \boxed{9}$.
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The inequality 2x - 3y < 7 represents the given graph, which is the correct option (B).
What is inequality?Inequality is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are not equal.
As per option (B), we have
2x - 3y < 7
Subtract 2x from both sides of the inequality:
-3y < 7 - 2x
Divide both sides of the inequality by -3.
y > (7 - 2x) / (-3)
y > (2/3)x - (7/3)
This represents a region in the coordinate plane that is above the line y = (2/3)x - (7/3).
Thus, the inequality 2x - 3y < 7 illustrates the given graph as shown.
Therefore, the correct answer is an option (B) 2x - 3y < 7.
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Find an equation for the surface with all points which are equidistant of (−1,0,0)and the plane x=1. Draw the surface.
The equation for the surface with all points which are equidistant of (−1,0,0)and the plane x=1 is x² - 2x + y² + z² = 2.
In mathematics, an equation is a statement that asserts the equality of two expressions. An equation typically consists of two sides separated by an equal sign (=).
To start with, let's consider the given information. We have been given that all points on the surface are equidistant from (-1,0,0) and the plane x=1. This means that the distance between any point on the surface and (-1,0,0) is the same as the distance between that point and the plane x=1.
We can use the formula for the distance between a point and a plane to derive an equation for the surface. The formula for the distance between a point (x1, y1, z1) and a plane Ax + By + Cz + D = 0 is given by:
distance = |Ax1 + By1 + Cz1 + D| / √(A² + B² + C²)
In our case, the plane is x=1, which can be written as x - 1 = 0. So, A=1, B=0, C=0, and D=-1. The point (-1,0,0) can be substituted as x1=-1, y1=0, and z1=0.
We can simplify the distance formula to:
distance = |x-1|
Now, since all points on the surface are equidistant from (-1,0,0) and x=1, we can set up the following equation:
|x-1| = √( (x+1)² + y² + z² )
Squaring both sides and simplifying, we get:
(x-1)² = x² + 2x + 1 + y² + z²
Rearranging and simplifying, we get the final equation for the surface:
x² - 2x + y² + z² = 2
This is the equation of the surface that contains all points that are equidistant from (-1,0,0) and x=1.
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