(a) The probability that a 0 is received is 66.7%
(b) By using the Bayes theorem, the probability that a 0 was transmitted, given that a 0 was received is 90%
Probability:
In statistics, probability deals with finding out the likelihood of the occurrence of an event.
Given,
Here the space probe near Neptune communicates with earth using bit strings. suppose that in its transmissions it sends a 1 one-third of the time and a 0 two-thirds of the time. when a 0 is sent, the probability that it is received correctly is 0.6 and the probability that it is received incorrectly (as a 1) is 0.4. when a 1 is sent the probability that it is received correctly is 0.8 and the probability that it is received incorrectly (as a 0) is 0.2.
Now, we have to find the probability of the following.
Here the probability that a 0 is received is calculated as,
=> 0.9 of 2/3(0 received when a 0 is sent) and 0.2 of 1/3(0 received when a 1 is sent). So
Now, the probability is
=> p = (0.9 x 2)/3 + (0.2 x 1)/3
=> p = 0.6667
When we convert this into percentage then we get, 66.67% probability that a 0 is received.
Similarly, by using the Bayes theorem, the probability that a 0 was transmitted, given that a 0 was received is calculated as,
Here we know that, Event A: 0 received and Event B: 0 transmitted.
Then 0.6667 = 66.67% probability that a 0 is received, which means that
P(A) = 66.7%
And the zero is transmitted two-thirds of time, which means that
P(B) = 66.7%
Where as if 0 is sent, then the probability that it is received correctly is 0.9, which means that
P(A|B) =0.9
Therefore, the probability is
=> 66.7 x 0.9/66.7
=> 0.9
We know that 0.9 = 90% probability that a 0 was transmitted, given that a 0 was received.
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help meeeeeeeeeeeeee pleaseeeeeeeeeeeeeeeeeeeeeeeeee
If the domain of f(x) = x2 - 4 is limited to {-1,0,1,2} what is the greatest value of the range?
The greatest value of the range is
If the domain of f(x) = x² - 4 is limited to {-1, 0, 1, 2} the greatest value of the range is 0
How to find the greatest value of the rangeThe range represents the output values
Using the given expression to solve it can be found that the output values are
for x = -1, f(x) = x² - 4 = -1² - 4 = -3
for x = 0, f(x) = x² - 4 = 0² - 4 = -4
for x = 1, f(x) = x² - 4 = 1² - 4 = -3
for x = 2, f(x) = x² - 4 = 2² - 4 = 0
The greatest value of the range is 0
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If 36 is the total number we are trying to get to how much do you add together???
To get 36 as a result, you can add different numbers, some examples that can work are 18+18, 24+12, 12+12+12, and 8+12+16.
What is addition?This is a basic mathematical operation that involves calculating the total of two, three, or more numbers. This implies in addition you combine or add numbers and in most cases you get a bigger number. Here are some examples:
5 + 6 = 11
254 + 25 = 279
0.2 + 1.2 = 1.4
How to find numbers whose total is 36?Use substraction to find possible numbers. Here is an example:
36 -18 = 18 which means 18 + 18 = 36
36 - 20 = 16 which means 20+16 = 36
Add small numbers until you get to the total. Here is an example:
2 + 4 + 8 + 6 + 3 + 7 + 5+ 1 = 36
Based on this, some numbers that you can use to get 36 are
18+18
24+12
12+12+12
8+12+16
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Regression analysis was applied between sales data (y in $1000s) and advertising data (x in $100s) and the following information was obtained. Ŷ = 12 + 1.8x n = 17 SSR = 225 SSE = 75 sb1 = .2683 Based on the above estimated regression equation, if advertising is $3000, then the point estimate for sales (in dollars) is?
The point estimate for sales when advertising is $3000 is $5412.
Based on the given information, the estimated regression equation is Ŷ = 12 + 1.8x, where Ŷ is the estimated value of the dependent variable (sales), x is the value of the independent variable (advertising), and the constants 12 and 1.8 are the coefficients of the regression equation.
If advertising is $3000, then the point estimate for sales (in dollars) is the value of Ŷ when x = 3000. Substituting this value into the estimated regression equation, we get:
Ŷ = 12 + 1.8 * 3000 = 12 + 5400 = 5412
So, the required amount is $5412.
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Help with a statistics math problem
The equation of the regression line of the given data is y = 1.28x + 3.69. By using a scatter plot, the required equation is calculated.
What is the regression line?A regression line describes the relationship between the dependent and independent variables of a set of data.
It is given by y = mx + b; where m is the estimated slope of the data and b is the estimated y-intercept of the data.
Calculation:To find the regression line, we need to draw the scatter plot for the given data.
The coordinates paired from the given data are:
(12, 18.11), (7, 14.08), (11, 17.97), (8, 15.55), (13, 17.93), (4, 7.71), (3, 4.93), (5, 10.77), and (9, 16.68).
The scatterplot for these points is shown below:
From the plot, consider any two points on the line.
Such as (x1, y1) = (3.35, 7.99) and (x2, y2) = (9.61, 16.02)
So, the equation of the regression line is
(y - y1) = {(y2 - y1)/(x2 - x1)} (x - x1)
⇒ (y - 7.99) = {(16.02 - 7.99)/(9.61 - 3.35)} (x - 3.35)
⇒ y - 7.99 = (8.03/6.26)(x - 3.35)
⇒ y - 7.99 = 1.282(x - 3.35)
⇒ y - 7.99 = 1.282x - 4.2947
⇒ y = 1.282x + 3.695
When rounded to two decimals, we get
y = 1.28x + 3.69
Therefore, the equation of the regression line of the given data is y = 1.28x + 3.69.
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Suppose that f is differentiable for all x and that f (x)≤2 for all x. If f(1)=2 and f(4)=8 then f(2) has the value equal to a.3 b.4 c.6 d.8
The value of the function f(x) which is differentiable at all x, at x = 2
Or f(2) = b. 4.
What is a function?A function can be defined as the outputs for a given set of inputs.
The inputs of a function are known as the independent variable and the outputs of a function are known as the dependent variable.
Given, The function 'f' is differentiable for all x.
Now, f(1) = 2 and f(4) = 8, So, f(x) can be 2x as it is satisfying the conditions.
∴ At x = 2 or f(2) = 2(2) = 4.
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what is the solution to the system of equations x-y=-1 and x-3=- -13?
Answer:
Write in Standard Form
1. The standard form of a linear equation is
Ax + By = Cx-y=-1
2. x = 16
Ig?
I'm not sure.
Let U be a Universal set with subsets A and B. U={1,2,3,4,5,6,7,8,9,10} A={1,3,5,7} B={4,5,6} Find A′. Use set roster notation. If there is more than one element in the set, list in ascending order, separated by a comma.
The complement of set A is-
A' = {2, 4, 5, 6, 7, 8, 9, 10}
What is a set? What is union of sets? What is complement of a set?A set is a collection of elements or numbers or objects, represented within the curly brackets { }. For example: {1,2,3,4} is a set of numbers.The union of two sets is a set containing all elements that are in A or in B (possibly both). For example, {1,2} ∪ {2,3} = {1,2,3}.The complement of a set is the set that includes all the elements of the universal set that are not present in the given set.We have -
U={1,2,3,4,5,6,7,8,9,10} A={1,3,5,7} B={4,5,6}A' = U - A
A' = {1,2,3,4,5,6,7,8,9,10} - {1,3,5,7}
A' = {2, 4, 5, 6, 7, 8, 9, 10}
Therefor, the complement of set A is -
A' = {2, 4, 5, 6, 7, 8, 9, 10}
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Find the third derivative of the function y=6x-x2/x2
An online customer service department estimates that about 15 percent of callers have to wait more than 8 minutes to have their calls answered by a person. The department conducted a simulation of 1,000 trials to estimate the probabilities that a certain number of callers out of the next 10 callers will have to wait more than 8 minutes to have their calls answered. The simulation is shown in the following histogram.Based on the simulation, what is the probability that at most 2 of the next 10 callers will have to wait more than 8 minutes to have their calls answered?
The probability that at most 2 of the next 10 callers have to wait more than 8 minutes is the sum of the probabilities of 0, 1, or 2 callers having to wait for more than 8 minutes which is 0.810
The probability that at most 2 of the next 10 callers having to wait can be defined using the expression :
P(X ≤ 2) = P(0) + P(1) + P(2)
Using the individual probabilities given by the histogram attached :
P(X ≤ 2) = 0.181 + 0.345 + 0.284
P(X ≤ 2) = 0.810
Therefore, the probability that at most 2 callers have to wait more than 8 minutes is 0.810
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Help me please Ahaheinendjdn
A potter has 9pounds of clay. He uses 3 10pound of clay for each bowl he makes.how many bowls can potter make?
Answer:
30 bowls
Step-by-step explanation:
9 lb / (3/10 lb/bowl) = 9 * 10/3 bowls = 30 bowls
Part C: Mr. Chin wants to add a rectangular desk that is 7 feet wide by 5 feet long. He says
he has enough space in the reading room to add the desk. Do you agree with Mr. Chin's
conjecture? Why or why not? Explain.
Not agree with the Mr. Chin's conjecture, because there may be space for the desk in the room but Mr. Chin not have the actual measurement, it may or may not fit the desk in room of given length and width.
What is conjecture?A good guess or an idea about a pattern is a conjecture. Speculate (about something) We can only speculate about the killer's thoughts. speculate that... He speculated that the population might double within a decade. speculate about something She speculated about the existence of a brand-new species.
A mathematical assertion that has not yet been thoroughly established is known as a conjecture. When one observes a pattern that is consistent across many cases, speculations result. However, just because a pattern applies to a lot of cases does not mean that it applies to all of them. To fully accept the mathematical observation, conjectures must be demonstrated. A theorem is formed when a conjecture is thoroughly demonstrated.
A crucial step in solving a problem is making a guess; It's more than just a tool for professionals in math. It is extremely uncommon for a problem's solution to be immediately apparent in everyday problem solving. Instead, the process of solving problems involves looking at the problem structure, looking at the cases, making a guess about the solution, and then proving that guess.
Hence, not agree with Mr. Chin's conjecture.
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What are 3 characteristics of exponential functions?
A GENERAL NOTE: CHARACTERISTICS OF THE GRAPH OF THE PARENT FUNCTION [tex]f(x) = b^{x}[/tex]
-It’s a one-to-one function
-The graph passes through y-intercept of the point (0,1)
-x-intercept: none
-The domain is all real numbers: (−∞,∞)
-The range is y > 0 or (0,∞)
-The graph is increasing
-The graph is asymptotic to the x-axis as x approaches negative infinity
-Horizontal asymptote: y=0
-The graph increases without bound as x approaches positive infinity
-Increasing if b > 1
-Decreasing if b < 1
-The graph is continuous
-The graph is smooth
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The graph on the left shows the parabola y=x2. What parabola does the graph on the right depict?
y-axis
0
2
x-axis
sixe-A
x-axis
Assessment
Answer:
y=x^2 + 1
Step-by-step explanation: the second parabola is only raised by 1 unit up, therefore the entire function shifts upwards by 1.
Statements
1. WXYZ is a
parallelogram with
diagonals
XZ and WY
intersecting at
point V.
2. WX // ZY and
XY // WZ
3. ZWXYZYZW
ZWXZ = ZYZX
4.XYZW
5. AWZVAYXV
6. WV YV
7. XZ bisects WY
Given
Reasons
Definition of a parallelogram
Opposite sides of a
parallelogram are congruent
Corresponding parts of
congruent triangles are
congruent
Definition of segment bisector
1. ZX ≅WY; WXYZ is a parallelogram. 1. Given
2. ZY≅ WX 2. The opposite sides of the
parallelogram are ≅
what is parallelogram ?There are 4 different kinds of parallelograms, including 3 unique kinds. The four types are rhombuses, squares, parallelograms, and rectangles.
There are not equal sides on a parallelogram. A parallelogram only has equal sides on its opposite sides.
what is segment bisector ?A line segment bisector splits a line segment into two equally sized pieces. It passes through the segment's halfway.
The following justifications are given for the choice made above:
The following is how the two-column proof is presented.
Statement Motive
1. ZX ≅WY; WXYZ is a parallelogram. 1. Given
2. ZY≅ WX 2. The opposite sides of the parallelogram are ≅
3. YX ≅ YX 3. The reflexive quality
4. Δ ZYX ≅ΔWXY 4. The SSS thm.
5 . ∠ZYX ≅ ∠WXY 5 . CPCTC
6 . m∠ZYX ≅m∠WXY 6. Def. of ≅
7. m∠ZYX + mWXY equal 180° 7. Parallelograms with consecutive s are additional
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each term in the following sequence is six more than the previous term what is the value of x+z? x,y,z,7,13
The following list has six more terms than the preceding one. What does x+z equal if it is -10?
What, for instance, does sequence mean?A series of numbers is an ordered list. The three dots signify moving forward in the set pattern. Each digit with in sequence is referred to as a word. The first term in the list is 1, followed by the second by 3, the third by 5, and so on.
t1=X\st2=y\st3=Z
t4=7\st5=13
common distinction Since d=6, t4=t3+d 7=t3+6 t3=7-6 t3=1=Z
Z=1\st3=t2+d
X + Z= (-11) + 1 X+Z = -10 because 1 = t2+6 t2= 1-6 t2= -5 = y y= -5 t2= t1+d -5= t1+6 t1= -5 -6 t1= -11 =X X = -11
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y = 6x + 7, what is the slope?
Answer: m= 6
Step-by-step explanation:
Answer:
6
Step-by-step explanation:
because slope is the amount of times your changing the line
Let T be the outcome of a roll with a fair die.
a. Describe the probability distribution of T ; that is, list the outcomes and the corresponding probabilities.
b. Determine E[T] and Var(T).
The probability distribution of rolling a die is
P(1) = 1/6, P(2) = 1/6, P(3) = 1/6, P(4) = 1/6, P(5) = 1/6, P(6) = 1/6
Given that,
Let T be the outcome of a roll with a fair die.
Then
T = {1, 2, 3, 4, 5, 6}
The probability is the ratio of the number of favorable outcomes to the total number of outcomes
The probability of getting 1 P(1) = 1/6
The probability of getting 2 P(2) = 1/6
The probability of getting 3 P(3) = 1/6
The probability of getting 3 P(3) = 1/6
The probability of getting 4 P(4) = 1/6
The probability of getting 5 P(5) = 1/6
The probability of getting 6 P(6) = 1/6
Therefore, the probability distribution of rolling a die has been found
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a jet plane can carry up to 200,000 liters of fuel. it used 130,000 liters of fuel during a flight. what percentage of the fuel capacity did it not use on this flight?
35% of the fuel Capacity did not use on this flight.
How to calculate percent change of any quantity?Step 1: Calculate the change i.e., subtract old value from the new value
Step 2: Divide that change by the old value.
Step 3: Convert that to a percentage (by multiplying by 100 and adding a "%" sign)
Percentage change = ((final value - initial value) ÷final value) ×100
Given: Total capacity in a fuel tank of a jet = 200000 liters
Used fuel =130000 liters
% Of fuel which is not used = ((total capacity of fuel tank - used fuel) ÷total capacity of fuel tank) ×100
[tex]=\frac{200000-130000}{200000}*100\\=\frac{70000}{2000}[/tex]
=35%
Therefore, 35% of the fuel capacity did not used in this flight.
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Lian is deciding which of two gyms to join. Each gym charges a monthly rate plus a one-time membership fee.
Lian correctly wrote and solved a system of linear equations by substitution to compare the costs of the memberships. In her work, she substituted an expression for one variable and solved for the other. This resulted in the equation 75 = 75. What can Lian conclude?
Both gyms charge the same monthly rate and the same membership fee.
Liam can conclude that the price and monthly rate of both the gyms are equal.
Given, Lian is deciding which of two gyms to join. Each gym charges a monthly rate plus a one-time membership fee.
Lian correctly wrote and solved a system of linear equations by substitution to compare the costs of the memberships. In her work, she substituted an expression for one variable and solved for the other. This resulted in the equation 75 = 75.
we have to mention that what Lian can conclude,
As, substituting the first equation in the second one Liam got the price which is same in both the gyms.
So, Liam can conclude that the price and monthly rate of both the gyms are equal.
Hence, Liam can conclude that the price and monthly rate of both the gyms are equal.
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what fraction is equivalent to 3/4; what fraction is equivalent to 1/3; what fraction is equivalent to 2/3; what fraction is equivalent to 2/5; what fraction is equivalent to 1/4; what fraction is equivalent to 2/6; equivalent fractions; what fraction is equivalent to 2/4
6/8 is an equivalent fraction to 3/4; 2/6 is equivalent to 1/3; 6/9 is equivalent to 2/3; 10/25 is equivalent to 2/5; 4/16 is equivalent to 1/4; the fraction of 1/3 is equivalent to 2/6; 8/16 is an equivalent fraction to 2/4.
A fraction can be referred to or considered as a divisional representation of a whole number or a percentage, usually represented as a bigger numerical. A fraction denotes the representation of a number in such a way that the meaning or the accuracy of the number is not disturbed, but is shown differently. The fractions equivalent to each other have been mentioned above.
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thirty-six students at a local middle school were randomly selected to participate in a taste test to select a new menu item for a cafeteria. each student was given a choice between the macaroni and cheese and the cajun potato barrels. from the sample, 20 students selected macaroni and cheese and 16 students chose the cajun potato barrels. each student was asked to rate the item they chose on a scale from 1 (extremely dislike) to 9 (extremely like). which of the following prevents this study from being a well-designed experiment?
The treatments (new menu item) were not randomly assigned to the students when 36 local middle school kids were chosen at random to take part.
Given that,
36 local middle school kids were chosen at random to take part in a taste test to choose a new menu item for a cafeteria. The Cajun potato barrels or the macaroni and cheese were the options offered to each pupil. 20 pupils chose the macaroni and cheese from the sampling, while 16 selected the Cajun potato barrels. On a scale from 1 (very dislike) to 9 (very much like), each student was asked to rank the object they selected (extremely like).
We have to find which of the following keeps this study from being an experiment with a sound design.
Each student was given the option of choosing either the macaroni and cheese or the Cajun potato barrels. Because the students' treatments were not distributed at random, those who preferred their cuisine chose it and gave it high marks. As a result, the study has selection bias.
Therefore, The treatments (new menu item) were not randomly assigned to the students when 36 local middle school kids were chosen at random to take part.
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A group of ten friends played a number guessing game. They were asked to pick a number
between 1 and 20. The person closest to the target number wins. The ten people made these
guesses:
The table with the absolute guessing errors should be completed as follows;
Guess 2 15 10 8 12 19 20 5 7 9
Absolute guessing error 12 1 4 6 2 5 6 9 7 5
A graph of the guess and absolute guessing error is shown in the image attached below.
Yes, the absolute guessing error is a function of the guess because every guess (input value) has one absolute guessing error (output value).
What is the margin of error?In Mathematics, the margin of error (MOE) is sometimes referred to as an absolute guessing error and it can be defined as a measure of the difference that exist between an observed value and a true value of the population parameter.
Mathematically, the absolute guessing error for the guesses made by these group of ten friends with respect to the actual number of 14 as follows;
Absolute guessing error = 14 - 2 = 12
Absolute guessing error = 14 - 15 = 1
Absolute guessing error = 14 - 10 = 4
Absolute guessing error = 14 - 8 = 6
Absolute guessing error = 14 - 12 = 2
Absolute guessing error = 14 - 19 = 5
Absolute guessing error = 14 - 20 = 6
Absolute guessing error = 14 - 5 = 9
Absolute guessing error = 14 - 7 = 7
Absolute guessing error = 14 - 9 = 5
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Determine if x and y are inversely proportional. If applicable, find the constant of variation.
The relationship between x and y is and the constant of variation is 24.
What is inversely proportional?
One kind of proportionality relationship is inverse proportion. If two quantities are inversely related, one will increase while the other will decrease. Building a wall would be an example of an inverse proportion in action.
Given that x and y are inversely proportion.
Assume that the relationship between x and y is
y = k/x where k is constant of variation.
Putting x = 3 and y = 8 in y = k/x :
8 = k/3
Multiply both sides by 3:
k = 8 × 3
k = 24
Putting x = 4 and y = 6 in y = k/x :
6 = k/4
Multiply both sides by 4:
k = 6 × 4
k = 24
The relationship is y = 24/x and the constant of variation is 24.
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Answer:
x and y are not inversely proportional.
Image:
A farmer uses a lot of fertilizer to grow his crops. The farmer's manager thinks fertilizer products from distributor A contain more of the nitrogen that his plants need than distributor B's fertilizer does. He takes two independent samples of four batches of fertilizer from each distributor and measures the amount of nitrogen in each batch. Fertilizer from distributor A contained 23 pounds per batch and fertilizer from distributor B contained 18 pounds per batch. Suppose the population standard deviation for distributor A and distributor B is four pounds per batch and five pounds per batch, respectively. Assume the distribution of nitrogen in fertilizer is normally distributed. Let μ1 and μ2 represent the average amount of nitrogen per batch for fertilizer's A and B, respectively. Which of the following is the approptiate conclusion at the 5% significance level?a. Reject H 0; we can conclude that the mean amount of fertilizer per batch for distributor A is greater than the amount of fertilizer per batch for distributor Bb. Reject H 0; we cannot conclude that the mean amount of fertilizer per batch for distributor A is greater than the amount of fertilizer per batch for distributor B.c.Fail to reject H 0; we can conclude that the mean amount of fertilizer per batch for distributor A is greater than the amount of fertilizer per batch for distributor Bd. Fail to reject H 0; we cannot conclude that the mean amount of fertilizer per batch for distributor A is greater than the amount of fertilizer per batch for distributor B
The appropriate conclusion at the 5% significance level is b. Reject H0; we cannot conclude that the mean amount of fertilizer per batch for distributor A is greater than the amount of fertilizer per batch for distributor B.
How to calculate test statistic?[tex]\bar{x_1}[/tex] = pounds per batch distributor A = 23
[tex]\bar{x_2}[/tex] = pounds per batch distributor B = 18
[tex]\sigma_{1}[/tex] = standard deviation distributor A = 4
[tex]\sigma_{2}[/tex] = standard deviation distributor B = 5
n = sample size = 4
α = significance level = 5% = 0.05
Test statistic is using z test to find the z value from two population, after that we can search p-value and then comparing the p-value with significance level.
If p-value is lower than significance level, the null hypothesis (H0) is accepted. But, if p-value is higher than significance level, the H0 is rejected.
Calculate test statistic by this formula,
Z = [tex]\frac{\bar{x_1}-\bar{x_2}}{\sqrt{\frac{\sigma^2_1 + \sigma^2_2}{n}}}[/tex]
= [tex]\frac{23-18}{\sqrt{\frac{4^2 + 5^2}{4}}}[/tex]
= [tex]\frac{5}{\sqrt{\frac{16 + 25}{4}}}[/tex]
= 1.56
Using z table for 1.56, we get 0.9406. Since is right tailed test. So, p value is
p-value = 1 - 0.9406
= 0.0594
Then, p-value is 0.0594 is higher than significance level 0.05. So, H0 is rejected.
Thus, the conclusion is reject H0, we cannot conclude the mean amount of fertilizer per batch distributor A is greater than distributor B.
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where do u get the 150 out of the answer (x+150 (x-10) in the equation x^2+140x-1500?
Answer:
See explanation.
Step-by-step explanation:
Given quadratic:
[tex]x^2+140x-1500[/tex]
To factor a quadratic in the form [tex]ax^2+bx+c[/tex], first find two numbers that multiply to [tex]ac[/tex] and sum to [tex]b[/tex].
As a = 1, b = 140 and c = -1500, then:
[tex]\implies a \cdot c=1 \cdot -1500=-1500[/tex]
[tex]\implies b=140[/tex]
Therefore, the two numbers are 150 and -10 as:
[tex]\implies 150 \cdot -10 = -1500[/tex]
[tex]\implies 150+(-10)=140[/tex]
Rewrite [tex]b[/tex] as the sum of these two numbers:
[tex]\implies x^2+(150-10)x-1500[/tex]
Distribute:
[tex]\implies x^2+150x-10x-1500[/tex]
Factor the first two terms and the last two terms separately:
[tex]\implies x(x+150)-10(x+150)[/tex]
Factor out the common term (x + 150):
[tex]\implies (x+150)(x-10)[/tex]
how to use the midpoint formula to find an endpoint
Therefore you can use midpoint to find the other endpoint (x2,y2), then by rewriting the (x2,y2)= (2a-x1,2b-y1) .
What location constitutes the middle?The exact midpoint between two other plane points needs to be found on occasion. For example, you might seek out a line that splits a particular line segment into two equal halves. This central region is known as the "halfway."
Here,
If you are aware of one endpoint (x1, y1) and the midpoint (a, b), but are unaware of the second endpoint (x2, y 2), you can solve the problem by rewriting the
=> a=[tex]\frac{x1+x2}{2}[/tex]
=> 2a=x1+x2
=>2a-x1=x2
=> b=[tex]\frac{y1+y2}{2}[/tex]
=>2b=y1+y2
=> 2b-y1=y2
So, the unknown endpoint can be found by
(x2,y2)=(2a-x1,2b-y1)
Therefore you can use midpoint to find the other endpoint (x2,y2), then by rewriting the (x2,y2)= (2a-x1,2b-y1) .
To know more about mid point, visit
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Write a polynomial function f of least degree that has rational coefficients, a
leading coefficient of 1, and zeros 3 and 3 – i
Answer:
f(x) = x³ -9x² +28x -30
Step-by-step explanation:
You want a monic polynomial f of least degree with zeros 3 and 3-i.
FactorsFor zero x=a, the polynomial has a factor (x -a). Complex zeros come in conjugate pairs, so another one is 3+i. This means the factored form is ...
f(x) = (x -3)(x -3 +i)(x -3 -i)
f(x) = (x -3)(x² -6x +9 +1) . . . . . combine the two right factors
f(x) = x³ -9x² +28x -30
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Additional comment
When multiplying polynomials mentally, it can be useful to consider the coefficients of the product terms in order of decreasing powers:
f(x) = (1)x³ +(-3 -6)x² +(10 +(-3)(-6))x +(-3)(10)
What is the area of a triangle whose vertices are J(-2,1), K(0.3), L(3.-4)?
Check the picture below, so that's the triangle hmmmm a bit non-right-triangle or irregular per se, so let's use Heron's formula on this one, so we'll have to first find the length of each side
[tex]~\hfill \stackrel{\textit{\large distance between 2 points}}{d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2}}~\hfill~ \\\\[-0.35em] ~\dotfill\\\\ J(\stackrel{x_1}{-2}~,~\stackrel{y_1}{1})\qquad K(\stackrel{x_2}{0}~,~\stackrel{y_2}{3}) ~\hfill JK=\sqrt{(~~ 0- (-2)~~)^2 + (~~ 3- 1~~)^2} \\\\\\ ~\hfill JK=\sqrt{( 2)^2 + ( 2)^2} \implies JK=\sqrt{ 8 }[/tex]
[tex]K(\stackrel{x_1}{0}~,~\stackrel{y_1}{3})\qquad L(\stackrel{x_2}{3}~,~\stackrel{y_2}{-4}) ~\hfill KL=\sqrt{(~~ 3- 0~~)^2 + (~~ -4- 3~~)^2} \\\\\\ ~\hfill KL=\sqrt{( 3)^2 + ( -7)^2} \implies KL=\sqrt{ 58 } \\\\\\ L(\stackrel{x_1}{3}~,~\stackrel{y_1}{-4})\qquad J(\stackrel{x_2}{-2}~,~\stackrel{y_2}{1}) ~\hfill LJ=\sqrt{(~~ -2- 3~~)^2 + (~~ 1- (-4)~~)^2} \\\\\\ ~\hfill LJ=\sqrt{( -5)^2 + (5)^2} \implies LJ=\sqrt{ 50 }[/tex]
now let's use those three lengths for Heron's
[tex]\qquad \textit{Heron's area formula} \\\\ A=\sqrt{s(s-a)(s-b)(s-c)}\qquad \begin{cases} s=\frac{a+b+c}{2}\\[-0.5em] \hrulefill\\ a=\sqrt{8}\\ b=\sqrt{58}\\ c=\sqrt{50}\\ s=\frac{\sqrt{8}+\sqrt{58}+\sqrt{50}}{2}\\\\ \qquad \frac{\sqrt{58}+7\sqrt{2}}{2} \end{cases}[/tex]
[tex]A=\sqrt{\frac{\sqrt{58}+7\sqrt{2}}{2}\left(\frac{\sqrt{58}+7\sqrt{2}}{2}-\sqrt{8} \right)\left(\frac{\sqrt{58}+7\sqrt{2}}{2}-\sqrt{58} \right)\left(\frac{\sqrt{58}+7\sqrt{2}}{2}-\sqrt{50} \right)} \\\\\\ ~\hfill {\Large \begin{array}{llll} A=10 \end{array}}~\hfill[/tex]
