In a group of 700 people, there must be at least two individuals who have the same first and last initials.
What is combinatorics?
Combinatorics is a branch of mathematics that focuses on counting, arranging, and organizing objects or elements in a systematic and discrete manner. It deals with the study of combinations, permutations, and other mathematical structures related to discrete objects.
To determine whether there must be two people with the same first and last initials in a group of 700 people, we can use the Pigeonhole Principle.
The Pigeonhole Principle states that if we have more pigeons than pigeonholes, then at least one pigeonhole must contain more than one pigeon. In this case, the pigeons represent individuals with different initials, and the pigeonholes represent the unique combinations of first and last initials.
In the given scenario, we have [tex]700[/tex] people and a finite number of possible combinations of first and last initials. Let's consider the number of possible combinations of initials. Since we have 26 letters in the English alphabet, there are 26 choices for the first initial and 26 choices for the last initial. This gives us a total of [tex]26 * 26 = 676[/tex] possible combinations.
Now, since we have 700 people, and the number of possible combinations (676) is less than the number of people, it is not possible for each person to have a unique combination of initials. By the Pigeonhole Principle, at least one combination of initials must be shared by more than one person.
Therefore, in a group of 700 people, there must be at least two individuals who have the same first and last initials.
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Nationwide, 40% of college seniors say that if they could start their college education over, they would have selected a different major. A student researcher in the education department selected a random sample of 50 seniors at Harvard and asked them if they the same question.
a. Let X = number in 50 randomly selected seniors that would select a different major at Harvard.
Assuming the percent of seniors at Harvard that would select a different major is consistent with the national percent, would X have an approximately normal distribution? If so, what would the mean and standard deviation be?
b. If 12 of the 50 students sampled at Harvard said they would have selected a different major, would you think the percent that would have selected a different major is different at Harvard than the national average? Be sure to clearly explain your answer.
Given: Nationwide, 40% of college seniors say that if they could start their college education over, they would have selected a different major. A student researcher in the education department selected a random sample of 50 seniors at Harvard and asked them if they the same question.
a. Mean is 0.40 (40%) and Standard deviation = 0.0775.
b. we cannot conclude that the percent that would have selected a different major is different at Harvard than the national average.
a. Assuming the percent of seniors at Harvard that would select a different major is consistent with the national percentage, X (the number in 50 randomly selected seniors that would select a different major at Harvard) would have an approximately normal distribution.
The mean would be the same as the population mean, which is 0.40 (40%).
The standard deviation would be calculated using the formula given below:
Standard deviation = sqrt[(p(1-p))/n], Where, p = population proportion (0.40), n = sample size (50).
Standard deviation = sqrt[(0.40 x 0.60)/50]
Standard deviation ≈ 0.0775
b. To determine if the percent that would have selected a different major is different at Harvard than the national average, we need to perform a hypothesis test.
Hypotheses:H_0: p = 0.40 (The proportion of seniors at Harvard who would have selected a different major is the same as the national percentage.)
H_a: p ≠ 0.40 (The proportion of seniors at Harvard who would have selected a different major is different from the national percentage.)
Since the sample size (50) is greater than 30 and the population standard deviation is unknown, we can use the z-test to test the hypothesis.
The formula for the test statistic is given below:
z = (p - P)/sqrt[(P(1 - P))/n], Where, p = sample proportion, P = population proportion, n = sample size.
z = (12/50 - 0.40)/sqrt[(0.40 x 0.60)/50]
z ≈ -1.84
Using a significance level of α = 0.05 and a two-tailed test, the critical values of z are ±1.96.
Since the calculated z-value (-1.84) is less than the critical value (-1.96), we fail to reject the null hypothesis.
We do not have sufficient evidence to conclude that the proportion of seniors at Harvard who would have selected a different major is different from the national percentage.
Therefore, we cannot conclude that the percent that would have selected a different major is different at Harvard than the national average.
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Given some scores on an entrance exam, which are roughly normally distributed, we are told they have a mean of 82 and a standard deviation (std dev) of 5. If some student received a 90 on the exam, what would there z-score be calculated as?
The z-score for a student who received a score of 90 on the entrance exam is 1.6.
What is the formula for calculating the area of a triangle?The z-score for a student who received a score of 90 on the entrance exam, with a mean of 82 and a standard deviation of 5, can be calculated as follows:
z = (x - μ) / σx is the student's score (90 in this case)μ is the mean (82)σ is the standard deviation (5)Plugging in the values:
z = (90 - 82) / 5Simplifying:
z = 8 / 5Calculating:
z = 1.6Therefore, the z-score for the student who received a score of 90 on the exam is 1.6.
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Find the value of each trigonometric ratio. Express your answer as a fraction in lowest terms .
Answer:
The answers are in the attachment above
good day mate
Angle ∠B is a right-angle then the measure of angle ∠A is 43.6 degrees and the measure of angle ∠C is 46.4 degrees.
What is a right-angle triangle?It is a type of triangle in which one angle is 90 degrees and it follows the Pythagoras theorem and we can use the trigonometry function. The Pythagoras is the sum of the square of two sides is equal to the square of the longest side.
The triangle ΔABC is given.
The angle ∠B is a 90°.
Then the value of the angle ∠C will be
[tex]\rm C = \sin ^{-1} \dfrac{21}{29}\\\\C = 46.40^o[/tex]
Then the value of the angle ∠A will be
∠A + ∠C = 90°
∠A + 46.40° = 90°
∠A = 43.6°
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Please help me with the question please ASAP ASAP please please ASAP please please help
* It’s NOT 24 if you get 24
Answer:
The tree is 8 foot long
hope this helps
good day mate
Answer:
8
Step-by-step explanation:
12:6 = x:4
2:1 = x:4
x=8
Suppose you toss a coin and will win $1 if it comes up heads. If it comes up tails, you toss again. This time you will receive $2 if it comes up heads. If it comes up tails, toss again. This time you will receive $4 if it is heads. Continue in this fashion for a total of 10 flips of the coin, after which you receive nothing if it comes up tails. What is the mathematical expectation for this game?
Answer:
5
Step-by-step explanation:
The winnings are in G.P. : 1, 2, 4, ..... till 10 toss.
[tex]$a_n = 1 \times 2^{n-1}\ \ \ \forall \ n = 1,2,3,4,....,10$[/tex]
[tex]$a_n$[/tex] denotes the winnings on the [tex]$n^{th}$[/tex] toss.
The probability of earning amount [tex]$a_n$[/tex] on the [tex]$n^{th}$[/tex] toss is = [tex]$\left(\frac{1}{2}\right)^n$[/tex]
∴ [tex]$E(X) = \sum_{n=1}^{10} \ a_n \times \left(\frac{1}{2}\right)^n $[/tex]
[tex]$=\sum_{n=1}^{10} \ 1 \times \frac{2^{n-1}}{2^n} $[/tex]
[tex]$=\sum_{n=1}^{10} \ \frac{1}{2}$[/tex]
Sum of the 1st n terms of the A.P. is :
[tex]$=\frac{n}{2}[2a+(n-1)d] $[/tex]
[tex]$=\frac{10}{2}[2\times \frac{1}{2}+(10-1)\times 0] $[/tex]
= 5
Therefore, E(X) = 5
Hence the expected value of the game is 5
Andrea and Tim are picking apples. Andrea picks 4 à pounds of apples. Tim picks
33 pounds of apples. How many total pounds of apples did Andrea and Tim
pick?
Answer:
37 pounds of apples.
Step-by-step explanation:
you add together the amount of apples that both people have gathered, which in this case would be 33+4 which makes 37, so your answer would be 37 apples.
The sweater was normally $50. It was on sale for $12 off. What was the percent of discount?
Answer:
The discount was 24%.
Step-by-step explanation:
24% of 50 is 12.
HELP WILL MARK BRAINLIEST
Answer:
The answer is a...............
the same ice cream shop is running a sale on cylindrical shaped tubs of ice cream. The cylinder has a radius of 4.5 cm and a height of 18 cm. Find the volume of a tub of ice cream
Answer:
you got this
Step-by-step explanation:
I believe in you
GET BRAINLESTTTTTT !!
A bag of marbles had 24 white marbles and the rest were blue. For a game, 5/6 of the white marbles were chosen, and 25 of the blue marbles were also chosen.
Use this information to answer the questions below.
If not enough information is given to answer a question, click on "Not enough information."
(a) How many of the bag's white marbles were not chosen?
(b) How many of the bag's white marbles were chosen?
(c) How many blue marbles were in the bag before the game?
Answer:
Step-by-step explanation:
We know that there are 24 white marbles in the starting.
a. How many were not chosen. => 1/6 of the white marbles.
1/6 * 24 = 4 marbles were not chosen.
b. 24 - 4 = 20 marbles were chosen
c. Not enough Info
Answer:
4,20,not enough info
Step-by-step explanation:
A survey of 150 college students was done to find out what elective course they were taking Lot A the set of those taking ort, B the set of those taking basketweaving, and C - the set of those taking canoeing. The study revealed the following information A-45 IAN B = 12 181 55 ANC-15 C-40 BC-2 Anno 2 How many students were not taking any of these electives?
A survey of 150 college students was done to find out what elective course they were taking Lot A the set of those taking ort, B the set of those taking basketweaving, and C - the set of those taking canoeing. The study revealed the following information A-45 IAN B = 12 181 55 ANC-15 C-40 BC-2 Anno 2
Based on the given information, we can calculate the number of students who were not taking any of the elective courses.
Let's break down the information provided:
A: The set of students taking Art elective = 45
B: The set of students taking Basket weaving elective = 12
C: The set of students taking Canoeing elective = 40
A ∩ B: The set of students taking both Art and Basket weaving electives = 2
A ∩ C: The set of students taking both Art and Canoeing electives = 15
B ∩ C: The set of students taking both Basket weaving and Canoeing electives = 0 (not specified)
A ∩ B ∩ C: The set of students taking all three electives = 2
To find the number of students who were not taking any of these electives, we need to subtract the total number of students taking at least one elective from the total number of students surveyed.
Total students surveyed = 150
Students taking at least one elective = A ∪ B ∪ C
Students taking at least one elective = A + B + C - (A ∩ B) - (A ∩ C) - (B ∩ C) + (A ∩ B ∩ C)
Students taking at least one elective = 45 + 12 + 40 - 2 - 15 - 0 + 2
Students taking at least one elective = 82
Students not taking any of these electives = Total students surveyed - Students taking at least one elective
Students not taking any of these electives = 150 - 82
Students not taking any of these electives = 68
Therefore, there were 68 students who were not taking any of the elective courses using set theory.
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A school janitor has mopped 1/3 of a classroom in 5 minutes. At what rate is he mopping?
simplify your answer and write it as a proper fraction, mixed number, or whole numer.
___ classrooms per minute
Given,A school janitor has mopped 1/3 of a classroom in 5 minutes.We have to find the rate at which he is mopping.Using the concept of unitary method,Rate of mopping 1 classroom in 5 × 3 = 15 minutes= 1/15 of a classroom in 1 minute.Rate of mopping 1/3 classroom in 5 minutes = (1/3) ÷ 5= 1/15 classroom per minuteHence, the required rate at which he is mopping is 1/15 classroom per minute.
Answer: 1/15.
To determine the rate at which the janitor is mopping, we can calculate the fraction of the classroom mopped per minute.
Given that the janitor mopped 1/3 of the classroom in 5 minutes, we can express this as:
(1/3) classroom / 5 minutes
To simplify this fraction, we divide the numerator and denominator by the greatest common divisor, which is 1:
(1/3) classroom / (5/1) minutes = (1/3) classroom × (1/5) minutes
Multiplying the numerators and the denominators gives us:
1 classroom × 1 minute / 3 × 5
Simplifying further:
1 classroom × 1 minute / 15
Therefore, the rate at which the janitor is mopping is 1/15 classrooms per minute.
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The given information is that the janitor mopped 1/3 of a classroom in 5 minutes. We have to find out at what rate is he mopping.
The rate of mopping is 1/15 classrooms per minute.
Let's try to solve the problem below. The given fraction is 1/3 of a classroom that was mopped in 5 minutes. We need to find the rate of mopping which can be calculated by dividing the fraction of the classroom mopped by the time it took to mop it. The rate of mopping can be found by performing the following calculation:
Rate of mopping = Fraction of the classroom mopped/Time taken to mop
= 1/3/5
= 1/15
So the rate of mopping is 1/15 classrooms per minute. This is the simplified answer.
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Help please
Mark True or False in the table on your answer document to indicate whether
each comparison is true
Answer:
1. false
2. false
3. true
Step-by-step explanation:
The high school soccer team can have no more
than 22 players on the roster. Write and solve an
inequality finding the number of players the
coach may choose if the coach already has 13
players.
Answer:
13 + x ≤ 22
x ≤ 9
Step-by-step explanation:
Total players = 22
Players available = 13
Players remaining = x
The inequality can be written as:
Players available + Players remaining ≤ Total players
13 + x ≤ 22
x ≤ 22 - 13
x ≤ 9
How can she might choose a random sample of five students from her class of 35 students?
Answer:
who?
Step-by-step explanation:
PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!!PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!
Answer:
1. D
2. B
3. C
4. A
Step-by-step explanation:
just know when the sentence says "each" or "per" next to a number, there need to be an x next to it.
In other words, I kinda just winged it :P
Please help me it’s due today
Ared candle is 8 inches tall and burns at a rate of inch per hour. 1 A blue candle is 6 inches tall and burns at a rate of inch per hour. 5 After how many hours will both candles be the same height? Enter your answer in the box.
Answer:
After 8 hrs both candles would've burned out completely, so they'll, more or less, be at the same height.
Find the indicated critical z value. Find the value of z a/2 that corresponds to a confidence level of 85%. a/2 Answer:
The critical z value for a confidence level of 85% is approximately 1.44.
To understand this better, let's delve into the concept of confidence level. A confidence level is a measure of the uncertainty or margin of error associated with an estimation. In this case, we are given a confidence level of 85%, which means that we want to be 85% confident that the true population parameter falls within a specific range.
To find the critical z value, we need to determine the z value that leaves a certain percentage in the tails of the standard normal distribution. Since the confidence level is 85%, we divide this value by 2 to get a/2, which is 0.85/2 = 0.425. This value corresponds to the area under the curve in one tail of the distribution.
Using a standard normal distribution table or a statistical software, we can find the z value that corresponds to a cumulative probability of 0.425. The closest z value to this probability is approximately 1.44. Therefore, the critical z value for a confidence level of 85% is approximately 1.44.
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The following data are cost (in cents) per ounce for nine different brands of sliced Swiss cheese (www .consumerreports.org):
29 62 37 41 70 82 47 52 49
a. Compute the variance and standard deviation for this data set. s2 = 279.111; s = 16.707
b. If a very expensive cheese with a cost per slice of 150 cents was added to the data set, how would the values of the mean and standard deviation change?
a. The variance of the data set is 279.111 and the standard deviation is 16.707.
b. Adding an expensive cheese increases the mean to 55.9 and the standard deviation to 48.53.
a. To compute the variance and standard deviation for the given data set, we can use the following formulas:
Variance (s²) = [(Σx²) - (Σx)² / n] - 1
Standard Deviation (s) = √(Variance)
Using the given data set: 29, 62, 37, 41, 70, 82, 47, 52, 49
Step 1: Calculate the mean (x) of the data set.
Mean (x) = (Σx) / n
Σx = 29 + 62 + 37 + 41 + 70 + 82 + 47 + 52 + 49 = 409
n = 9 (number of data points)
Mean (x) = 409 / 9 = 45.44 (rounded to two decimal places)
Step 2: Calculate the sum of squares (Σx²) for the data set.
Σx² = 29² + 62² + 37² + 41² + 70² + 82² + 47² + 52² + 49² = 151509
Step 3: Substitute the values into the variance formula and calculate the variance.
Variance (s²) = [(Σx²) - (Σx)² / n] - 1
Variance (s²) = (151509 - (409)² / 9) / (9 - 1) = 279.111 (rounded to three decimal places)
Step 4: Calculate the standard deviation by taking the square root of the variance.
Standard Deviation (s) = √(Variance)
Standard Deviation (s) = √279.111 = 16.707 (rounded to three decimal places)
b. If a very expensive cheese with a cost per slice of 150 cents was added to the data set, the values of the mean and standard deviation would change. To find the new mean and standard deviation, we need to incorporate the additional data point and recalculate.
New Σx = Σx + 150 = 409 + 150 = 559 (updated sum of the data points)
New n = n + 1 = 9 + 1 = 10 (updated number of data points)
New Mean (x) = New Σx / New n = 559 / 10 = 55.9 (rounded to one decimal place)
The addition of the expensive cheese increased the mean.
To calculate the new variance and standard deviation, we follow the same steps as in part a using the updated Σx and n values:
New Σx² = Σx² + (150)² = 151509 + 22500 = 174009
New Variance (s²) = [(New Σx²) - (New Σx)² / New n] - 1 = (174009 - (559)² / 10) / (10 - 1) = 2355 (rounded to three decimal places)
New Standard Deviation (s) = √New Variance = √2355 = 48.53 (rounded to two decimal places)
Therefore, the addition of the expensive cheese increased both the mean and the standard deviation of the data set.
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help, i will give brainliest
Answer:
The answer is J
Step-by-step explanation:
If you start from Quinn's location (8,7), and move south 3 units, you get to (8,4). Then you move to the west 4 units and that gets you to (4,4).
hope this helped! :)
Graph each set of numbers on the number line and order the numbers from greatest to least 0.5, -1, -1/4, 0
Answer:
0.5, 0, -1/4, -1
Part 1: Given cosine of theta is equal to radical 3 over 2 comma determine three possible angles θ on the domain [0,[infinity]). Part 2: Given θ = 495°, convert the value of θ to radians and find sec θ.
The required answer is sec θ = -√2.
Explanation:-
Part 1: Given cosine of theta is equal to radical 3 over 2 on the domain [0,[infinity]).
To determine three possible angles θ, the cosine inverse function which is a cos and since cosine function is positive in the first and second quadrant. Therefore conclude that, cosine function of θ = radical 3 over 2 implies that θ could be 30 degrees or 330 degrees or 390 degrees. So, θ = {30, 330, 390}.Part 2:To convert 495° to radians, multiply by π/180°.495° * π/180° = 11π/4To find sec θ, we use the reciprocal of the cosine function which is sec.
Therefore, sec θ = 1/cos θ.To find cos 11π/4, the reference angle, which is 3π/4. Cosine is negative in the third quadrant so the final result is sec θ = -√2.
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If water flows through the pipe at a rate of 2L per seconds. How much water will pass through the pipe in 1/2hour
Answer:
3600 Liters
Step-by-step explanation:
The given rate is 2L and we have to find how much water will flow in 1/2 hours at the same rate
1/2 hour is the same thing as 30 mins
multiply by 60 to convert minutes to seconds
[tex]3omin * \frac{60sec}{1min} = 1800 s[/tex]
since the rate is 2L per second we multiply 1800 by 2 and get 3600 Liters
AC and BD are diameters of the given circle. OC is 6 centimeters and ABO = CBO. what is AO. A 3/ B 4/ C 5/ D 6
Answer:
A
Step-by-step explanation:
Which is closest to the height of a pine tree?
1. 9 inches
2. 9 Yards
3. 9 Miles
Answer:
2/b
Step-by-step explanation:
9 yards
Consider the system of equations 2x + 10y + 42 -1 4x + 18y + 10z 0 (a) If A is the coefficient matrix, find A-1. (b) Solve the system using A-1. (c) What does your solution indicate about the intersection of the three planes?
(a) The inverse matrix A⁻¹ is:
| -9/2 5/2 |
| 1 -1 |
(b) The solution to the system is x = -9/2 and y = 1. z is undetermined.
(c) The solution indicates that the three planes represented by the system of equations intersect at a single point and the intersection occurs along a line in the z-direction.
(a) The coefficient matrix, A, is given by:
A = | 2 10 42 |
| 4 18 10 |
We can use matrix inversion. A matrix is invertible if its determinant is non-zero. Let's calculate the determinant of matrix A:
det(A) = (2 * 18) - (10 * 4) = 36 - 40 = -4
Since the determinant is non-zero (-4 ≠ 0), the matrix A is invertible. Now, we can find the inverse of A:
A⁻¹ = (1/det(A)) * adj(A)
Where adj(A) denotes the adjugate of matrix A. To calculate the adjugate, we need to find the cofactor matrix of A and then take its transpose:
Cofactor matrix of A:
| 18 -4 |
| -10 2 |
Transpose of the cofactor matrix:
| 18 -10 |
| -4 2 |
Now, divide the transpose by the determinant:
A⁻¹ = (1/-4) * | 18 -10 |
| -4 2 |
Simplifying:
A⁻¹ = | -9/2 5/2 |
| 1 -1 |
Therefore, the inverse matrix A⁻¹ is:
| -9/2 5/2 |
| 1 -1 |
(b) We have the equation AX = B, where X is the column vector of variables (x, y, z), A is the coefficient matrix, and B is the column vector of constants.
The coefficient matrix A is:
A = | 2 10 42 |
| 4 18 10 |
The column vector B is:
B = | 1 |
| 0 |
Now, we can solve for X using A-1:
X = A⁻¹ * B
Substituting the values:
X = | -9/2 5/2 | * | 1 |
| 1 -1 | | 0 |
Multiplying the matrices:
X = | (-9/2 * 1) + (5/2 * 0) |
| (1 * 1) + (-1 * 0) |
Simplifying:
X = | -9/2 |
| 1 |
Therefore, the solution to the system is x = -9/2 and y = 1. The value of z is not determined from this calculation.
(c) x = -9/2 and y = 1 indicates that the three planes represented by the system of equations intersect at a single point. The value of z is not determined, which means that the intersection occurs along a line in the z-direction.
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Saul's Corner Market sells large candy bars for $2 each. A bag of pretzels is $3. On Tuesday, the store sold twice
as many candy bars as Monday but the same number of pretzels. On Wednesday, the store sold the same
number of candy bars as Monday but only half the amount of pretzels. On Thursday, the store sold three times
as much as they sold on Tuesday. On Friday, the store sold 12 fewer candy bars and 5 more bags of pretzels
compared to Wednesday.
Let c represent the number of candy bars sold and p represent the number of pretzels sold on Monday. Write
an expression for each day to show the amount of money that the store brought in from candy bars and
pretzels.
1. Monday
2. Tuesday
3. Wednesday
4. Thursday
5. Friday
6. Monday through Friday
Answer:
Step-by-step explanation:
Monday: c and p
Tuesday: 2c+p
Wednesday: 2c+p/2
Thursday: 6c+3p/2
Friday: (2c-12)+(p/2+5)
Adding them all up, you get, for Monday-Friday: (13c-12)+(9p/2+5)
I'm not sure how to explain this to you though, you just follow the directions they gave, to multiply and subtract.
Hope this is helpful!
The point with coordinates $(6,-10)$ is the midpoint of the segment with one endpoint at $(8,0)$. Find the sum of the coordinates of the other endpoint.
Answer:
[tex](4,-20)[/tex]
Step-by-step explanation:
Let [tex]P(x,y),\,Q(u,v)[/tex] be two points then midpoint of [tex]PQ[/tex] is given by [tex](\frac{x+u}{2},\frac{y+v}{2})[/tex]
Put midpoint as [tex](6,-10)[/tex] and [tex](u,v)=(8,0)[/tex]
Therefore,
[tex](\frac{x+u}{2},\frac{y+v}{2})=(6,-10)\\\\(\frac{x+8}{2},\frac{y+0}{2})=(6,-10)\\\\\frac{x+8}{2}=6,\,\frac{y}{2}=-10\\\\x+8=12,\,y=-20\\x=12-8,\,y=-20\\x=4,\,y=-20[/tex]
So, the other point is [tex](4,-20)[/tex]
Find the area of the rectangle.
Answer:
Step-by-step explanation:
the area of the rectangle is : -7y² ( -2y^4+y²-1)
14y^6-7y^4+7y² unit