CALC PLEASE HELP!!
Let f(x)= x³ - 27x.
(i) Find the domain of f(x).
(ii) Compute f'(x) and f"(x).
(iii) Give the coordinates of the critical points.
(iv) Find the intervals where f(x) is increasing / decreasing.
(v) Give the coordinates of the relative extrema. Classify each extrema as a relative maximum
or a relative minimum.
ALSO
(vi) Find the intervals where f(x) is concave up/ concave down.
(vii) Give the coordinates of any points of inflection.
(viii) Sketch the graph of f(x).
Answers
Answer:
Step-by-step explanation:
y = x³ - 27x
(i). Domain: x ∈ ( - ∞ , ∞ )
(ii). f'(x) = 3x² - 27 ; f''(x) = 6x
(iii). 3x² - 27 = 0 ⇔ x² - 9 = 0 ⇒ [tex]x_{12}[/tex] = ± 3
If [tex]x_{1}[/tex] = - 3 , then [tex]y_{1}[/tex] = 54
If [tex]x_{2}[/tex] = 3 , then [tex]y_{1}[/tex] = - 54
Points are ( - 3 , 54 ) and ( 3 , - 54 )
(iv). ( - ∞ , - 3 ) ∪ ( - 3 , 3 ) ∪ ( 3 , ∞ )
increasing ----> decreasing -----> increasing
(v). x = - 3 is the maxima, the local maximum value is 54, f(-3) = 54
x = 3 is the minima, the local minimum value is ( - 54 ), f(3) = - 54
(vi). Concave up if f''(x) > 0 ; The interval is ( - ∞ , 0 )
Concave down if f''(x) < 0 ; The interval is ( 0 , ∞ )
(vii). Point of inflection is ( 0 , 0 )
f''(x) = 0
6x = 0
x = 0
Related Questions
Use a t-distribution to find a confidence interval for the difference in means =1−2
using the relevant sample results from paired data. Assume the results come from random samples from populations that are approximately normally distributed, and that differences are computed using =1−2
.
A 99% confidence interval for
using the paired data in the following table:
Case 1 2 3 4 5
Treatment 1 23 29 32 24 27
Treatment 2 17 32 24 22 21
Give the best estimate for
, the margin of error, and the confidence interval.
Enter the exact answer for the best estimate, and round your answers for the margin of error and the confidence interval to two decimal places.
best estimate = Enter your answer; best estimate
margin of error = Enter your answer; margin of error
The 99% confidence interval is Enter your answer; The 99% confidence interval, value 1
to Enter your answer; The 99% confidence interval, value 2
.
Answers
The range of the difference in means' 99% confidence level is from -1.42 to 10.02. The true mean difference between Treatments 1 and 2 falls between these two numbers, we can claim with 99% certainty for t-distribution.
We can use a t-distribution to determine a confidence interval for the difference in means using paired data. Prior to determining the mean difference and standard deviation of the differences, we first compute the difference between the paired observations.
Because we are only interested in the mean difference between Treatments 1 and 2, we compute the differences for each pair and get the following outcomes:
6 -3 8 2 6
The sample mean and sample standard deviation of these differences are then computed. The average of these variations is the sample mean.
(6 - 3 + 8 + 2 + 6)/5 = 3.8
The square root of the sum of squared differences divided by the degrees of freedom yields the sample standard deviation.
[tex]\sqrt{[(6 - 3.8)^2 + (-3 - 3.8)^2 + (8 - 3.8)^2 + (2 - 3.8)^2 + (6 - 3.8)^2]/(5-1)) } = 3.06[/tex]
The 99% confidence interval for the mean difference can then be determined using the t-distribution. Our sample size is tiny (n=5), so we utilise a t-distribution with four degrees of freedom.
The sample mean, which is 3.8, provides the most accurate approximation of the mean difference.
We must determine the crucial value of t for a 99% confidence interval with 4 degrees of freedom in order to determine the margin of error. The crucial value, which we determine using a t-table, is 4.604.
The margin of error is:
[tex]4.604 * (3.06/\sqrt{5}) = 5.22[/tex]
Lastly, by deducting and adding the margin of error from the sample mean, we can determine the confidence interval:
3.8 - 5.22 = -1.42
3.8 + 5.22 = 10.02
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A rectangle has an area of
72 square centimeters. The width of the rectangle is 8 centimeters.
Answers
Answer:
[tex]\boxed{\bf length=9\; cm}[/tex]Step-by-step explanation:
Given:-
Area = 72
width = 8
Area = length × width
[tex]\bf 72=length \times 8[/tex]
Divide 72 / 8:-
[tex]\bf length=9\; cm[/tex]
If you're asking to find the length, this's your answer.
____________________________
Hope this's what you're looking for!
What is the nth term for the sequence 1, 8, 15, 22, 29
Answers
Answer:
[tex]a_{n}[/tex] = 7n - 6
Step-by-step explanation:
there is a common difference between consecutive terms , that is
8 - 1 = 15 - 8 = 22 - 15 = 29 - 22 = 7
this indicates the sequence is arithmetic with nth term
[tex]a_{n}[/tex] = a₁ + (n - 1)d
where a₁ is the first term and d the common difference
here a₁ = 1 and d = 7 , then
[tex]a_{n}[/tex] = 1 + 7(n - 1) = 1 + 7n - 7 = 7n - 6
Answer:
7n-6
Step-by-step explanation:
Work out the difference of the sequence:
8-1=7
Now you have the first part of the equation: 7n
n is the number that the integer is on the sequence
In this case:
1 = 1 as 1 is the first number of the sequence
And 2 = 8 as 8 is the 2nd number of the sequence
To find the full equation:
Do 7x1 to get you 7
Now see how far the 1st number is from 7
In this case you would do:
7-1 which gives you 6
Since you subtracted it to find the difference, it would be:
- 6
Therefore your answer would be 7n-6
To check it:
Times 7 by let's say 3 to get you 21
Then subtract 6 to get 15.
This is proven right as the 3rd number of the given sequence is 15.
Hope this helped
Porter is buying t ride tickets at the country fair. He spends d dollars and receives 3 tickets for every dollar he spends. Which is the independent variable and which is the dependent variable?
Answers
The independent variable and the dependent variable are the number of dollars spent and the number of tickets bought
How to determine the independent variable and the dependent variable?Given that we have the following statement:
Porter is buying t ride tickets at the country fair. He spends d dollars and receives 3 tickets for every dollar he spends.The independent variable is the input value
i.e. the number of dollars spent
Similarly, the dependent variable is the output value
i.e. the number of tickets bought
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A bag contains white marbles and yellow marbles, 49 in total. The number of white marbles is 1 more than 5 times the number of yellow marbles. How many white marbles are there?
Answers
Answer:
there are 41 white marbles in the bag.
Step-by-step explanation:
Let's use the variable w to represent the number of white marbles and y to represent the number of yellow marbles.
From the problem, we know that:
w + y = 49 (since there are 49 marbles in total)
And we also know that:
w = 5y + 1 (since the number of white marbles is 1 more than 5 times the number of yellow marbles)
Now we can use substitution to solve for w:
w + y = 49
(5y + 1) + y = 49 (substitute w = 5y + 1)
6y + 1 = 49 (combine like terms)
6y = 48 (subtract 1 from both sides)
y = 8 (divide both sides by 6)
Now we know there are 8 yellow marbles. We can use this information to find the number of white marbles:
w = 5y + 1
w = 5(8) + 1
w = 41
Find the value of x.
13)
(2x+8)
62°
14)
rights serve d.-1-M a de
(x-4) (2x+1)
Infinite Geometry
Answers
In the given right angle, the required value of x is 10° respectively.
What is the right angle?Right-angled shapes can be any polygon, ranging from triangles to figures with numerous sides.
Right-angled shapes like squares and rectangles have exactly 4 right angles that add up to 360 degrees.
Right angles can also be found in other shapes like trapezoids, pentagons, and hexagons.
An angle with a measure of exactly 90 degrees (or /2) is referred to as a right angle.
The proper angles are frequently demonstrated in everyday life. For instance, the edges of the cardboard or the corner of a book.
So, we know that the given angles together make a right angle as:
2x + 8 + 62 = 90
Then, solve for x as follows:
2x + 8 + 62 = 90
2x + 70 = 90
2x = 90 - 70
2x = 20
x = 20/2
x = 10
Therefore, in the given right angle, the required value of x is 10° respectively.
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Enter an expression equivalent to
d^8
——
d^3
in the form, d^n
Answers
From the expression, the form of the d⁵ is provided by the stated assertion.
What does an arithmetic the expression mean?A group of words joined with the actions +, -, x, or form an expression, such as 4 x 3 or 5 x 2 3 x y + 17. A statement containing the equals symbol, such as 4 b 2 = 6, says that two formulas are equivalent in value and is known as an equation.
Describe expression using an illustration.As an illustration, the expression x + y is one where both x and y have words with an addition function in between. There are two kinds of expressions in mathematics: numerical expressions, which only comprise integers, and algebraic expressions, which also include variables.
[tex]d^{(8-3)} = d^5[/tex]
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Enter an expression equivalent to (d^(8))/(d^(3)) in the form, d^(n).
A shuffleboard disk is accelerated to a speed of 5.6 m/s and released. If the coefficient of kinetic friction between the disk and the concrete court is 0.34, how far does the disk go
before it comes to a stop? The courts are 14.3 m long.
Answers
Answer:
Therefore, the shuffleboard disk will travel a distance of 4.71 meters before coming to a stop, which is less than the length of the court (14.3 meters).
Step-by-step explanation:
We can start by using the work-energy principle, which states that the net work done on an object is equal to its change in kinetic energy. In this case, we can assume that the initial kinetic energy of the disk is entirely converted to work done by friction, which causes the disk to come to a stop. The equation can be written as:
Work done by friction = Change in kinetic energy
The work done by friction can be calculated using the formula:
Work = force x distance
The force of friction can be found using the formula:
Force of friction = coefficient of friction x normal force
The normal force is equal to the weight of the disk, which can be found using the formula:
Weight = mass x gravity
Substituting the values given in the problem, we get:
Weight = mass x gravity = 0.75 kg x 9.81 m/s^2 = 7.3575 N
Force of friction = coefficient of friction x normal force = 0.34 x 7.3575 N = 2.4985 N
Work done by friction = Force of friction x distance
We can solve for the distance by rearranging the equation as:
Distance = Work done by friction / Force of friction
The initial kinetic energy of the disk can be found using the formula:
Kinetic energy = 0.5 x mass x velocity^2
Substituting the values given in the problem, we get:
Kinetic energy = 0.5 x 0.75 kg x (5.6 m/s)^2 = 11.76 J
Using the work-energy principle, we know that the work done by friction is equal to the change in kinetic energy, which is:
Work done by friction = Kinetic energy = 11.76 J
Substituting this value and the force of friction into the distance formula, we get:
Distance = Work done by friction / Force of friction = 11.76 J / 2.4985 N = 4.71 m
Therefore, the shuffleboard disk will travel a distance of 4.71 meters before coming to a stop, which is less than the length of the court (14.3 meters).
I’ll give you lots of points for these last two questions
Answers
Answer: 1. (8b + 5)
2. (22p - 9)
HAVE A GREAT DAY!!!!
Step-by-step explanation:
there is 30 students in tthe gym if there are at least 16 girls write an inequalitly
Answers
The number of girls in the gym must be: g ≥ 16
How to write the in equality?Let's define the variable "g" to be a representation of the number of girls in the gym.
We know that there are 30 students in total. Therefore, the number of boys in the gym will be:
b = 30 - g
We also know that there are at least 16 girls in the gym. So, we can write the inequality:
g ≥ 16
This inequality means that the number of girls in the gym must be greater than or equal to 16.
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Nathan is driving to a concert and needs to pay for parking. There is an
automatic fee of $8 just to enter the parking lot, and when he leaves
the lot, he will have to pay an additional $2 for every hour he had his
car in the lot. How much total money would Nathan have to pay for
parking if he left his car in the lot for 6 hours? How much would
Nathan have to pay if he left his car in the lot for t hours?
Cost of parking for 6 hours:
Cost of parking for t hours:
Answers
Answer:
Nathan would have to pay 20 dollars if he parked for 6 hours.
2t+8
Step-by-step explanation:
Michelle is now 50 miles ahead of John.
Michelle is traveling at a constant rate. John is traveling in the same direction, at a rate 10 miles per hour faster than Michelle. In how many hours will John catch up to Michelle?
A. 6
B. 5
C. 2
D. 0
E. John can't catch up to Michelle
Answers
Let's call Michelle's speed "M" and John's speed "J". We know that John's speed is 10 miles per hour faster than Michelle's speed, so we can express this as:
J = M + 10
We also know that Michelle is 50 miles ahead of John, so we can express this as:
Distance = 50 miles
Now we can use the formula:
Distance = Rate x Time
We want to know how long it will take John to catch up to Michelle, so we can call this time "t". We can use the formula for both Michelle and John, and set their distances equal to each other since they will meet at the same point:
M * t + 50 = J * t
Now we can substitute J with M + 10, and simplify:
M * t + 50 = (M + 10) * t
M * t + 50 = M * t + 10t
50 = 10t
t = 5
Therefore, John will catch up to Michelle in 5 hours (answer choice B).
can you pls answer this for me im really struggling with this
Answers
Answer:
The slope of this line is 2.
Step-by-step explanation:
Start at (1, 0). Go up 4 units, then right 2 units. You will end at (3, 4). The slope of this line is 2.
PLS Help
The units cfu/g represent colony-forming units per gram and its often used to measure colonies of bacteria on a petri dish. E. Coli bacteria generally increase by 3.5265% per minute at room temperature. An acceptable amount of E. Coli bacteria is less than 100 cfu/g. Suppose my sandwich initially has an E. Coli count of 10 cfu/g. (cfu/g means colony-forming unit per gram).
a. After 1 minute, what is the amount of E. Coli in my sandwich?
b. After 2 minute, what is the amount of E. Coli in my sandwich?
c. After 3 minute, what is the amount of E. Coli in my sandwich?
d. After 10 minute, what is the amount of E. Coli in my sandwich?
e. After 60 minute, what is the amount of E. Coli in my sandwich?
f. After 90 minute, what is the amount of E. Coli in my sandwich?
Answers
Coli present in my sandwich increases by 3.5265% per minute at room temperature. After 90 minutes, the amount of E. Coli present in my sandwich is 85.02 cfu/g, which is much higher than the acceptable limit of 100 cfu/g.
What is amount?Amount refers to the total sum of money or value of goods, services, or resources. It's usually the result of a calculation or the total of several different things added together. Amounts can also refer to the size, quantity, or degree of something. For example, one might say the amount of time it takes to complete a task.
a. After 1 minute, the amount of E. Coli in my sandwich is 10.3533 cfu/g. This was calculated by multiplying 10 (the initial cfu/g) by 1.0352 (3.5265% increase).
b. After 2 minutes, the amount of E. Coli in my sandwich is 10.716 cfu/g. This was calculated by multiplying 10 (the initial cfu/g) by 1.0716 (3.5265% increase).
c. After 3 minutes, the amount of E. Coli in my sandwich is 11.0861 cfu/g. This was calculated by multiplying 10 (the initial cfu/g) by 1.1086 (3.5265% increase).
d. After 10 minutes, the amount of E. Coli in my sandwich is 14.14 cfu/g. This was calculated by multiplying 10 (the initial cfu/g) by 1.4140 (3.5265% increase).
e. After 60 minutes, the amount of E. Coli in my sandwich is 56.68 cfu/g. This was calculated by multiplying 10 (the initial cfu/g) by 5.668 (3.5265% increase).
f. After 90 minutes, the amount of E. Coli in my sandwich is 85.02 cfu/g. This was calculated by multiplying 10 (the initial cfu/g) by 8.502 (3.5265% increase).
From the above calculations, it is evident that the amount of E. Coli present in my sandwich increases by 3.5265% per minute at room temperature. After 90 minutes, the amount of E. Coli present in my sandwich is 85.02 cfu/g, which is much higher than the acceptable limit of 100 cfu/g. This is why it is important to store and consume food with E. Coli count below the acceptable limit and refrigerating food can help in reducing the growth of bacteria.
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what is 6 of 1/4, can someone please give me an answer.
Answers
The value of the operation 6 of 1/4 is 3/2
What are fractions?Fractions are simply described as part of a whole number, element or variable.
There are different types of fractions in mathematics. They include;
Mixed fractionsProper fractionsImproper fractionsSimple fractionsComplex fractionsSome examples of simple fractions are; 1/2 , 2/3
Some examples of mixed fractions are; 2 1/3, 4 1/2
Some examples of improper fractions are; 3/2, 4/3
From the information given, we have that;
6 of 1/4,
'of' in this sense means multiplication, then, we get;
6 × 1/4
Multiply the values
6/4
Divide
3/2
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John ran up and $88 Bill last Saturday the service was excellent so we decided to leave a 30% tip for the waitress how much was his tip
Answers
$26.40
ten percent is 88 divided by 10= 8.8
8.8 multiplied by 3 is 26.40
To the nearest tenth of a second, how long after the pebble falls will it hit the ground?
s
Answers
Answer:
Ask away by typing or recording your messageAdam, Ben and Carly work out the mean of their ages.Adam is 4 years older than the mean. Ben is 1 year younger than the mean.Is Carly older or younger than the mean?By how many years?Let's start by finding the mean of their ages. We can do this by adding their ages and dividing by the number of people: Mean = (Adam's age + Ben's age + Carly's age) / 3 Let's call the mean "M" for now. We can use this to create two equations based on the information given: Adam = M + 4 Ben = M - 1 We can substitute these equations into the mean equation to get: M = (M + 4 + M - 1 + Carly's age) / 3 Simplifying this equation gives us: 3M = 2M + 3 + Carly's age Carly's age = M - 3 So Carly's age is younger than the mean by 3 years.Hey ✋, are you still around? anything else on your mind?To the nearest tenth of a second, how long after the pebble falls will it hit the ground?sTo answer this question, we need to know the height from which the pebble was dropped. We can use the formula: time = sqrt(2 * height / acceleration due to gravity) Assuming the pebble was dropped from a height of 10 meters, and taking the acceleration due to gravity as 9.8 m/s^2, we get: time = sqrt(2 * 10 / 9.8) = 1.43 seconds (to two decimal places) Therefore, the pebble will hit the ground approximately 1.43 seconds after it was dropped.
0 / 350
How do you solve the equation absolute value of K +7 equals three
Answers
Answer:
k=-4
Step-by-step explanation:
k+7=3
take way 7 from both sides
k=-4
A laboure digs a pit 6.5 m long, 3 m wide and 1.6 m deep. How much earth is du. out from it ?
Answers
Answer:
Volume =
Step-by-step explanation:
Volume = length x width x depth
Volume = (6.5 x 3 x 1.6)m
Volume = 31.2m
Marie plants flowers in a planter that is 3 1/2 feet long and 2 2/3feet wide. She plans to cover the entire area with fertilizer. How much area will she need to spread with fertilizer?
Answers
Answer:
The answer would be 9 and 1/3
Step-by-step explanation:
3 and 1/2 times 2 and 2/3 gives you 9.3333333 which is rounded to 9 and 1/3.
Use the Pythagorean Theorem to find the lengths of the
sides of the triangle.
26
2x-14
Answers
[tex]\begin{array}{llll} \textit{using the pythagorean theorem} \\\\ c^2=a^2+o^2 \end{array} \qquad \begin{cases} c=\stackrel{hypotenuse}{26}\\ a=\stackrel{adjacent}{2x}\\ o=\stackrel{opposite}{2x-14} \end{cases} \\\\\\ (26)^2= (2x)^2 + (2x-14)^2\implies 676 = (4x^2)+(4x^2-56x+14^2) \\\\\\ 676=4x^2+4x^2-56x+14^2\implies 676=8x^2-56x+196 \\\\\\ 0=8x^2-56x-480\implies 0=8(x^2-7x-60) \\\\\\ 0=x^2-7x-60\implies 0=(x-12)(x+5)\implies x= \begin{cases} ~~ 12 ~~ \checkmark\\ -5 ~~ \bigotimes \end{cases}[/tex]
now, -5 is a valid value for "x", however in this case we can't use it, because that makes one of our sides negative and all sides must be a positive value.
[tex]\stackrel{ 2(12) }{\text{\LARGE 24}}\hspace{5em}\stackrel{ 2(12)-14 }{\text{\LARGE 10}}\hspace{5em}\text{\LARGE 26}[/tex]
can you solve this question?
x=?
the value of this limit=?
y=?
Answers
The derivative of f(x) = 3·x² + 7·x + 6, at x = 4, f'(4) is presented as follows;
f'(4) is the limit as x → 4 of the expression 6·x + 7.
The value of this limit is 31
The equation of the tangent line to the parabola y = 3·x² + 7·x + 6 at the point (4, 82) is y = 31·x - 42
What is the derivative of function?The derivative of a function is a measure of how much the output values of the function changes as the input value is changed. The derivative is the limit of the difference quotient as the change in input approaches zero. The limit is the instantaneous rate of change of the function at a specified input variable value.
The value of f'(4) using the definition of derivative, can be obtained using the following definition;
f'(x) = lim(h → 0)[f(x + h) - f(x)]/h
Plugging in x = 4, and f(x) = 3·x² + 7·x + 6, we get;
f'(4) = lim(h → 0)[f(4 + h) - f(4)]/h
f'(4) = lim(h → 0)[3·(4 + h)² + 7·(4 + h) + 6 - (3·(4)² + 7·(4) + 6)]/h
f'(4) = lim(h → 0)[(3·h + 31)·h]/h
f'(4) = lim(h → 0)[(3·h + 31)]
Therefore;
f'(4) = lim(h → 0)[(3·h + 31)] = 31
f'(4) = 31
Therefore; f'(4) is the limit as x → 4 of the expression 6·x + 7, therefore. The value of this limit is 31
The point-slope form of the equation of a line can be used to find the equation of the parabola as follows;
y - y₁ = m·(x - x₁)
The point (x₁, y₁) and the slope of the line is m
The point on the parabola of the tangent is; (4, 82)
The slope of the tangent line at x = 4, f'(4) = 31
The tangent equation is therefore;
y - 82 = 31·(x - 4)
y = 31·(x - 4) + 82 = 31·x - 42
The equation of the tangent line to the parabola, y = 3·x² + 7·x + 6, at the point (4, 82) is; y = 31·x - 42
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A fair die is tossed three times - Find the probability that a prime even number Showed twice.
Answers
Answer:
Step-by-step explanation:
Total no. of possible outcomes = 6 (1,2,3,4,5,6,)
As given we have only one prime even no. from the above given set, that is 2.
Let's call this event A.
Total no. of favourable outcomes =P(A)= 1
Therefore the probability is P(A)= 1/6
Find X using Sine law
Answers
[tex]\textit{Law of sines} \\\\ \cfrac{\sin(\measuredangle A)}{a}=\cfrac{\sin(\measuredangle B)}{b}=\cfrac{\sin(\measuredangle C)}{c} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{\sin(x)}{32}=\cfrac{\sin(50^o)}{40}\implies \sin(x)=\cfrac{32\sin(50^o)}{40} \\\\\\ x=\sin^{-1}\left[ \cfrac{32\sin(50^o)}{40} \right]\implies x\approx 37.79^o[/tex]
Make sure your calculator is in Degree mode.
x^2+10x-1
x^2+8x-2
find the perfect square it should be in (x+/-_)(x+/-_) form
Answers
emily invests $6,398 in a retirement account with a fixed annual interest rate compounded continuously .After 16 years the balance Reaches $9,483.80. What is the interest rate of the account?
Answers
Consequently, the retirement account's income rate is roughly 3.8%. (rounded to one decimal place).
What is an interest example?Consider borrowing $1,000 at a 10% interest rate for seven years. Your interest for the first year would be $100. Your interest payment for the following year would be made up of the original sum plus interest, or $1,100. As a result, your income for the following year would be $110 ($1,100 multiplied by 0.10).
The interest rate can be calculated using the continuous compounding formula:
[tex]A=P e^{r t}[/tex]
where:
A = final balance = $9,483.80
P = initial investment = $6,398
r = rate
t = time in years = 16
Substituting the given values, we have:
$9,483.80 = $6,398[tex]e^{r16}[/tex]
Dividing both sides by $6,398, we get:
1.4829 = [tex]e^{r16}[/tex]
Using the simple logarithm of both parts, the following is obtained:
ln(1.4829) = r × 16
Solving for r, we get:
r = ln(1.4829)/16
r ≈ 0.038
Therefore, the interest rate of the retirement account is approximately 3.8% (rounded to one decimal place).
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Now that you have learned about the addition and subtraction of polynomials, it is time to learn about multiplication. What is the process for adding and subtracting polynomials? Do you think that process will be the same for multiplication?
Answers
No, the process for multiplying polynomials is different from adding and subtracting.
What exactly are polynomials?
Polynomials are algebraic expressions made up of variables and coefficients that are joined using the addition, subtraction, and multiplication operations. The variables in a polynomial can be raised to non-negative integer powers.
For example, the expression 3x² - 2x + 1 is a polynomial, where 3, -2, and 1 are the coefficients, and x², x, and 1 are the variables with their respective powers.
Now,
The process for adding and subtracting polynomials involves combining like terms. To add or subtract two polynomials, we simply combine the coefficients of the same degree terms.
For example, to add the polynomials 2x² + 3x + 4 and 4x² - 2x - 1, we group the like terms and add the coefficients:
(2x² + 4x²) + (3x - 2x) + (4 - 1) = 6x² + x + 3
To subtract the polynomial 4x² - 2x - 1 from the polynomial 2x² + 3x + 4, we change the sign of the second polynomial and then combine the like terms:
(2x² + 3x + 4) - (4x² - 2x - 1) = 2x² + 3x + 4 - 4x² + 2x + 1 = -2x² + 5x + 5
The process for multiplying polynomials is different from adding and subtracting. When we multiply two polynomials, we need to distribute each term of polynomials, and then combine the like terms.
For example, to multiply the polynomials (x + 2) and (x - 3), we use the distributive property:
(x + 2)(x - 3) = x(x - 3) + 2(x - 3) = x² - 3x + 2x - 6 = x² - x - 6
As we can see, the process for multiplying polynomials is different from adding and subtracting, but all three operations involve combining like terms in some way.
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In the given Fig. PQR is a triangle, right angled at Q. If XY || QR, PQ = 6 cm, PY = 4 cm and PX : XQ = 1 : 2. Calculate the lengths of PR and QR.
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Basic Proportionality Theorem (BPT): If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points then the other two sides are divided in the same ratio. This is also known as Thales theorem.
Given:
[tex]\angle Q= 90^\circ , XY \ || \ QR, PQ = 6 \ \text{cm}, PY = 4 \ \text{cm} \ \text{and} \ PX : XQ = 1 : 2[/tex]
Since, [tex]XY \ || \ QR[/tex],
[tex]PX/XQ = PY/YR[/tex]
[ By Thales theorem (BPT)]
[tex]\dfrac{1}{2} = PY/YR[/tex] [tex][PX : XQ = 1 : 2][/tex]
[tex]\dfrac{1}{2} = 4 /(PR - PY)[/tex]
[tex][YR= PR - PY][/tex]
[tex]\dfrac{1}{2} = 4 /(PR - 4)[/tex]
[tex]PR - 4 = 2 \times 4[/tex]
[tex]PR - 4 = 8[/tex]
[tex]PR = 8 +4[/tex]
[tex]PR = 12 \ \text{cm}[/tex]
In right [tex]\Delta PQR[/tex],
[tex]PR^2 = PQ^2 + QR^2[/tex]
[ By Pythagoras theorem]
[tex]12^2 = 6^2 + QR^2[/tex] [tex][\text{Given} : PQ= 6 \ \text{cm}][/tex]
[tex]144 = 36 + QR^2[/tex]
[tex]144 - 36 + QR^2[/tex]
[tex]108= QR^2[/tex]
[tex]QR =\sqrt{108} =\sqrt{3\times36} = 6\sqrt{3} \ \text{cm}[/tex]
Hence, the lengths of PR and QR is 12 cm and [tex]6\sqrt{3}[/tex] cm.
In the diagram below, ABC~ DBE. If AD = 24, DB = 12, and DE = 4, what is the length of
AC?
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Answer:
Step-by-step explanation:
because 110
Find the value of t for a t-distribution with 45 degrees of freedom such that the area to the right of t equals 0.010. Round your answer to three decimal places, if necessary.
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The value of t for a t-distribution with 45 degrees of freedom such that the area to the right of t equals 0.010 is approximately -2.326.
With its bell-shaped structure and heavier tails, the t-distribution, commonly referred to as the Student's t-distribution, is a kind of probability distribution that resembles the normal distribution. When there are insufficient samples or unknown variances, it is used to estimate population parameters. T-distributions have broader tails than normal distributions because they are more likely to contain extreme values.
To find the value of t for a t-distribution with 45 degrees of freedom such that the area to the right of t equals 0.010, we can use a t-table or a calculator. Using a calculator, we can use the inverse t-distribution function. The inverse t-distribution function gives us the value of t for a given probability and degrees of freedom.
Using this function, we have:
t = invT(0.010, 45) ≈ -2.326
Rounding this to three decimal places gives us the answer:
t ≈ -2.326
Therefore, the value of t for a t-distribution with 45 degrees of freedom such that the area to the right of t equals 0.010 is approximately -2.326.
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