The area of the region that is bounded by the given curve and lies in the specified sector is A = 2(e^(7π/12) - e^(π/12))
The polar curve r = e^(θ/2) represents a spiral that starts from the origin and gets farther away as it unwinds. We want to find the area of the region that lies inside this spiral and inside the sector defined by the angles θ = π/6 and θ = 7π/6.
To solve the problem, we need to find the points where the curve intersects the sector, which are given by plugging in the values of θ:
r(π/6) = e^(π/12)
r(7π/6) = e^(7π/12)
Then we can set up the integral for the area inside the sector:
A = 1/2 ∫[π/6, 7π/6] (r(θ))^2 dθ
Substituting the equation for r:
A = 1/2 ∫[π/6, 7π/6] e^θ/2 dθ
Using the power rule for integration:
A = 2(e^(7π/12) - e^(π/12))
This is the exact value of the area inside the sector and inside the spiral. If we want a decimal approximation, we can use a calculator or computer software to evaluate it.
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find the closed formula for 3,6,11,18 by relating them to a well known sequence. assume the first term given is
The closed formula for this particular sequence is an = n² + 2.
Take note that the odd numbers 3, 5, 7, 9, and 11 are separate consecutive terms. This shows that the first n odd numbers can be added to the initial term, az, to get the nth term. Hence, the following is how we may represent the nth term a = az + 1 + 3 + 5 + ... + (2n-3) (2n-3). We may utilize the formula for the sum of an arithmetic series to make the sum of odd integers simpler that is 1 + 3 + 5 + ... + (2n-3) = n².
As a result, we get a = az + n^2 - 1. In conclusion, the equation for the series (an)n21, where a1 = az and an is the result of adding the first n odd numbers to az, is as a = az + n^2 - 1. We have the following for the given series where a1 = az = 3.
So, the closed formula for this particular sequence is an = n² + 2.
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Your question is incomplete. The complete question is:
Find the closed formula for the sequence (an)n21. Assume the first term given is az. an = 3, 6, 11, 18, 27... Hint: Think about the perfect squares.
A large equilateral triangle pyramid stands in front of the city's cultural center. Each side of the base measures 40 feet and the slant height of each lateral side of the pyramid is 50 feet.
A painter can paint 100 square feet of the pyramid in 18 minutes.
How long does it take the painter to paint 75% of the pyramid?
Answer:
A large equilateral triangle pyramid stands in front of the city's cultural center. Each side of the base measures 40 feet and the slant height of each lateral side of the pyramid is 50 feet.
A painter can paint 100 square feet of the pyramid in 18 minutes.
How long does it take the painter to paint 75% of the pyramid?
Step-by-step explanation:
The total surface area of the pyramid can be calculated using the formula for the lateral surface area of a pyramid:
Lateral surface area = (1/2) × perimeter of base × slant height
Since the base is an equilateral triangle, the perimeter is 3 times the length of one side:
Perimeter of base = 3 × 40 feet = 120 feet
Lateral surface area = (1/2) × 120 feet × 50 feet = 3000 square feet
To paint 75% of the pyramid, the painter needs to paint:
0.75 × 3000 square feet = 2250 square feet
Since the painter can paint 100 square feet in 18 minutes, the time required to paint 2250 square feet can be calculated as:
2250 square feet ÷ 100 square feet per 18 minutes = 225 ÷ 10 × 18 minutes = 405 minutes
Therefore, the painter would need 405 minutes or 6 hours and 45 minutes to paint 75% of the pyramid.
Use the rational zeros theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. f(x) = x^3 - x^2 - 37x - 35 Find the real zeros of f. Select the correct choice below and; if necessary, fill in the answer box to complete your answer. (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any rational numbers in the expression. Use a comma to separate answers as needed.) There are no real zeros. Use the real zeros to factor f. f(x)= (Simplify your answer. Type your answer in factored form. Type an exact answer, using radicals as needed. Use integers or fractions for any rational numbers in the expression.)
By using rational zeros theorem, we find that there are no real zeros of the polynomial function f(x) = x^3 - x^2 - 37x - 35, so we cannot factor f(x) over the real numbers.
To find the real zeros of the polynomial function f(x) = x^3 - x^2 - 37x - 35, we can use the rational zeros theorem, which states that any rational zeros of the function must have the form p/q, where p is a factor of the constant term (-35) and q is a factor of the leading coefficient (1).
The possible rational zeros of f are therefore ±1, ±5, ±7, ±35. We can then test each of these values using synthetic division or long division to see if they are zeros of the function. After testing all of the possible rational zeros, we find that none of them are actually zeros of the function.
Therefore, we can conclude that there are no real zeros of the function f(x) = x^3 - x^2 - 37x - 35.
However, we could factor it into linear and quadratic factors with complex coefficients using the complex zeros of f(x). But since the problem only asks for factoring over the real numbers, we can conclude that the factored form of f(x) is:
f(x) = x^3 - x^2 - 37x - 35 (cannot be factored over the real numbers)
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One side of the triangle is 4 cm, and the sum of the other two sides is equal to a whole number of cm. What is the smallest possible perimeter of the triangle?
F. 9 cm
G. 10 cm
H. 11 cm
J. 15 cm
K. 17 cm
Answer:
9 cm
Step-by-step explanation:
By the Triangle Inequality, any two sides of a triangle must be greater than the remaining side.
In order to minimize the perimeter, we will assume that 4 cm is the longest side.
Thus, the two remaining sides must be greater than 4.
Since we are given that the sum of the two remaining sides is a whole number, the smallest whole number value greater than 4 is 5.
Hence, the smallest perimeter possible 9 cm.
What is the slope of the line in the following graph?
Answer:
1/3
Step-by-step explanation:
using rise over run fron the two dots, we can find 2/6, which simplifies down to 1/3
What is the difference between the questionnaire and an interview?
Answer: Questionnaire refers to a research instrument, in which a series of question, is typed or printed along with the choice of answers, expected to be marked by the respondents, used for survey or statistical study. It consists of aformalisedd set of questions, in a definite order on a form, which are mailed to the respondents or manually delivered to them for answers. The respondents are supposed to read, comprehend and give their responses, in the space provided.
A ‘Pilot Study’ is advised to be conducted to test the questionnaire before using this method. A pilot survey is nothing but a preliminary study or say rehearsal to know the time, cost, efforts, reliability and so forth involved in it.
The interview is a data collection method wherein a direct, in-depth conversation between interviewer and respondent takes place. It is carried out with a purpose like a survey, research, and the like, where both the two parties participate in the one to one interaction. Under this method, oral-verbal stimuli are presented and replied by way of oral-verbal responses.
It is considered as one of the best methods for collecting data because it allows two way exchange of information, the interviewer gets to know about the respondent, and the respondent learns about the interviewer. There are two types of interview:
Personal Interview: A type of interview, wherein there is a face to face question-answer session between the interviewer and interviewee, is conducted.
Telephonic Interview: This method involves contacting the interviewee and asking questions to them on the telephone itself.
Parts A-D. What is the value of the sample mean as a percent? What is its interpretation? Compute the sample variance and sample standard deviation as a percent as measures of rotelle for the quarterly return for this stock.
The sample mean is 2.1, the sample variance is 212.5% and the standard deviation is 14.57%
What is the sample mean?a. The sample mean can be computed as the average of the quarterly percent total returns:
[tex](11.2 - 20.5 + 13.2 + 12.6 + 9.5 - 5.8 - 17.7 + 14.3) / 8 = 2.1[/tex]
So the sample mean is 2.1%, which can be interpreted as the average quarterly percent total return for the stock over the sample period.
b. The sample variance can be computed using the formula:
[tex]s^2 = sum((x - mean)^2) / (n - 1)[/tex]
where x is each quarterly percent total return, mean is the sample mean, and n is the sample size. Plugging in the values, we get:
[tex]s^2 = (11.2 - 2.1)^2 + (-20.5 - 2.1)^2 + (13.2 - 2.1)^2 + (12.6 - 2.1)^2 + (9.5 - 2.1)^2 + (-5.8 - 2.1)^2 + (-17.7 - 2.1)^2 + (14.3 - 2.1)^2 / (8 - 1) = 212.15[/tex]
So the sample variance is 212.15%. The sample standard deviation can be computed as the square root of the sample variance:
[tex]s = \sqrt(s^2) = \sqrt(212.15) = 14.57[/tex]
So the sample standard deviation is 14.57%.
c. To construct a 95% confidence interval for the population variance, we can use the chi-square distribution with degrees of freedom n - 1 = 7. The upper and lower bounds of the confidence interval can be found using the chi-square distribution table or calculator, as follows:
upper bound = (n - 1) * s^2 / chi-square(0.025, n - 1) = 306.05
lower bound = (n - 1) * s^2 / chi-square(0.975, n - 1) = 91.91
So the 95% confidence interval for the population variance is (91.91, 306.05).
d. To construct a 95% confidence interval for the standard deviation (as percent), we can use the formula:
lower bound = s * √((n - 1) / chi-square(0.975, n - 1))
upper bound = s * √((n - 1) / chi-square(0.025, n - 1))
Plugging in the values, we get:
lower bound = 6.4685%
upper bound = 20.1422%
So the 95% confidence interval for the standard deviation (as percent) is (6.4685%, 20.1422%).
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Find the value of x. If your answer is not an integer, leave it in simplest radical form. The diagram is not drawn to scale.
NOTE: Enter your answer and show all the steps that you use to solve this problem in the space provided. Use the 30°-60°-90° Triangle Theorem to find the answer.
The value of the x is 5√3 after we successfully do the application of the 30°-60°-90° Triangle theorem.
What is Triangle theorem?The 30°-60°-90° Triangle Theorem states that in such a triangle, the side opposite the 30° angle is half the length of the hypotenuse, and the side opposite the 60° angle is the product of the length of the hypotenuse and the square root of 3 divided by 2.
Using this theorem, we can write:
y = hypotenuse
Opposite of 30° angle = 5 = hypotenuse/2
Opposite of 60° angle = x = hypotenuse × (√(3)/2)
Solving for the hypotenuse in terms of y from the first equation, we get:
hypotenuse = 5×2 = 10
Substituting this value into the third equation, we get:
x = 10 × (√(3)/2) = 5 × √(3)
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find the following answer
According to the Venn diagram the value of [tex]n(A ^ C \cap B ^ C) = {3}[/tex] so the number of elements in that set is 1.
What is Venn diagram ?
A Venn diagram is a diagram that shows all possible logical relations between a finite collection of different sets. It is usually represented as a rectangle or a circle for each set and the overlapping areas between them, showing the common elements that belong to more than one set. Venn diagrams are widely used in mathematics, logic, statistics, and computer science to visualize the relationships between different sets and help solve problems related to set theory.
According to the question:
To solve this problem, we first need to understand the notation used.
n(A) denotes the set A and the numbers within the braces {} indicate the elements in set A. For example, n(A)={7,4,3,9} means that the set A contains 7, 4, 3, and 9.
n(AnB) denotes the intersection of sets A and B, i.e., the elements that are common to both A and B. For example, n(AnB)={4,3} means that the sets A and B have 4 and 3 in common.
^ denotes intersection of sets
cap denotes the intersection of sets
Now, we need to find the elements that are common to sets A and C, and sets B and C. We can do this by taking the intersection of A and C, and the intersection of B and C, and then taking the intersection of the two resulting sets.
[tex]n(A ^ C) = n(A \cap C) = {3,9}[/tex]
[tex]n(B ^ C) = n(B\cap C) = {3,5}[/tex]
Now, we take the intersection of [tex]n(A ^ C)[/tex] and [tex]n(B ^ C)[/tex]:
[tex]n(A ^ C \cap B ^ C) = {3}[/tex]
Therefore, the answer is 1.
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Maximize z = 3x₁ + 5x₂
subject to: x₁ - 5x₂ ≤ 35
3x1 - 4x₂ ≤21
with. X₁ ≥ 0, X₂ ≥ 0.
use simplex method to solve it and find the maximum value
Answer:
See below.
Step-by-step explanation:
We can solve this linear programming problem using the simplex method. We will start by converting the problem into standard form
Maximize z = 3x₁ + 5x₂ + 0s₁ + 0s₂
subject to
x₁ - 5x₂ + s₁ = 35
3x₁ - 4x₂ + s₂ = 21
x₁, x₂, s₁, s₂ ≥ 0
Next, we create the initial tableau
Basis x₁ x₂ s₁ s₂ RHS
s₁ 1 -5 1 0 35
s₂ 3 -4 0 1 21
z -3 -5 0 0 0
We can see that the initial basic variables are s₁ and s₂. We will use the simplex method to find the optimal solution.
Step 1: Choose the most negative coefficient in the bottom row as the pivot element. In this case, it is -5 in the x₂ column.
Basis x₁ x₂ s₁ s₂ RHS
s₁ 1 -5 1 0 35
s₂ 3 -4 0 1 21
z -3 -5 0 0 0
Step 2: Find the row in which the pivot element creates a positive quotient when each element in that row is divided by the pivot element. In this case, we need to find the minimum positive quotient of (35/5) and (21/4). The minimum is (21/4), so we use the second row as the pivot row.
Basis x₁ x₂ s₁ s₂ RHS
s₁ 4/5 0 1/5 1 28/5
x₂ -3/4 1 0 -1/4 -21/4
z 39/4 0 15/4 3/4 105
Step 3: Use row operations to create zeros in the x₂ column.
Basis x₁ x₂ s₁ s₂ RHS
s₁ 1 0 1/4 7/20 49/10
x₂ 0 1 3/16 -1/16 -21/16
z 0 0 39/4 21/4 525/4
The optimal solution is x₁ = 49/10, x₂ = 21/16, and z = 525/4.
Therefore, the maximum value of z is 525/4, which occurs when x₁ = 49/10 and x₂ = 21/16.
Question 6 of 10
Based only on the information given in the diagram, which congruence
theorems or postulates could be given as reasons why ACDE AOPQ?
Check all that apply.
AA
A. AAS
B. ASA
C. LL
OD. HL
E. LA
F. SAS
Therefore, A, B, C, and F are the proper responses as the congruence theories or postulates based on the data.
what is triangle ?Having three straight sides and three angles where they intersect, a triangle is a closed, two-dimensional shape. It is one of the fundamental geometric shapes and has a number of characteristics that can be used to study and resolve issues that pertain to it. The triangle inequality theory states that the sum of a triangle's interior angles is always 180 degrees, and that the longest side is always the side across from the largest angle. Triangles can be used to solve a wide range of mathematical issues in a variety of disciplines and can be categorised based on the length of their sides and the measurement of their angles.
given
We can use the following congruence theories or postulates based on the data in the diagram:
A. ASA
B. AAS
C. LL (corresponding angles hypothesis)
F. SAS
Therefore, A, B, C, and F are the proper responses as the congruence theories or postulates based on the data.
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I will mark you brainiest!
Given the coordinates shown and given that SU = 10, what are the coordinates of U if STUV is a kite?
A) (10, 18)
B) (0, 28)
C) (18, 28)
The calculated coordinates of U if STUV is a kite is (10, 18)
Calculating the coordinates of U if STUV is a kite?From the question, we have the following parameters that can be used in our computation:
The figute of a kite
Also, we have
S = (0, 18)
And the distance SU to be
SU = 10
If the quadrilateral STUV is a kite, then the coordinates S and U are on the same horizontal level (according to the figure)
So, we have
U = (0 + 10, 18)
Evaluate
U = (10, 18)
Hence, the coordinates of U if STUV is a kite is (10, 18)
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PLEASE HELP MEEE
whoever answers right gets brainliest!!!
Answer:
[tex]30\leq x[/tex] AND [tex]x \leq 106[/tex]
Notice the valid answer is the one with the AND since need to be both at the same time.
Step-by-step explanation:
Is the one is market, you add 6 to each side and you obtain that answer
[tex]24 \leq x-6\leq 100[/tex]
[tex]24 +6\leq x\leq 100+6[/tex]
[tex]30\leq x\leq 106[/tex]
The number 0 is an element of the set of natural numbers.
OA. True
B. False
SUBI
it is false. 0 is not a natural number. it is a whole number
how do you find the simplest radical form for this please help me i got a (f) and i really need help that’s why i’m up this late trying to do all of my missing assignments.
Answer:
[tex]14 {y}^{2} \sqrt{ {x}^{3} {z}^{9} }[/tex]
Step-by-step explanation:
[tex] \sqrt{196 {x}^{3} {y}^{4}{z}^{9} }= \sqrt{196} \times \sqrt{ {x}^{3} } \times \sqrt{ {y}^{4} } \times \sqrt{ {z}^{9} } \\ \sqrt{196} = 14 \\ \sqrt{ {x}^{3} } = {x}^{ \frac{3}{2} } \\ \sqrt{ {y}^{4} } = {y}^{2} \\ \sqrt{ {z}^{9}} = {z}^{ \frac{9}{2} }[/tex]
A fractional exponent is not necessarily simpler so just take out the 1st and 3rd parts of the term which simplify nicely:
[tex] \sqrt{196 {x}^{3} {y}^{4}{z}^{9} } = 14 {y}^{2} \sqrt{ {x}^{3} {z}^{9} } [/tex]
the figure below shows the change of a population over time. which statement best describes the mode of selection depicted in the figure?
The statement that best describes the mode of selection depicted in the figure is (b) Directional Selection, changing the average color of population over time.
The Directional selection is a type of natural selection that occurs when individuals with a certain trait or phenotype are more likely to survive and reproduce than individuals with other traits or phenotypes.
In the directional selection of evolution, the mean shifts that means average shifts to one extreme and supports one trait and leads to eventually removal of the other trait.
In this case, one end of the extreme-phenotypes which means that the dark-brown rats are being selected for. So, over the time, the average color of the rat population will change.
Therefore, the correct option is (b).
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The given question is incomplete, the complete question is
The figure below shows the change of a population over time. which statement best describes the mode of selection depicted in the figure?
(a) Disruptive Official, favoring the average individual
(b) Directional Selection, changing the average color of population over time
(c) Directional selection, favoring the average individual
(d) Stabilizing Selection, changing the average color of population over time
A home has gone up in value over several
decades and is now worth 1354% of its
original sale price of $23,000. What is the
value now?
Answer:
$31,142
Step-by-step explanation:
To convert a percentage into a decimal, you move the decimal two places to the left. 1354% converted into a decimal is 13.54.
$23,000 * 13.54 = $31,142
Let G
be a group. Say what it means for a map φ:G→G
to be an automorphism. Show that the set-theoretic composition φψ=φ∘ψ
of any two automorphisms φ,ψ
is an automorphism. Prove that the set Aut(G)
of all automorphisms of the group G
with the operation of taking the composition is a group.
a) An automorphism of a group G is a bijective map φ:G→G that preserves the group structure. That is, φ(ab) = φ(a)φ(b) and φ(a⁻¹) = φ(a)⁻¹ for all a, b ∈ G.
b) The set-theoretic composition φψ of any two automorphisms φ, ψ is an automorphism, as it preserves the group structure and is bijective.
c) The set Aut(G) of all automorphisms of G, with the operation of composition of maps, is a group. This is because it satisfies the four group axioms: closure, associativity, identity, and inverses. Therefore, Aut(G) is a group under composition of maps.
An automorphism of a group G is a bijective map φ:G→G that preserves the group structure, meaning that for any elements a,b∈G, we have φ(ab) = φ(a)φ(b) and φ(a⁻¹) = φ(a)⁻¹. In other words, an automorphism is an isomorphism from G to itself.
To show that the set-theoretic composition φψ is an automorphism, we need to show that it satisfies the two conditions for being an automorphism. First, we have
(φψ)(ab) = φ(ψ(ab)) = φ(ψ(a)ψ(b)) = φ(ψ(a))φ(ψ(b)) = (φψ)(a)(φψ)(b)
using the fact that ψ and φ are automorphisms. Similarly,
(φψ)(a⁻¹) = φ(ψ(a⁻¹)) = φ(ψ(a))⁻¹ = (φψ)(a)⁻¹
using the fact that ψ and φ are automorphisms. Therefore, φψ is an automorphism.
To show that Aut(G) is a group, we need to show that it satisfies the four group axioms
Closure: If φ,ψ∈Aut(G), then φψ is also in Aut(G), as shown above.
Associativity: Composition of maps is associative, so (φψ)χ = φ(ψχ) for any automorphisms φ,ψ,χ of G.
Identity: The identity map id:G→G is an automorphism, since it clearly preserves the group structure and is bijective. It serves as the identity element in Aut(G), since φid = idφ = φ for any φ∈Aut(G).
Inverses: For any automorphism φ∈Aut(G), its inverse φ⁻¹ is also an automorphism, since it is bijective and preserves the group structure. Therefore, Aut(G) is closed under inverses.
Since Aut(G) satisfies all four group axioms, it is a group under composition of maps.
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Calculate the following limits?
The answer of the given question based on the limits the answers are as follows, (a) lim f(x) = 1 , (b) lim f(x) = 3 , (c) lim f(x) = 3.
What is Graph?A graph is visual representation of data that shows the relationship between two or more variables. Graphs can be used to display wide variety of information, including numerical data, functions, and networks. The most common types of graphs like line graphs, bar graphs, scatter plots, and pie charts.
Graphs are widely used in many fields, like science, economics, engineering, and social sciences, to help people understand and analyze complex data. They are powerful tool for visualizing trends, patterns, and relationships, and are often used to communicate findings to wider audience.
a) The limit of f(x) as x approaches 2 from the left:
We can see from the graph that as x approaches 2 from the left, f(x) approaches 1. Therefore, we can write:
lim f(x) = 1
x→2-
b) The limit of f(x) as x approaches 2 from the right:
Similarly, as x approaches 2 from the right, f(x) approaches 3. Therefore:
lim f(x) = 3
x→2+
c) The limit of f(x) as x approaches 2:
Since the limit from the left and the limit from the right exist and are equal, we can say that the limit of f(x) as x approaches 2 exists and equals the common value of the left and right limits. Therefore:
lim f(x) = 3
x→2
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If 5 is increased to 9, the increase is what percentage of the original number
Answer: It's a 80% increase
Step-by-step explanation:
f the random walk starts in the center, on average how many steps does it take to return to the center?
Total number of steps taken by an average man in a year while walking with 7192 steps a day is equals to 2,625,080 steps/year.
Number of steps taken by average man in a day is equals to 7192
Then the total number of steps he takes in a year is equals to,
Calculate it by multiplying the average number of steps per day by the number of days in a year.
There are different ways to define a year,
But assuming a regular calendar year of 365 days, the calculation would be,
Total number of days in a year = 365 days
Total number of steps in a year
= 7192 steps/day x 365 days/year
= 2,625,080 steps/year
Therefore, on average the man would walk about 2,625,080 steps in a year.
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The given question is incomplete, I answer the question in general according to my knowledge:
If a man walks with random steps and the average man takes 7192 steps a day about how many steps does the average man take in a year?
If n is an integer and n > 1, then n! is the product of n and every other positive integer that is less than n. for example, 5! = 5 x 4 x 3 x 2 x 1a. Write 6! in standard factored formb. Write 20! in standard factored formc. Without computing the value of (20!)2, determine how many zeros are at the end of this number when it is written in decimal form. Justify your answer
a. 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720. b. 20! = 20 x 19 x 18 x ... x 2 x 1. c. There are 16 zeros at the end of the decimal representation of (20!)2.
a. 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720
b. 20! = 20 x 19 x 18 x ... x 2 x 1. To write this in factored form, we can identify the prime factors of each number and write the product using exponents. For example, 20 = 2² x 5, so we can write 20! as:
20! = (2² x 5) x 19 x (2 x 3²) x 17 x (2² x 7) x 13 x (2 x 2 x 3) x 11 x (2³) x (3) x (2) x 7 x (2) x 5 x (2) x 3 x 2 x 1
Simplifying, we get:
20! = 2¹⁸ x 3⁸ x 5⁴ x 7² x 11 x 13 x 17 x 19
c. The number of zeros at the end of (20!)² in decimal form is determined by the number of factors of 10, which is equivalent to the number of factors of 2 x 5. Since there are more factors of 2 than 5 in the prime factorization of (20!)², we only need to count the number of factors of 5. There are four factors of 5 in the prime factorization of 20!, which contribute four factors of 10 to the square. Therefore, (20!)²ends in 8 zeros when written in decimal form.
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given a function f(x), find the critical values and use the critical values to find intervals of increasing/deacreasing, maxes and mins.
The critical values, the intervals of increasing or decreasing and the maximum and minimum points of the f(x) is (-1.5, -16), x < -1.5 and x = -1.5 and for b (4,6) and (2,10), (2,4).
A) Critical values
We will find out the critical value by solving for f ' (x) = 0
therefore, taking the derivative of given function we get,
f ' (x) = 4(2x) + 12 = 0
= 8x + 12 = 0
therefore, 8x = -12
x = -12/8
x= -1.5
x = -1.5 is the only critical value in x-coordinate. Now to determine the y-coordinate, simply put the value of x in the function f(x) = 4x2 + 12x - 7
we get, f(-1.5) = 4(-1.5)2 + 12 (-1.5) - 7
= 4(2.25) - 18 - 7
= 9 - 25 = -16
therefore, the critical value of the function f(x) = 4x2 + 12x - 7 is (-1.5, -16)
f(x) =x3 - 9x2 + 24x - 10.
Intervals of increasing and decreasing function is i.e. f decreases for
x < -1.5.
Therefore, f has minimum value at x = -1.5.
B) Critical values
We will find out the critical value by solving for f ' (x) = 0
therefore, taking the derivative of given function we get,
f '(x) = 3x2 - 9(2x) + 24
= 3x2 - 18x + 24 = 0
therefore, 3 ( x2 - 6x + 8) = 0
i.e x2 - 6x + 8 = 0
(x-4) (x-2) = 0
So, x = 4 or x = 2 are the two critical values in x-coordinate. Now to determine the y-coordinate, simply put the values of x in the function f(x) =x3 - 9x2 + 24x - 10
we get, Substituting x = 4
f(4) = 43 - 9 (4)2 +24 (4) -10
= 64 - 144 + 96 - 10
= 6
Now, Substituting x = 2
f(2) = 23 - 9(2)2 + 24(2) - 10
= 8 - 36 + 48 - 10
= 10
Therefore, the critical values of the function f(x) =x3 - 9x2 + 24x - 10 are (4,6) and (2,10).
Intervals of increasing and decreasing functions is f decreases in (2,4).
therefore, f has minimum at x = 4 and maximum at x = 2.
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Complete question:
For each function determine: i) the critical values ii) the intervals of increasing or decreasing iii) the maximum and minimum points.
a. f(x) = 4x²+12x–7 (3 marks)
b. F(x) = x°-9x²+24x-10 (3 marks)
the graph shows the preimage shaded in grey and the image outlined in black. what is the scale factor of the dilation?
The scale factor of dilation of the shaded in gray to the shaded in black is 3
Calculating the scale factor dilationGiven that
The preimage = shaded in gray
The image = shaded in black
From the graph, we have the following values on the image and the preimage
The preimage = shaded in gray = 4
The image = shaded in black = 12
The scale factor of dilation is then calculated as
Scale factor = shaded in black/shaded in gray
So, we have
Scale factor = 12/4
Evaluate
Scale factor = 3
Hence, the scale factor dilation is 3
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WILL MARK AS BRAINLIEST!!!!!!!!!!!!!!
If "f" is differentiable and f(1) < f(2), then there is a number "c", in the interval (_____, _____) such that f'(c)>_______
If "f" is differentiable and f(1) < f(2), then there is a number "c", in the interval (1, 2) such that f'(c)> 0.
How do we know?Applying the Mean Value Theorem for derivatives, if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number c in the interval (a, b) such that:
f'(c) = (f(b) - f(a)) / (b - a)
In the scenario above, we have that f is differentiable, and that f(1) < f(2).
choosing a = 1 and b = 2.
Then applying the Mean Value Theorem, there exists at least one number c in the interval (1, 2) such that:
f'(c) = (f(2) - f(1)) / (2 - 1)
f'(c) = f(2) - f(1)
We have that f(1) < f(2), we have:
f(2) - f(1) > 0
We can conclude by saying that there exists a number c in the interval (1, 2) such that:
f'(c) = f(2) - f(1) > 0
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Suppose we want to choose 5 letters, without replacement, from 15 distinct letters
[tex]\text{order does not matter}[/tex]
[tex]\text{sample space}= \text{15 letters}[/tex]
[tex]\text{no repetition}[/tex]
[tex]\text{P(A)}= \text{15C5}= \text{3003 ways}[/tex]
how many one-to-one functions are there from a set with five elements to sets with the following number of ele- ments? a) 4 b) 5 c) 6 d) 7
a) Number of one-to-one functions are equal to the zero, because n< m.
b) Number of one-to-one functions are equal to the ⁵P₅ = 120.
c) Number of one-to-one functions are equal to the ⁶P₅ = 720.
c) Number of one-to-one functions are equal to the ⁷P₅ = 2250.
One to one function is a special form of function that defined from one set to another and maps every element of the range to exactly one element of its domain unique output. As we know a set A has m elements and set B has n elements, then
Number of one-to-one functions from set A to Set B = P(n,m) or ⁿPₘ , n≥ m and number of one-to-one functions from set A to Set B = 0 , n< m.Now, we have a domain set with five elements, m = 5
a) Here, another set (co-domain) has 4 elements, n = 4. So, Number of one-to-one functions = 0 , n<m.
b) number of elements in another set,n= 5
So, Number of one-to-one functions = ⁵P₅ = 5!/(5 - 5 )! ( permutation formula)
= 5!/0! = 120
c) Number of elements in another set, n= 6
So, Number of one-to-one functions= ⁶P₅
= 6!/(6 - 5)!
= 6!/1! = 720
d) Number of elements in another set, n
= 7
So, Number of one-to-one functions
= ⁷P₅ = 7!(7 - 5)!
= 7!/2! = 2250
Hence, required value is 2250.
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Help please! I have no idea!!!! PLEASEE
To highlight the line y = 0 on the graph in black/grey, draw a straight line passing through all points whose y-coordinate is 0.
What is graph?
In mathematics, a graph is a visual representation of a set of data, typically as a set of points or lines on a coordinate plane. Graphs are used to represent various types of data, such as numerical values, functions, relationships, and patterns.
Assuming that the graph is a coordinate plane with the x-axis and y-axis, do the following:
To highlight the point (9, 8) on the graph in red, locate the point (9, 8) on the coordinate plane and mark it with a red color.
To highlight the point (20, f(20)) on the graph in green, you need to know the value of f(20) first. Once you have that value, locate the point (20, f(20)) on the coordinate plane and mark it with a green color.
To highlight the line y = 5 on the graph in blue, draw a straight line passing through all points whose y-coordinate is 5. This line should be parallel to the x-axis and should be marked with a blue color.
Therefore To highlight the line y = 0 on the graph in black/grey, draw a straight line passing through all points whose y-coordinate is 0. This line should be parallel to the x-axis and should be marked with a black/grey color.
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10.5.PS-18 Question content area top Part 1 The diagram shows a track composed of a rectangle with a semicircle on each end. The area of the rectangle is square meters 11200. What is the perimeter of the track? Use 3.14 for pi.
Show your solution ( 3. ) C + 18 = 29
Answer:
Show your solution ( 3. ) C + 18 = 29
Step-by-step explanation:
To solve the equation C + 18 = 29, we want to isolate the variable C on one side of the equation.
We can start by subtracting 18 from both sides of the equation:
C + 18 - 18 = 29 - 18
Simplifying the left side of the equation:
C = 29 - 18
C = 11
Therefore, the solution to the equation C + 18 = 29 is C = 11.
