By using trigonometric formulas and integration rules, we conclude that the integral is approximately equal to - 7.764 units.
How to determine the value of a definite integral
In this problem we must determine the solution to a definite integral of trigonometric equations, which requires the use of trigonometric formula and integration rules to be found. The integral can be simplified by using the following formulas:
sin⁴ x = (1 / 8) · (3 - 4 · cos 2x + cos 4x)
sin² x = (1 /2) · (1 - cos 2x)
Then, the integral is simplified into this form:
[tex]I = \pi \int \limits_{\frac {\pi}{6}}^{\frac{3\pi}{4}} \sin^{4} x \,dx + 4\pi \int \limits_{\frac {\pi}{6}}^{\frac{3\pi}{4}} \sin^{2} x \,dx[/tex]
[tex]I = \frac{\pi}{8} \int \limits_{\frac {\pi}{6}}^{\frac{3\pi}{4}} (3 - 4\cdot \cos 2x + \cos 4x) \,dx + 4\pi \int \limits_{\frac {\pi}{6}}^{\frac{3\pi}{4}} (1 - \cos 2x) \,dx[/tex]
[tex]I = \frac{3\pi}{8}\cdot x - \frac{\pi}{4} \cdot \sin 2x + \frac {\pi}{32}\cdot \sin 4x - 4\pi \cdot x - 2\pi\cdot \sin 2x[/tex], for x ∈ [π / 6, 3π / 4]
And the definite integral is:
I = 2.159 + 1.466 - 0.085 - 23.028 + 11.724
I = - 7.764
The definite integral of f(x) = sin⁴ x + 4 · sin² x for x ∈ [π / 6, 3π / 4] is approximately equal to - 7.764.
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Construct a polynomial function with the stated properties. Reduce all fractions to lowest terms.
Third-degree, with zeros of -2,-1, and 5, and a y-intercept of -10.
Answer:
y = x³ -2x² -13x -10
Step-by-step explanation:
You want a polynomial with x-intercepts -2, -1, and 5, and a y-intercept of -10.
Polynomial factorsA polynomial with zero x = p will have a factor of (x -p). The three given zeros mean factors of the desired polynomial will be ...
p(x) = (x -(-2))(x -(-1))(x -5) = (x +2)(x +1)(x -5)
Expanding this gives ...
p(x) = (x +2)(x² -4x -5) = x³ -2x² -13x -10
This polynomial has a constant of -10, which will be its y-intercept. No additional scaling is needed.
The polynomial function is ...
y = x³ -2x² -13x -10
Need help thanks. Due today
Answer:
The answer would be D(x/y)= -(11/8) / -(11/4)
Step-by-step explanation:
x + 9.27 = 21
I need help answer this questions
Answer:
x = 11.73 is what i got
Step-by-step explanation:
Plot (−3 1/2, −1 1/4) on the coordinate plane.
The point is located at 3 1/2 points to the left of the origin and 1 1/4 points below the origin
How to plot the point on a coordinate plane?The coordinate of the point on a coordinate plane is given as
Point = (-3 1/2, -1 1/4)
Rewrite the above coordinate as follows:
(x, y) = (-3 1/2, -1 1/4)
Split the coordinates into x and y coordinates
So, we have the following equations
x = -3 1/2
y = -1 1/4
The above means that:
(-3 1/2, -1 1/4) is located at 3 1/2 points to the left of the origin (-3 1/2, -1 1/4) is located 1 1/4 points below the originNext, we plot the points
See attachment for the plot of the points on a coordinate plane
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Find the x-and y-intercepts of the line:3x - 4y = 12A.x-int: (-4,0)y-int: (0,3)B. x-int: (4,0)y-int: (0, -3)C. x-int: (0,4)y-int:(-3,0)D. x-int: (0,-4)y-int: (3,0)
To find the x-intercept, simply but y=0 in the equation given and solve for x
That is;
3x - 4y = 12
3x - 4(0) = 12
3x = 12
Divide both-side of the equation by 3
x= 4
The x-intercept is;
(4, 0)
To find the y-intercept, simply put y=0 in the equation given and then solve for y
That is;
3x - 4y = 12
3(0) - 4y = 12
-4y = 12
Divide both-side of the equation by -4
y = -3
The y-intercept is (0, -3)
Hence, the correct option is B.
x-int: (4,0)
y-int: (0, -3)
A package of 5 pairs of insulated gloves costs $28.955. What is the unit price of the pairs of gloves ?
If a package of 5 pairs of insulated gloves costs $28.955,then the unit price of the pairs of gloves is $5.791
A package of 5 pairs of insulated gloves costs $28.955
The price of 5 pairs of insulated gloves = $28.955
Here we have to use the unitary method
The price of 1 pair of insulated gloves = The price of 5 pairs of insulated gloves / 5
Substitute the values in the equation and divide it
The price of 1 pair of insulated gloves = 28.955/5
= $5.791
Hence, if a package of 5 pairs of insulated gloves costs $28.955,then the unit price of the pairs of gloves is $5.791
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Solve 7|14i-10|+9 = 37
Answer:
i = 1, (3/7)
Step-by-step explanation:
7|14i-10|+9 = 37,
-9 -9
------------------------
7|14i - 10| = 28
÷7 ÷7
---------------------
|14i - 10| = 4
14i - 10 = 4
+10 +10
------------------
14i = 14
÷14 ÷14
--------------
i = 1
|14i - 10| = -4
14i - 10 = -4
+10 +10
---------------------
14i = 6
÷14 ÷14
--------------
i = (3/7)
I hope this helps!
Provide the correct reason for the statement in line 3. (please help)
Answer:
c) subtraction property of =
Step-by-step explanation:
step 3 show the movement of +15 moving to the other side of the = by subtracting 15 on each side, leaving you with 3x-16
????????????????????
Answer:
536
Explanation:
rom the given table:
If an individual and spouse enroll in the low deductible plan, the monthly premium will be $536.
Study Island Math Help Worth 40 PTS
Answer:
B
Step-by-step explanation:
using the rules of exponents
• [tex]a^{m}[/tex] × [tex]a^{n}[/tex] = [tex]a^{(m+n)}[/tex]
• [tex]a^{-m}[/tex] = [tex]\frac{1}{a^{m} }[/tex]
• [tex](a^m)^{n}[/tex] = [tex]a^{mn}[/tex]
given
[tex]5^{8}[/tex] × [tex](5^{12}) ^{-4}[/tex]
= [tex]5^{8}[/tex] × [tex]5^{(12(-4))}[/tex]
= [tex]5^{8}[/tex] × [tex]5^{-48}[/tex]
= [tex]5^{(8-48)}[/tex]
= [tex]5^{-40}[/tex]
= [tex]\frac{1}{5^{40} }[/tex]
I want to place an order of 253,625 books. How can I estimate the number to make sure it’s enough books
The estimate where the number of books is enough is 254,000
What is the estimation of a number?The estimation of numbers is the approximation of the said number.
This means that the estimation of a number is close to the said number, but it has a slight different value
How to estimate the number of books?The number of books is given as
Number of books = 253,625
From the question, we understand that:
You want to make sure that the number of books is enough
This means that we estimate the number of books to a number greater than 253,625
A good way to do this is to approximate to the nearest thousand
When the number is approximated to the nearest thousand, we have
Estimate = 254,000
Hence, the estimate is 254,000
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write the sum in sigma notation [tex]\frac{1}{3} +\frac{1}{15} +\frac{1}{35} +\frac{1}{63}[/tex]
Sum in the sigma notation of the given series is equal to
∑[1 /(2n +1)(2n + 3)] , n =0,1,2,3.
As given in the question,
Given series :
(1 /3) + ( 1/ 15) + ( 1/ 35) + ( 1/ 63)
Simplify the given series
= [1/ (1 ×3)] + [1/ (3 ×5)] + [1/ (5 ×7)] + [1/ (7×9)]
1, 3, 5, 7 can be expressed as ( 2n + 1) , where n = 0, 1, 2, 3
3, 5, 7, 9 can be expressed as ( 2n + 3) , where n = 0, 1, 2, 3
Express the given series as a sum in the sigma notation form
(1 /3) + ( 1/ 15) + ( 1/ 35) + ( 1/ 63)
=∑ [ 1/ (2n +1) (2n + 3) ] where n = 0,1,2,3
Therefore, sum in the sigma notation of the given series is equal to
∑[1 /(2n +1)(2n + 3)] , n =0,1,2,3.
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helppppppp 20 points
C) Triangle ABC is equilateral.
E) Segment AB is the same length as segment AC because they are both radii of the same circle centered at point A.
F) Segment AB is the same length as segment BC because they are both radii of the same circle centered at point B.
What are Euclid's Axioms?Euclidean geometry, sometimes referred to as Euclid's geometry is the study of solid and flat shapes based on many axioms and theorems. The assumptions of the evident universal truths that have not been verified are known as Euclid's axioms or common conceptions.In the given figure,
A and B are the centres of the two intersecting circles.
In a circle having centre at A, we have
AC = AB = radius
In a circle having centre at B, we have
BC = BA = radius
According to Euclid's first axiom, things that are equal to the same thing are equal to one another.
Therefore, AC = AB = BC
Hence, the triangle is an equilateral triangle.
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Given the initial inequality 2 > -4, identify whether the operation preserves the inequality symbol or reverses the inequalitysymbol.Add -2 to both sides.
ANSWER:
Preserves the inequality
STEP-BY-STEP EXPLANATION:
We have the following expression:
[tex]2>-4[/tex]We add -2 on both sides and we have
[tex]\begin{gathered} 2-2>-4-2 \\ 0>-6 \end{gathered}[/tex]This is true, therefore the equality symbol remains the same
You reflect the triangle below over the x-axis.
The coordinates of P' are ( , )
Answer:
P' (- 5, 3 )
Step-by-step explanation:
under a reflection in the x- axis
a point (x, y ) → (x , - y ) , then
P (- 5, - 3 ) → P' (- 5, 3 )
Construct a box-and-whisker plot for the given data.
The test scores of 40 students are listed below.
25 35 43 44 47 48 54 55 56 57
59 62 63 65 66 68 69 69 71 72
72 73 74 76 77 77 78 79 80 81
81 82 83 85 89 92 93 94 97 98
By median , test scores of 40 students are listed below. So five number summary is 25, 58,72,81,98
What are median in math?
The middle number is obtained by sorting all the data points and choosing the center one (or if there are two middle numbers, taking the mean of those two numbers). Example: Because the number 4 is in the middle when the numbers are ordered (1, 4, 7), it is the median of 4, 1, and 7.First we need to sort the data from Smallest to largest data value which is allready sorted
n = 40
Five number summary can be calculated as
Minimum = 25
Q1 can be calculated as
Lower Half of 40 values is 20, so median of 20 values will give us 25th percentile so Median = (20+1)/2 = 10.5th here 10th value is 57 and 11th value is 59 so Q1 = (57+59)/2 = 58
Median = (n+1)/2 = (40+1)/2 = 20.5th value and 20th and 21st value is 72 so median = (72+72)/2 = 72
Upper 20 values after median will provide us Q3 or 75th percentile which can be calculated as 20+(20+1)/2 = 30.5th value so 30th value is 81 and 31st value is also 81. So Q3 = (81+81)/2 = 81
Max = 98
So five number summary is 25, 58,72,81,98
And correct boxplot for this data is A. Boxplot A is correct which shows correct five number summary.
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The water pressure on a submerged object is given by P = 64d, where P is the pressure in pounds per square foot, and d is the depth of the object in feet. a. solve the formula for d. b. Find the depth of a submerged object if the pressure is 672 pounds per square foot.
The water pressure on a submerged object is given by P = 64d, then the
a. The answer to the subject of the formula, "d," is d = P/64.
b. The depth of a submerged object if the pressure is 672 pounds per square foot is 10. 5 feet
A function can be defined as an expression, law or rule that shows the relationship between two variables.
These variables are namely;
Independent variableDependent variableGiven the role as;
P = 64d
Where;
P stands for pounds per square foot of pressure.
d represents the object's depth in feet.
Let's now make the variable "d" the focus of the formula.
The coefficient of the variable "d," which is 64, is used to divide both sides of the equation. So that the variable can stand on its own, this is done.
(a) We get,
P/64 = 64d/64
d = P/ 64
(b) P equals 672 pounds per square foot if there is pressure.
When the variable's value is substituted into the formula, we get;
d = P/64
Then,
d = 672/64
Discover the quotient,
d = 10. 5 ft.
The value is 10.5 feet .
Therefore,
a. The answer to the subject of the formula, "d," is d = P/64.
b. The depth of a submerged object if the pressure is 672 pounds per square foot is 10. 5 feet.
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I really need to know how to solve for this question!! I have no idea what to do
Given:
Cost of Truck = $16000
Down payment amount = $3000
EMI = $250
Duration of EMI = 60 Months
Required:
a) Total amount financed
b) Total finance charge paid at the end of 60 months
Explanation:
Total cost of Truck = $16000
The total amount taken from other source = Total Amount - Down payment
=$16000-$3000
=$13000
EMI = $ 250
[tex]Total\text{ }amount\text{ }paid\text{ }at\text{ }the\text{ }end\text{ }of\text{ }60\text{ }months=250\times60[/tex][tex]Total\text{ }amount\text{ }paid\text{ }at\text{ }the\text{ }end\text{ }of\text{ }60\text{ }months=15000[/tex]Total amount paid at the end of 60 months = $15000
The total amount financed = Actual price -Down payment
=$16,000- $3000
=$13000
a) Total amount financed =3$18000
The total finance charged = Actual price - amount paid monthly for 60 months.
=$16000-$15000
=$1000
b)The total finance charged =1$20
Answer
Hence the total amount finance is 13000 dollar.
The total finance charged is 1000 dollar.00
15x=75Write which property of equality you used.
1) Given this equality,
15x = 75 Divide both sides by 15
x=5
2) Then we can write that since x = 5 and 15x = 75 we have here is the Division Property of Equality.
Because in this case, we divided both sides by 15, which is different from zero to find equivalent equality.
3) So
The polynomial passes through the
X-intercepts of (-5, 0), (-3, 0), and (1, 0). The end behavior of the graph is down, up.
Finally, it passes through the point of (3, 14).
The equation of the polynomial is P(x) = 6.86(x + 5) (x + 3) (x - 1)
Given,
x intercepts which the polynomial passes though = (-5, 0), (-3, 0), and (1, 0).
Point = (3, 14).
We have to find the polynomial equation;
A polynomial is defined as;-
P(x) = a(x - c₁) (x - c₂)....(x - cₙ)
Here,
a is a constant
c₁, c₂,...cₙ are the zeros of the polynomial
Here,
-5, -3 and 1 are the zeros of the given polynomial
(x + 5), (x + 3) and (x - 1) are the factors of the given polynomial.
Then the equation of polynomial be like;
P(x) = a(x + 5) (x + 3) (x - 1)
This polynomial passes through point (3, 14)
That is,
14 = a(3 + 5) (3 + 3) (3 - 1)
14 = a x 8 x 6 x 2
14 = 96a
a = 96/14
a = 6.86
Substitute this in equation of polynomial;
That is,
P(x) = 6.86(x + 5) (x + 3) (x - 1)
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Solve for x.
√x + 20 = 9
Answer:
x = 61
Step-by-step explanation:
sqrt = square root
sqrt(x + 20) = 9 square both sides to cancel out sqrt
x + 20 = 81 subtract 20 to get x alone
x = 61
Jane, Alan, and Ravi sent a total of 146 text messages over their cell phones during the weekend. Ravi sent 8 fewer
messages than Jane. Alan sent 4 times as many messages as Ravi. How many messages did they each send?
Jane, Alan, and Ravi sent a total of 146 text messages over their cell phones during weekend, Ravi sent 8 fewer messages than Jane. Alan sent 4 times as many messages as Ravi. So as per question data,
Jane sent= 31 messages
Ravi sent= x-8= 31-8 = 23 messages
Alan sent= 4(x-8) = 92 messages
What operations are used here?The best way to solve an expression is from left to right if all the operations are the same (for instance, only addition, only subtraction, only multiplication, or only division). However, we must adhere to the order of operations for expressions including several operations.
The rule that dictates how to solve an expression involving numerous operations is known as the order of operations.
PEMDAS can help you remember that sequence. In PEMDAS, each letter denotes a mathematical operation.
Let, Jane sent x number of messages
Then, Ravi sent x-8 number of messages
and, Alan sent 4(x-8) number of messages
Total number of messages by them = 146
now,
x+(x-8)+4(x-8)=146
6x-40=146
x=31
so, Jane sent= 31 messages
Ravi sent= x-8= 31-8 = 23
Alan sent= 4(x-8) = 92
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3) Recall problem 48 from your notes: Half an hour after leaving her house, Leah was 25 miles from
home. Leah then merged onto the freeway, and was able to travel at a constant speed for the next
several hours. Four hours after leaving her house, she was 280 miles from home.
a.
Find the model, m(t) that gives Leah's distance from home d in terms of the number of hours
she has been on the freeway, t.
b.
Suppose that Rachel lives 5 miles away from the same freeway entrance that Leah took. Rachel
leaves her house, and 10 minutes later is on the freeway. After 3.7 hours, Rachel is also 280 miles
away from her home.
i. What is Rachel's average speed on the freeway?
ii. Suppose that Rachel travels at a constant speed once she gets on the freeway. Write a
function k = f(t) that describes the number of miles she is from home, k, in terms of
the number of hours since she left home, t.
c. Assuming they are both traveling at a constant rate, who is driving faster? Give two different
answers to how you can see this
i. intuitively from the information provided.
ii. Algebraically from how you defined the functions.
Answer:
What is the role of ethnocentrism in defining the concept of “progress”
Step-by-step explanation:
Answer this question find angle a,b,c,d,e,f
We can determine the values of the angles as follows:
*By opposite by angle we will have that angle f is equal to:
[tex]f=180-44-78\Rightarrow f=58[/tex]angle f has a measurement of 58°.
*By supplementary angles we will have that angle e is equal to:
[tex]e=90-64\Rightarrow e=26[/tex]angle e has a measurement of 26°.
*As is indicated (Two lines have the same markings indicating they are parallel) we will have that the measurement of angle b is 78°.
*By complementary angles, the measurement of angle a is:
[tex]a=180-78\Rightarrow a=102[/tex]angle a has a measurement of 102°.
*We will have by alternate exterior angles that c has a measurement of 58°.
*By supplementary angles, we will have that the measurement of angle d equals:
[tex]d=180-58\Rightarrow d=122[/tex]angle d has a measurement of 122°.
180
94
112
90
68
56
100
(x+18)° is vertical opposite 112° and vertical opposite angles are equal meaning
⇒(x+18)°=112°
x+18°=112°
x=112°-18°
∴x=94°
(2y)° is in a straight line with 112° meaning their sum is 180°
[tex]2y+112=180\\2y=180-112\\2y=68\\\frac{2y}{2} =\frac{68}{2} \\y=34[/tex]
∴y=34°
What is the value of x in the equation x^2 + 2 = 0
Answer:
[tex]{ \tt{ {x}^{2} + 2 = 0 }} \\ \\ { \tt{ {x}^{2} = - 2 }} \\ \\ { \tt{x = \sqrt{ - 2} }}[/tex]
- At this point, we can't operate the square root. Let's introduce the imaginary part of a value
From complex numbers;
[tex]{ \boxed{ \tt{{ \red{note} \: {\rightarrow{i \times i = - 1 = {i}^{2} }}}}}}[/tex]
Therefore;
[tex]{ \tt{x = \sqrt{2 \times {i}^{2} } }} \\ \\ { \boxed{ \rm{ \: x = i \sqrt{2} \: \: }}}[/tex]
Help PLEASE!!! How do I solve this?
Two similar boxes A and B are given with box B created by dividing box A's dimensions by 2.
Recall that the scale factor, k is the ratio of lengths of two corresponding sides of similar figures.
Since the dimensions of A were divided by 2 to form B, it follows that the scale factor of box A to box B is:
[tex]k=\frac{2}{1}[/tex]Also, recall the Similar Solids Theorem
It follows that the ratio of the volumes of the given solids is:
[tex]\frac{\text{Volume of A}}{\text{Volume of B }}=k^3=(\frac{2}{1})^3=\frac{2^3}{1^3}=\frac{8}{1}[/tex]Let the volume of box A be V₁ and let the volume of box B be V₂, it, therefore, implies that:
[tex]\frac{V_1}{V_2}=\frac{8}{1}[/tex]Substitute the value for the volume of box A into the equation, V₁=64:
[tex]\begin{gathered} \frac{64}{V_2}=\frac{8}{1} \\ Cross-Multiply_{} \\ \Rightarrow8V_2=64 \\ Divide\text{ both sides by 8:} \\ \Rightarrow\frac{8V_2}{8}=\frac{64}{8}\Rightarrow V_2=8\text{ cubic meters} \end{gathered}[/tex]Hence, the volume of box B is 8 m³.
what is the period of the graph of y=5sin(4piex)+3
The period of the graph is π/2.
What is the period of the graph?The length of one function wave is referred to as the period. One entire wave in this situation is 180 degrees, or radians. Without using a graph, you can get this simply dividing 2 by the frequency, which in this case is 2.
Given: 5 sin (4πx)+3
Period = 2π / 4
⇒ period = π/2
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the numbers between 1 to 2022 are written on a piece of paper.Caitlin circles the even numbers with red circles and nicole circles the multiples of 5 with yellow circles.How many numbers are circled with only one color?
The number of numbers circled with only one color found using arithmetic progression formula are 809 numbers
What is an arithmetic progression?An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms in the sequence is constant
The given parameters are;
The set of the numbers written on the paper = Numbers from 1 to 2022
Numbers circled with red circles by Caitlin = All even numbers between 1 and 2022
Numbers Nicole circles using yellow = Numbers that are multiples of 5
Required; The number of numbers that are circled with one colour only
Solution; The even numbers are numbers that are a multiple of 2
The number of even numbers between 1 and 2022 are found using arithmetic progression formula as follows;
[tex] a_{n} = a_{1} + (n - 1)\cdot d[/tex]
Where;
[tex]a_{1}[/tex] = The first term = 2 (First even number)
d = The common difference = 2
n = The number of terms
Given that 2022 is an even number, we have;
[tex] a_{n} = 2022 = 2 + (n - 1)\cdot 2[/tex]
Which gives;
[tex] \displaystyle {n = \frac{(2022 - 2)}{2} + 1 = 1011} [/tex]
We note that 10 is the lowest common multiple of 2 and 5, which gives;
Multiples of 10 have two colors, therefore, the numbers that have only one color is given by the difference between the number of even numbers and the number of multiples of 10 as follows;
The number of multiples of 10 are;
[tex] \displaystyle {n = \frac{(2020 - 10)}{10} + 1 = 202} [/tex]
Which gives;
The numbers having only one colour = 1011 - 202 = 809
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A pet store has 3 bird cages with 2 birds in each cage, and 10 aquariums with 4 fish in each aquarium. Which of the following expressions represents how many birds and fish the pet store owns?
(3+2)+(10 + 4)
(3+2) x (10+4)
(3 x 2) × (10 x 4)
(3 x 2) + (10 x 4)
Answer:
(3 × 2) + (10 × 4) is the correct expression.