f(x) = x + 1 is the monic quadratic polynomial used here
Given,
The monic quadratic polynomial f(x)
The remainder when f(x) divided by x - 1 is 2
The remainder when f(x) divided by x - 3 is 4
We have to find f(x).
Monic Quadratic Polynomial;-
A monic polynomial in algebra is a univariate polynomial with a single variable and a leading coefficient of 1. Its leading coefficient is the highest degree nonzero coefficient.
Here,
The remainder when f(x) divided by x - 1 is 2
Assume f(x) / x - 1 = 1
Then,
f(x) = (x - 1) × 1 + 2
f(x) = x - 1 + 2
f(x) = x + 1
Next,
The remainder when f(x) divided by x - 3 is 4
Assume f(x) / x - 3 = 1
Then,
f(x) = (x - 3) × 1 + 4
f(x) = x - 3 + 4
f(x) = x + 1
That is,
The monic quadratic polynomial used here is, f(x) = x + 1
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In a tug-of-war, Team A is pulling to the left with 6 Newtons and Team B is pulling to the right with
4 Newtons. What is the net force?
Answer: 2n!
Step-by-step explanation:
6N -4 N =2N
Use the function below to find F(3).
Answer:
c) 4/3
Step-by-step explanation:
Line j has an equation of y= 1 /5 x+7. Perpendicular to line j is line k, which passes through the point (– 1,– 3). What is the equation of line k
Answer:
y = -5x -8
Step-by-step explanation:
The slope must be the negative reciprocal(-5). So we have y = -5x + b, where we need to find b. To do that, we will plug in the known point. This gives us [tex]-3 = -1 * -5 + b[/tex], or[tex]-3 = 5 + b[/tex]. We subtract 5 from both sides to get b = -8.
what is the seventh partial sum for the sequence - 13,22,31,40....
?
The nth partial sum of an infinite sequence is the sum of first n terms starting from the first one. The partial sum of the sequence is given as 280.
What is an arithmetic sequence?An arithmetic sequence is a kind of sequence of numbers where the difference of any two consecutive terms are the same.
This difference is known as the common difference of the sequence.
Given that,
The sequence is 13,22,31,40....
The first term of the sequence is a₁ = -13.
The difference between consecutive terms is,
22 - 13 = 9
31 - 22 = 9
And, 40 - 31 = 9
Since, the difference between two consecutive terms are same, the given sequence is arithmetic.
The sum of first n terms of an arithmetic sequence is given as,
Sₙ = n/2(2a₁ + (n - 1)d)
Substitute n = 7 to obtain,
S₇ = 7/2(2 × 13 + (7 - 1) × 9)
= 7/2 × 80
= 280
Hence, the seventh partial sum of the given sequence is 280.
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Rewrite the following equation in slope-intercept form.
–10x − 11 = –12y
Answer:
y = 5/6x+11/12
Step-by-step explanation:
divide by -1 on both sides to get 10x+11=12y
divide by 12 on both sides
-4(v + 1) + 6v = 2(v + 2) - 8
Answer:
0 , All real numbers are solutions.
Step-by-step explanation:
Let's solve your equation step-by-step.
−4(v+1)+6v=2(v+2)−8
Step 1: Simplify both sides of the equation.
−4(v+1)+6v=2(v+2)−8
(−4)(v)+(−4)(1)+6v=(2)(v)+(2)(2)+−8(Distribute)
−4v+−4+6v=2v+4+−8
(−4v+6v)+(−4)=(2v)+(4+−8)(Combine Like Terms)
2v+−4=2v+−4
2v−4=2v−4
Step 2: Subtract 2v from both sides.
2v−4−2v=2v−4−2v
−4=−4
Step 3: Add 4 to both sides.
−4+4=−4+4
0=0
Answer:
no solution
Step-by-step explanation:
-4(v + 1) + 6v = 2(v + 2) - 8
-4v - 4 + 6v = 2v + 4 - 8
2v - 4 = 2v - 4
0 = 0
no solution
in randomized, double-blind clinical trials of a new vaccine, were randomly divided into two groups. subjects in group 1 received the new vaccine while subjects in group 2 received a control vaccine. after the second dose, of subjects in the experimental group (group 1) experienced as a side effect. after the second dose, of of the subjects in the control group (group 2) experienced as a side effect. does the evidence suggest that a higher proportion of subjects in group 1 experienced as a side effect than subjects in group 2 at the level of significance? question content area bottom part 1 verify the model requirements. select all that apply. a. the data come from a population that is normally distributed. b. the samples are independent. your answer is correct.c. the sample size is more than 5% of the population size for each sample. d. the samples are dependent. e. and your answer is correct.f. the sample size is less than 5% of the population size for each sample. your answer is correct. part 2 determine the null and alternative hypotheses. : equals : greater than part 3 find the test statistic for this hypothesis test.\
A recipe calls for 2 1/4 cups of sugar for 1 1/2 dozen cookies. What amount of sugar is needed for one dozen cookies?
The required number of cups of sugar for 1 dozen of cookies is 3 cups.
Given that,
A recipe calls for 2 1/4 cups of sugar for 1 1/2 dozen cookies. What amount of sugar is needed for one dozen cookies? is to be determined.
The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
here,
For
2 1/4 cups = 9 / 2 cups
number of cookies = 3/2 dozen
For.
1 dozen cookies = 9 / 2 / [ 3 / 2] = 3 cups of sugar
Thus, the required number of cups of sugar for 1 dozen of cookies is 3 cups.
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The sum of a number and 7 times its reciprocal is 16/3. Find the number
Answer:
7/3 or 3
Step-by-step explanation:
Step by step explanation is shown in the picture attached.
Hope it helps
a local zoo found that if the price of admission was $15$15, the attendance was about 15501550 customers per day. when the price of admission was dropped to $9$9, attendance increased to about 29002900 per day. write a linear equation for the attendance in terms of the price, p. ( a
Linear equation of attendance in terms of the price is A = -225p + 4925
What is linear equation?
An equation is said to be linear if the maximum power of the variable is consistently 1. Another name for it is a one-degree equation. A linear equation with one variable has the conventional form Ax + B = 0. In this case, the variables x and A are variables, while B is a constant. A linear equation with two variables has the conventional form Ax + By = C. Here, the variables x and y, the coefficients A and B, and the constant C are all present.
Attendance as a function of ticket price.
A(p) = mp + b
We have two points (p, A): (15, 1550), (9, 2900)
Slope: m = (2900-1550)/(9-15) = 1350/-6 = -225
A = -225p + b
Using (15, 1550) as (p, A) find b
1550 = -225(15) + b
1550 = -3375 + b
4925 = b
A = -225p + 4925
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two gamblers are playing a coin toss game. gambler a has (n 1) fair coins; b has n fair coins. what is the probability that a will have more heads than b if both flip all their coins
Gambler A loses his comparative advantage the more coins that are tossed for both Gamblers. But Gambler A will always maintain the advantage and has quite a heavy advantage when a few coins are tossed.
What is the binomial distribution?
The discrete probability distribution of the number of successes in a series of n independent experiments is called the binomial distribution with parameters n and p in probability theory and statistics.
Let X be the count of successes in the first n flips by A, Y the indicator that the last flip by A is also a success, and Z the count of successes in n flips by B.
Because Y is independent of X and Z, then:
P(X + Y > Z) = P(Y = 1) P(X ≥ Z) + P(Y = 0) P(X > Z)
= 1/2(P(X = Z) + 2P(X > Z))
P(X > Z) = P(Z > X)
Gambler A loses his comparative advantage the more coins that are tossed for both Gamblers. But Gambler A will always maintain the advantage and has quite a heavy advantage when a few coins are tossed.
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suppose a student completes an experiment with an average value of 2.5 ml and a calculated standard deviation of 0.66 ml. what is the minimum value within a 1 sd range of the average?
The minimum value within a 1 sd range of the average as calculated from the given data is 1.84 ml.
standard deviation range = [average - SD, average + SD]
Here ,
the maximum value =
average - SD = 2.5 - 0.66 = 1.84 ml
and the maximum value =
average + SD = 2.5+ 0.66 = 3.16 ml
Both the standard deviation and range are indicators of how widely distributed a set of data is. Given that they are both measures of variance, each number conveys to us in its own particular way how dispersed the data are.
Although the range and standard deviation do not directly correlate, there is a general method that can be used to connect these two statistics. The range rule for standard deviation is another name for this connection.
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NO LINKS! PLEASE HELP ME!
Answer:
4860 bacteria
Step-by-step explanation:
The given scenario can be modelled using an exponential function with base e.
[tex]\boxed{\begin{minipage}{9 cm}\underline{General form of an Exponential Function with base $e$}\\\\$y=Ae^{kx}$\\\\where:\\\phantom{ww}$\bullet$ $A$ is the initial value ($y$-intercept). \\ \phantom{ww}$\bullet$ $e$ is Euler's number. \\ \phantom{ww}$\bullet$ $k$ is some constant.\\\end{minipage}}[/tex]
Define the variables:
Let y be number of bacteria.Let x be the time (in hours)Given the initial number of bacteria is 20, then A = 20.
[tex]\implies y=20e^{kx}[/tex]
If the bacteria triples its number in 16 hours, then when x = 16:
[tex]\implies y=3 \times 20= 60[/tex]
Substitute x = 16 and y = 60 into the equation and solve for k:
[tex]\begin{aligned}y&=20e^{kx}\\\implies 60&=20e^{16k}\\\dfrac{60}{20}&=e^{16k}\\3&=e^{16k}\\\ln 3&=\ln e^{16k}\\ \ln3&=16k \ln e\\\ln3&=16k\\k&=\dfrac{1}{16} \ln 3\end{aligned}[/tex]
Therefore, the equation that models the given scenario is:
[tex]\large\boxed{y=20e^{\left(\frac{1}{16}x \ln 3\right)}}[/tex]
To find how many bacteria there are after 80 hours, substitute x = 80 into the equation and solve for y:
[tex]\implies y=20e^{\left(\frac{1}{16}(80) \ln 3\right)}[/tex]
[tex]\implies y=20e^{\left(5 \ln 3\right)}[/tex]
[tex]\implies y=20e^{\left(\ln 3^5\right)}[/tex]
[tex]\implies y=20e^{\left(\ln 243\right)}[/tex]
[tex]\implies y=20 \cdot 243[/tex]
[tex]\implies y=4860[/tex]
Therefore, there are 4860 bacteria after 80 hours.
-----------------------------------------------------------------------------------------------
Please note that I have solved this question so that you can calculate the number of bacteria for any number of hours, part hours, etc.
However, there is a much simpler way of solving this particular question.
If the number of bacteria triples every 16 hours, then it will triple 5 times within 80 hours since 80 ÷ 16 = 5.
Therefore:
Hour 0: 20Hour 16: 20 × 3 = 60Hour 32: 60 × 3 = 180Hour 48: 180 × 3 = 540Hour 64: 540 × 3 = 1620Hour 80: 1620 × 3 = 4860As you can see, the result is the same as the previous method.
Solve for m 3/5m > -12 help I’ll give 30 points
Answer:
m > -20
Step-by-step explanation:
Multiply the 5 on both sides. Now you have 3m > -60. Divide both sides by 3. Now you have m > -20.
Voila
in a survey of 500 residents, 300 were opposed to the use of photo-cops for issuing traffic tickets. find the margin of error that corresponds to a 95% confidence interval for the proportion of residents that are opposed to the use of photo-cops for issuing traffic tickets.
The margin of error that corresponds to a 95 % confidence interval for the proportion of residents that are opposed to the use of photo - cops for issuing traffic tickets is 0.04.
Given:
in a survey of 500 residents, 300 were opposed to the use of photo-cops for issuing traffic tickets.
Sample size n = 500
Number of success X = 300
Sample proportion p = X/n
= 300/500
= 30/50
= 3/5
= 0.6
z score at 95% = 1.96
Margin of error = z * [tex]\sqrt{p(1-p)/n}[/tex]
= 1.96 * [tex]\sqrt{0.6(1-0.6)/500}[/tex]
= 0.04
Therefore the margin of error that corresponds to a 95 % confidence interval is 0.04.
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Of the 47 plays attributed to a playwright, 11 are comedies, 20 are tragedies, and 16 are histories. If one play is selected at random, find the odds against selecting a history
The odds against selecting a history are 31/16.
Define probability.Probability is the concept that describes the likelihood of an event occurring. In many situations, it is necessary to foresee how something will turn out in the actual world. We may be aware of the result of an event or not. When this occurs, we state that there is a possibility that the event will occur.
Probability is the ratio of the most likely outcomes to all other possible outcomes of an event. The number of successful outcomes for an experiment with 'n' outcomes may be expressed using the symbol x. The following formula can be used to determine the likelihood of an event.
Probability(Event) = favorable outcome/Total outcome = x/n
Number of playwrights = 47
Number of histories = 16
Therefore, the odds against selecting a history are
(47 - 16) / 16
= 31/16
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given that $x$, $\frac{1}{x}$, $y$, $\frac{1}{y}$, $z$ and $\frac{1}{z}$ are all integers, how many distinct values of $x y z$ are possible?
Distinct value of x, y, z are will be 3
Consider the function will be f(x) = x3 - x2 - 2x +2 = ( x - 1 ) (x2 - 2 )
we have three real roots and ,
x + y + z = - 1/ (-1)
= 1 ,
1/x + 1/y + 1/z = -2 /(-2)
= 1
Consider the function
f(x) = x3 - x2 - 40 + 20.
Set the derivation equal 0 we get x = 4 and x = -10/3
are two local extrema. When x= 4 , f(x)<0,when x = -10/3 . f(x) >0 there are three real roots . we have x + y + z = 1 and 1/x + 1/y + 1/z = - (-40)/ 20 = 2
consider the function
f(x) = x3 - x2- 40x + 40/3 .
we still have same local extrema as (2) and we have x= 4 , f(x) < 0 and we
x= -10/3 , f(x) > 0 so there are three real roots. We have x+ y + z = 1 and 1/x + 1/y + 1/z = -(-40)/40/3 = 3
So, there are 3 distinct values
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what is the sum of all integers $n$ that satisfy $$-3n < 7n-20 < 3n~?$$ in other words, what is the sum of all integers $n$ that satisfy both of the inequalities $$-3n<7n-20\text{~~and~~}7n-20<3n~?$$
3+4=7 is the total of all possible integer values of n.
What are integers?Integers include zero, a positive natural number, and a negative integer represented by a minus sign.
The negative numbers are the inverse additions of the comparable positive numbers.
In mathematical notation, the letter Z is widely used to denote a set of numbers.
An integer is a whole number that can be positive, negative, or zero and is not a fraction.
Integer examples include -5, 1, 5, 8, 97, and 3,043.
So, the two inequalities are resolved independently. We add 3n to both sides of the first inequality, -3n7n-20, to find the solution:
Then, we add 20 to both sides to get 010n-20:
20<10n.
Lastly, multiplying both sides by 10 yields 2n, indicating that n must be bigger than 2.
We deduct 3n from both sides of the second inequality, 7n-203n:
Add 20 on both sides of 4n-200 to obtain:
4n<20.
Lastly, n must be less than 5 since the result of dividing both sides by 4 is n5.
Only the numbers 3 and 4 are greater than 2 and less than 5.
Therefore, 3+4=7 is the total of all possible integer values of n.
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From Arcadia, how much farther is it to Lakewood than to Bluepoint?
5.38 mi
12.6 mi
12.88 mi
Bluepoint
Arcadia
Lakewood
Dayton
The distance from Lakewood than to Bluepoint is 17.98 miles.
What is Unitary Method?The unitary technique involves first determining the value of a single unit, followed by the value of the necessary number of units.
For example, Let's say Ram spends 36 Rs. for a dozen (12) bananas.
12 bananas will set you back 36 Rs. 1 banana costs 36 x 12 = 3 Rupees.
As a result, one banana costs three rupees. Let's say we need to calculate the price of 15 bananas.
This may be done as follows: 15 bananas cost 3 rupees each; 15 units cost 45 rupees.
Given:
Distance between Bluepoint to Arcadia = 5.38 miles.
Distance between Arcadia to Lakewood= 12.6 miles.
Distance between Lakewood to Dayton = 12.88 miles.
So, the distance from Lakewood than to Bluepoint
= 12.6 +5 .38
= 17.98 miles
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ANSWER ASAP!!!!!!!!!!!!!!!
Choose the scatter plot that suggests a linear relationship between x and y.
Answer:
The 4th scatter plot indicates a linear relationship
Step-by-step explanation:
The first and third scatter plot graphs are too noisy to indicate a proper trend. The second scatter plot is quadratic and the fourth one is linear.
A triangle has an area of 32 mm² in the base of 8 mm what is the height ?
The height of the triangle is 32 mm
How to find the height of a triangle?A triangle is a polygon with three sides. The sum of angles in a triangle is 180 degrees. The area of a triangle can be represented as follows:
area of a triangle = 1 / 2 bh
where
b = baseh = heightTherefore, the triangle has an area of 32 mm² and base of 8 mm. Let's find the height of the triangle as follows:
area of a triangle = 1 / 2 bh
32 = 1 / 2 × 8 × h
4h = 32
divide both sides by 4
h = 32 / 4
h = 8 mm
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Compute the requested average. Tim worker receives the following income monthly from a second job. Jan. Feb. Mar. Apr. May june $200 $150 $250 $260 $285 $380 what is his average pay? (to the nearest hundredth. ).
use the graphs of f and g to describe the transformation from the graph of f the the graph of f to the graph of g.
f(x) = -3x + 4 ; g(x) = f(x) + 1
The value of the function f(x) = -3x+4 is -3x+5
What is function?It is defined as a special type of relationship, and they have a predefined domain and range according to the function every value in the domain is related to exactly one value in the range.
Given a function, f(x) = -3x+4
g(x) is described as f(x) +1
g(x) = -3x+4+1
g(x) = -3x+5
The graph of f(x) will move 1 unit farther when transformed into g(x).
Hence, The value of the function f(x) = -3x+4 is -3x+5
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Fred deposits $6,500 in a saving account that pays 0.5% interest, compounded quarterly. Round each answer to the nearest cent.
a. Find the first quarter's interest.
b. Find the first quarter's ending balance.
c. Find the second quarter's interest.
d. Find the second quarter's ending balance.
e. Find the third quarter's interest.
f. Find the third quarter's ending balance.
g. Find the fourth quarter's interest.
h. What is the balance at the end of one year?
i. How much interest does the account earn in the first year?
The amount of interest earned quarterly at 0.5% interest rate compounded quarterly are;
a. Interest for the first quarter = $8.125
b. First quarter ending balance = $6,508.125
c. Interest for the second quarter = $8.13515625
d. Second quarter ending balance = $6,516.26015625
e. Interest for the third quarter = $8.1432520325
f. Third quarter closing balance = $6,524.40548145
g. Interest for the fourth quarter = $8.15550685975
h. The amount of interest earned in a year is about $32.56
What is an interest earned on an amount?An interest is the amount a borrower pays to a lenders as to make use of the lender's funds.
The amount Fred deposited, P = $6,500
The interest paid by the account, r = 0.5% compounded quarterly
a. The first quarter interest can be found by the formula;
[tex]C.I. = P\cdot \left(1 + \dfrac{r}{4} \right)^{4\times \dfrac{1}{4} }- P[/tex]
Therefore;
[tex]C.I. = 6500\times \left(1 + \dfrac{\frac{0.5}{100} }{4} \right)^{4\times \dfrac{1}{4} }- 6500 = 8.125[/tex]
The interest after the first period is $8.125
b. The first quarter ending balance is therefore;
A = P + C.I.
A = $6,500 + $8.125 = $6,508.125
c. The second quarter's interest can be found from the formula;
[tex]C.I. = P\cdot \left(1 + \dfrac{r}{4} \right)^{4\times t} - P[/tex]
Here, P = The balance from the first quarter = $6,508.125
t = 1/4 for one quarter
Therefore;
[tex]C.I. = 6508.125\times \left(1 + \dfrac{\frac{0.5}{100} }{4} \right)^{4\times \dfrac{1}{4} }- 6508.125 = 8.13515625[/tex]
The second quarter's interest is $8.13515625
d. The second quarter ending balance is therefore;
A = $6,508.125 + $8.13515625 = $6,516.26015625
The second quarter ending balance is $6,516.26015625
e. The third quarters interest is therefore;
[tex]C.I. = 6516.26015625\times \left(1 + \dfrac{\frac{0.5}{100} }{4} \right)^{4\times \dfrac{1}{4} }- 6516.26015625 = 8.1432520325[/tex]
The third quarter's interest is about $8.1433
f. The third quarter's ending balance is therefore;
A ≈ $6516.26015625 + $8.1432520325 = $6,524.40548145
The third quarter's ending balance is $6,524.40548145
g. The fourth quarter's interest can be found as follows;
[tex]C.I. = 6,524.40548145 \times \left(1 + \dfrac{\frac{0.5}{100} }{4} \right)^{4\times \dfrac{1}{4} }- 6524.40548145 = 8.15550685975[/tex]
The fourth quarter's interest is; $8.15550685975
h. The balance at the end of one year is therefore;
Balance = C.I. fourth quarter + Principal for the fourth quarter
Therefore;
Balance = $6,524.40548145 + $8.15550685975 = $6,532.56098831
i. The interest the account earns in a year is therefore;
$6,532.56098831 - $6,500 = $32.5609883
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please help me please
Answer:
For the first one
Part a:
115x + 685y >= 2300
Part b:
See explanation below
Second one:
1.5x + 1.0y >= 2000
Rest of explanation below
Step-by-step explanation:
First one.
Let x be the number of skimboards sold and y be the number of longboards sold.
Then revenue from sales of x units at $115 and y units at $685 would be
115x + 685y
This must be at lest the overhead to make zero loss
So the inequality is
115x + 685y ≥ 2300 (Answer a)
Part b
Any combination of x and y which satisfies this inequality will provide a profit. I have provided a graph of this inequality using an online graphing calculator. I am not sure what your teacher wants you to use.
Any point in the shaded region represents a combination of x and y which satisfies that inequality. However since you cannot sell a fraction of a surfboard, only points which represent whole numbers can be valid. Also only values of x and y >=0 can be chosen. So any point in the shaded region which has x and y non-negative and x and y whole numbers will generate a profit
Again I am not sure whether this question is asking for a specific value or possible values.
A specific value can be (x=5, y = 10) which is well within the shaded region. This represents 5 skimboards and 10 longboards with a revenue of 5 x 115 + 10 x 685 = $ 7425
Or you could sell 0 skimboards and 4 longboards giving you a revenue of
4 x 685 = $2,740
Or 20 skimboards and zero long boards which is a revenue of $2300 which breaks even with the overhead
Second Question: Modeling
Using x for number of hot dogs sold and y for number of sodas, the inequality is
(a) 1.0x + 1.25y >= 2000
(b) Graph attached
(c) 5 solution points plotted
Here are 5 values that fill fit the inequality
1000 hot dogs, 1000 sodas (Point A)400 hot dogs, 1400 sodas (Point B)600 hot dogs, 1200 sodas (Point C)0 hot dogs, 1800 sodas (Point D)1600 hot dogs, 800 sodas (Point E)Hope that helps. Sorry, best I could do
if a baseballs cost $4.50 how much will 5 cost
Answer:
22.5
Step-by-step explanation
22.5 is the correct answer because 4.50 times 5 is 22.5 :))))
Answer:
22.5
if one baseball cost $ 4.50 , we multiply it by 5 , or add 4.50+4.50+4
.50+4.50+4.50 same thing.
If the bond angle between two adjacent hybrid orbitals is 109. 5°, which is the hybridization?.
109.5° is the bond angle between two nearby hybrid orbitals. sp3 will be the hybridization.
The hybridization would be sp3. This is because the bond angle of 109.5° is characteristic of hybridized orbitals with four sp3 orbitals, which form an angle of 109.5°. sp3 hybridization occurs when one s orbital and three p orbitals mix to form four sp3 hybrid orbitals.
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Find the 8th term of the arithmetic sequence 3x-13x−1, -2x+6−2x+6, -7x+13, ...−7x+13,...
Answer:
11x +98 just assume it to be an arithmetic progression
Newton is thinking of a number that is greater than 0.4 and less than 0.5. The greatest digit in the number is in the hundredths place. What might Newton’s number be?
Answer:
0.49
Step-by-step explanation:
Answer:
0.49
Step-by-step explanation:
0.4 < x < 0.5
A sample of size
=n83
is drawn from a population whose standard deviation is
=σ42
Part 1 of 2
(a) Find the margin of error for a
95%
confidence interval for
μ. Round the answer to at least three decimal places.
The margin of error for a
95%
confidence interval for
μ
is
. Part 2 of 2
(b) If the confidence level were
99%
, would the margin of error be larger or smaller?
The margin of error would be higher at a 99% confidence level.
Confidence level is the probability, or degree of certainty, that the confidence interval would include the real population parameter when a random sample is drawn numerous times.
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