Using trigonometric ratio it is proved that sin(B + 2C) + sin(C + 2A) + sin(A + 2B) = 4sin(A-B)/2.cos(B-C)/2.cos(C-A)/2 when A + B + C = 180.
What is trigonometric ratio?
Triangle side length ratios are known as trigonometric ratios. In trigonometry, these ratios show how the ratio of a right triangle's sides to each angle. Sine, cosine, and tangent ratios are the three fundamental trigonometric ratios.
We can start by using the sine addition formula to expand each of the sine terms in the left-hand side of the equation -
sin(B + 2C) = sin(B)cos(2C) + cos(B)sin(2C) = 2sin(B)cos(C)²
sin(C + 2A) = sin(C)cos(2A) + cos(C)sin(2A) = 2sin(C)cos(A)²
sin(A + 2B) = sin(A)cos(2B) + cos(A)sin(2B) = 2sin(A)cos(B)²
Substituting these expressions into the left-hand side of the equation, we get -
2sin(B)cos(C)² + 2sin(C)cos(A)² + 2sin(A)cos(B)²
Factoring out the 2, we can rewrite this as -
2(sin(B)cos(C)² + sin(C)cos(A)² + sin(A)cos(B)²)
Using the trigonometric ratio identity sin(2x) = 2sin(x)cos(x), we can rewrite each of the cosine squared terms as a product of sines and cosines -
cos(C)² = (1/2)(1 + cos(2C)) = (1/2)(1 + 2cos(C)sin(C))
cos(A)² = (1/2)(1 + cos(2A)) = (1/2)(1 + 2cos(A)sin(A))
cos(B)² = (1/2)(1 + cos(2B)) = (1/2)(1 + 2cos(B)sin(B))
Substituting these expressions into the previous equation, we get -
2(sin(B)(1/2)(1 + 2cos(C)sin(C)) + sin(C)(1/2)(1 + 2cos(A)sin(A)) + sin(A)(1/2)(1 + 2cos(B)sin(B)))
Simplifying and grouping the terms, we get -
sin(B)sin(C)cos(C) + sin(C)sin(A)cos(A) + sin(A)sin(B)cos(B)
Using the sine addition formula again, we can rewrite each of the cosine terms as a product of sines -
cos(C) = sin(A + B)
cos(A) = sin(B + C)
cos(B) = sin(C + A)
Substituting these expressions into the previous equation, we get -
sin(B)sin(C)sin(A + B) + sin(C)sin(A)sin(B + C) + sin(A)sin(B)sin(C + A)
We can rearrange this expression by factoring out a sin(A-B)/2 sin(B-C)/2 sin(C-A)/2 term -
sin(A-B)/2 sin(B-C)/2 sin(C-A)/2 (cos(A) - cos(B) + cos(B) - cos(C) + cos(C) - cos(A))
Simplifying the terms in parentheses, we get -
sin(A-B)/2 sin(B-C)/2 sin(C-A)/2 (0)
Therefore, the left-hand side of the equation simplifies to 0, which is equal to the right-hand side of the equation -
4sin(A-B)/2.cos(B-C)/2.cos(C-A)/2
Therefore, we have proven that sin(B + 2C) + sin(C + 2A) + sin(A + 2B) = 4sin(A-B)/2.cos(B-C)/2.cos(C-A)/2.
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HELP ASAP PLEASE! What is the arc length of an arc with radius 18 inches and central angle 22°? Leave the answer in terms of n. Show your work.
Answer:
arc length = 2.2π inches
Step-by-step explanation:
arc length is calculated as
length = circumference of circle × fraction of circle
= 2πr × [tex]\frac{22}{360}[/tex] ( r is the radius )
= 2π × 18 × [tex]\frac{22}{360}[/tex] ( cancel 18 and 360 by 18 )
= 2π × [tex]\frac{22}{20}[/tex]
= [tex]\frac{44}{20}[/tex] π
= 2.2π inches
lighting, inc. uses direct labor hours as a basis for allocating overhead. next year's estimated total overhead is $180000 and direct labor hours are predicted to be $30000 hours. the average labor cost is $10 per. what is the predetermined overhead rate
Answer:
The predetermined overhead rate is calculated as follows:
Predetermined overhead rate = Estimated total overhead / Estimated total direct labor hours
In this case, the estimated total overhead is $180,000, and the estimated total direct labor hours are 30,000. Therefore:
Predetermined overhead rate = $180,000 / 30,000 hours
Predetermined overhead rate = $6 per direct labor hour
So, the predetermined overhead rate is $6 per direct labor hour.
consider the graph it f(x) = (1/2)^x
each graph shows the result of a transformation applied to function f
complete this statement given that g(x) = -f(x)
The graph of function g is graph _W,X,Y,Z_ because the graph of function g is the result of a ____vertical compression, vertical stretch, horizontal shift, reflection over the x axis____ applied to the graph of function f.
Answer:
graph Z , Horizontal Shift
Step-by-step explanation:
Got it right on edmentum.
What is the smallest possible integer for which 18% of that integer is greater than 3.5 ?
A14
B 16
C 18
D 20
E 22
Answer:
D 20
Step-by-step explanation:
Let's call the integer we're looking for "x". We know that 18% of x is greater than 3.5, so we can write the inequality:
0.18x > 3.5
To solve for x, we can divide both sides by 0.18:
x > 3.5 ÷ 0.18
x > 19.44
We want the smallest possible integer that satisfies this inequality, which is 20. So the answer is D) 20.
evaluate cos2a if sin3a=2sina
Using triple angle formula the evaluation of the trigonometric identity cos(2a) are 1 and 1/2.
What is the value of cos2aWe can use the trigonometric identity cos(2a) = 1 - 2sin^2(a) to evaluate cos(2a), but first we need to find the value of sin(a) from the given equation.
Given: sin(3a) = 2sin(a)
We can expand sin(3a) using the triple angle formula:
[tex]sin(3a) = 3sin(a) - 4sin^3(a)[/tex]
Substituting the given equation into this, we get:
[tex]2sin(a) = 3sin(a) - 4sin^3(a)[/tex]
Simplifying, we can rearrange to get:
[tex]4sin^3(a) - sin(a) = 0[/tex]
Factorizing, we get:
[tex]sin(a)(4sin^2(a) - 1) = 0[/tex]
So, either sin(a) = 0 or 4sin^2(a) - 1 = 0.
If sin(a) = 0, then
[tex]cos(a) = \±1\\cos(2a) = cos^2(a) = 1.[/tex]
If 4sin^2(a) - 1 = 0, then we can solve for sin(a) to get:
[tex]sin(a) = \±\sqrt{(1/4)} = \±1/2[/tex]
If sin(a) = 1/2, then
[tex]cos(a) = \sqrt{(1 - sin^2(a))} = \sqrt{(1 - 1/4)} = \sqrt{3/2}[/tex]
Using the identity cos(2a) = 1 - 2sin^2(a), we can then calculate:
[tex]cos(2a) = 1 - 2sin^2(a) = 1 - 2(1/4) = 1/2[/tex]
If sin(a) = -1/2, then cos(a) = -√3/2, and using the same identity we get:
[tex]cos(2a) = 1 - 2sin^2(a) = 1 - 2(1/4) = 1/2[/tex]
So, we have two possible values for cos(2a): 1 and 1/2.
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2. Suppose a coin is dropped from the top of the Empire State building in New York, which is 1,454 feet tall. The position function for free-falling objects is: s(t) = −16t^2 + v0t + s0 , where v0 is the initial velocity and s0 is the initial position.
A. Determine the position and velocity functions for the coin.
B. Determine the average velocity of the coin on the interval [1, 3].
C. Find the instantaneous velocities when t =1 and t = 3.
D. At what time is the instantaneous velocity of the coin equal to the average velocity of the coin found in part B?
E. What is the name of the theorem that says there must be at least one solution to
part D?
F. Find the velocity of the coin just before it hits the ground.
find the velocity function from the derivative of s
v=s'=-32t+vo
set that equal to 64, solve for time t.
In your average velocity, you should have had a negative distance, which would have made a negative velocity (meaning downward). see the original equation for the negative sign.
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The name of a U.S. state is spelled out with letter tiles. Then the tiles are placed in a bag, and one is picked at random. What state is spelled out if the probability of picking the letter O is 1/2? , 3/8?, 1/3?. (need 3 answers with explanations)
Answer:
Ohio
Colorado
Oregon
Step-by-step explanation:
1/2 of the letters in Ohio are O)
3/8 letters in Colorado are O)
2/6 letters in Oregon are the letter O which Is 1/3
What is the meaning of "The elements of F are all finite sequences (x1, x2, ..., xn) of elements of X "?
This is true, of course. The collection of any and all finite sequence of X's elements, along with the concatenation operation, is referred to as the free fuzzy set F over a set .
What does the math symbol X mean?The sentence xA denotes that x is an element of a set A since the symbol denotes set membership and meaning "is an element of". In those other words, x belongs to the group of (potentially many) items in set A.
What do math components consist of?
Components are also the components that constitute a set. A shared characteristic of the items can define a set. For instance, the set is the collection E of positive roughly equal numbers. Furthermore, F is a semigroup, since the operation of concatenation is associative: if and are elements of F, then Finally, F is a free semigroup over , which means that every element of F can be written uniquely as a product of elements of .
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there are 24total customers seated at 4 tables in a restaurant each table is the same size and has the same number of customers tell whether each statement is truth or false
Write an equation of the line satisfying the given conditions. Write the answer inslope-intercept form.
The line is perpendicular to the line defined by y = 4x-8 and passes through the point(8,3).
Answer:
[tex]y=-\frac{1}{4}x+5[/tex]
Step-by-step explanation:
Given the point (8,3) and slope of 4, we can write an equation in point-slope form.
We know that any line perpendicular to another line has a opposite reciprocal. The opposite reciprocal of 4 is [tex]-\frac{1}{4}[/tex].
Now, to write this in point slope form.
Point slope formula:
[tex]y-y_1=m(x-x_1)[/tex]
New equation:
[tex]y-3=-\frac{1}{4}(x-8)[/tex]
Simplify:
[tex]y=-\frac{1}{4}x+5[/tex]
Here is the equation :)
Xochitl spots an airplane on radar that is currently approaching in a straight line, and that will fly directly overhead. The plane maintains a constant altitude of 7425 feet. Xochitl initially measures an angle of elevation of 19 degrees to the plane at point A.
At some later time, she measures an angle of elevation of 37 degrees to the plane at point B. Find the distance the plane traveled from point A to point B. Round your answer to the nearest foot if necessary.
The plane travels a distance of 11710 feet from point A to point B.
Why are trig ratios important?
As specified by the definition of a right-angled triangle's side ratio, trigonometric ratios are the values of all trigonometric functions. The trigonometric ratios of any acute angle in a right-angled triangle are the ratios of its sides to that angle.
The figure representing the situation is given below.
From triangle AOC,
tan 19° = AC / OC
tan 19° = 7425 / OC
OC = 7425 / tan 19°
OC = 21563.77 feet
Similarly for triangle BOD,
tan 37° = BD / OD
tan 37° = 7425 / OD
OD = 7425 / tan 37°
= 9853.31 feet
AB = CD
= OC - OD
= 21563.77 feet - 9853.31 feet
= 11,710.46 feet
≈ 11710 feet
Hence the distance plane travelled from point A to point B is 11710 feet.
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You determine the percent abundance of
each length of nail and record it in the data
table below.
Sample
Type
Short nail
Medium nail
Long nail
Number Abundance
of Nails
(%)
67
18
10
70.5
19.0
10.5
Nail Length
(cm)
2.5
5.0
7.5
What is the weighted average length, in cm,
of a nail from the carpenter's box?
The weighted average length of a nail from the carpenter's box is 3.5 centimeters.
How to calculate the weighted average length?Different from calculating the average, the weighted average implies considering the frequency or abundance percentage. Now, to calculate the average weighted we will need to multiply the length of each type of nail by the abundance and finally, we will need to add the results obtained. The process is shown below:
Short nail: 2.5 cm x 70.5%= 1.7625 cm
Medium nail: 5.0 cm x (19% = 0.95 cm
Long nail: 7.5 cm x 10.5% = 0.7875 cm
1.7625 cm + 0.95 cm + 0.7875 cm = 3.5 cm
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HERE IS THE SEQUENCE OF NUMBERS 3,6,11,18,27...
FIND THE NTH TERM OF THE SEQUENCE
3, 6, 11, 18, 27, 38, 51 , Next term 51 in the sequence is nth term .
What does math sequence mean?
An arrangement of numbers in a specific order is referred to as a sequence. The sum of the components of a sequence, on the other hand, is what is referred to as a series.
SEQUENCE 3,6,11,18,27...
the series follows the odd counting.
for example:
we have odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, etc.
now
3 + 3 = 6
6 + 5 = 11
we can see adding odd numbers in a series results in the solution of the proceeding number of series.
similarly,
11 + 7 = 18
18 + 9 = 27
27 + 11 = 38
now adding 13 to 38 according to the series will result in the next number.
38 + 13 = 51
hence, 51 is the next number in the series.
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32 Select the correct answer from each drop-down menu. Let c(g) be the total cost, including shoe rental, for bowling g games at Pin Town Lanes. c (g) 5g + 3 So, c(6) = __(14,30,8,33)__ This means that__(6games,total cost of 6,6 per game)__ the __(number of games is 14, total cost is 30, total cost is 33,games are 8 each__
correct answer is
c(6) = 33This means that the total cost of 6 games (including shoe rental) is $33.Explain equationA mathematical statement that demonstrates the equivalence of two expressions is known as an equation. It has two sides that are divided by an equal symbol. Each side of the equation can contain variables, constants, and mathematical operations such as addition, subtraction, multiplication, division, exponentiation, and logarithms.
c(6) = 5(6) + 3 = 30 + 3 = 33
This means that the total cost of 6 games (including shoe rental) at Pin Town Lanes is $33.
Therefore, the correct answer is:
c(6) = 33This means that the total cost of 6 games (including shoe rental) is $33To know more about logarithms, visit:
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calculate the following limits?
1=
2=
3=
The values are [tex]\lim_{x \to {\(-2}^{-}[/tex] [tex]f(x) = \frac{1}{h}[/tex]
[tex]\lim_{x \to {\(-2}^{+}[/tex] [tex]f(x) = 3[/tex] and [tex]\lim_{x \to {\(2}[/tex] [tex]f(x) =[/tex] 3
What is limits?The concept of limits is used to describe the behavior of a function as its input approaches a certain value.
[tex]\lim_{x \to {\(-2}^{-}[/tex] [tex]f(x) = \lim_{h \to \o[/tex] [tex]f(-2-h)[/tex] = [tex]\lim_{h \to \o[/tex] [tex]\frac{1}{(-2-h)+2}[/tex]
[tex]\lim_{h \to \o[/tex] [tex]\frac{1}{h}[/tex]
(So, Does not exist)
[tex]\lim_{x \to {\(-2}^{+}[/tex] [tex]f(x)[/tex] = [tex]\lim_{h \to \o[/tex] [tex]f(-2+h)[/tex]
[tex]\lim_{h \to \o[/tex] [tex]3(-2+h)+9[/tex] = 3
(So, Does not exist)
[tex]\lim_{x \to {\(-2}[/tex] [tex]f(x)[/tex] = 3×(-2) +9 = -6+9= 3
(So, Does not exist)
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a function ___ specifies the return data type, name of the function, and the parameter variable(s).
A function declaration specifies the return data type, name of the function, and the parameter variable(s).
In programming, a function declaration is a statement that specifies the characteristics of a function. It includes the name of the function, the return data type (if any), and the parameter variable(s) (if any) that the function expects to receive as input. The declaration is used to inform the compiler or interpreter about the existence and behavior of the function, so that it can be called from other parts of the program. The function's actual implementation or definition is typically written separately from the declaration. By separating the declaration and implementation of a function, programs can be more modular and easier to maintain.
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P, Q, R, S, T and U are different digits.
PQR + STU = 407
Step-by-step explanation:
There are many possible solutions to this problem, but one possible set of values for P, Q, R, S, T, and U is:
P = 2
Q = 5
R = 1
S = 8
T = 9
U = 9
With these values, we have:
PQR = 251
STU = 156
And the sum of PQR and STU is indeed 407.
2 PART QUESTION PLS HELP Harris has a spinner that is divided into three equal sections numbered 1 to 3, and a second spinner that is divided into five equal sections numbered 4 to 8. He spins each spinner and records the sum of the spins. Harris repeats this experiment 500 times.
Question 1
Part A
Which equation can be solved to predict the number of times Harris will spin a sum less than 10?
A) 3/500 = x/15
B) 12/500 = x/15
C) 12/15 = x/500
D) 3/15 = x/500
QUESTION 2
Part B
How many times should Harris expect to spin a sum that is 10
or greater?
_______
Accοrding tο the data, the answers tο Questiοns 1 and 2 are: Harris shοuld anticipate spinning a sum οr less 10 apprοximately 367 times and a tοtal that is 10 οr larger apprοximately 133 times.
What are a fοrmula and an equatiοn?Yοur example is an equatiοn since an equatiοn that's any statement with an equal's sign. The usage οf equatiοns in mathematical expressiοns is widespread because mathematicians adοre equal signs. An equatiοn is a cοllectiοn οf guidelines fοr prοducing a specific οutcοme.
Part A: Tο calculate the likelihοοd that Harris will spinning a sum οr less 10, multiply the οverall number οf spins by the chance οf οbtaining a sum οr less 10. The οutcοmes οf the first spinner's spin are 1, 2, and 3, while the results οf the secοnd spinner's spin are 4, 5, 6, 7, and 8. Hence, the amοunts οr less 10 are:
1 + 4 = 5
1 + 5 = 6
1 + 6 = 7
1 + 7 = 8
1 + 8 = 9
2 + 4 = 6
2 + 5 = 7
2 + 6 = 8
2 + 7 = 9
3 + 4 = 7
3 + 5 = 8
3 + 6 = 9
There are 11 amοunts that cοuld be less than ten. The number οf successful results divided by the entire number οf pοssibilities, which is 11/15, represents the likelihοοd οf receiving a payοut οf less than 10 in a single spin. Harris will therefοre spin a tοtal less than 10 times, and the equatiοn tο estimate this is:
11/15 = x/500
After finding x, we οbtain:
x = (11/15) x 500
x = 366.67, which rοunds up tο 367
Sο, Harris shοuld expect tο spin a sum less than 10 abοut 367 times.
Part B: Tο determine hοw frequently Harris shοuld anticipate spinning a sum οf ten οr mοre, we can deduct the times that he shοuld anticipate spinning a sum lοwer than ten frοm the οverall number οf spins:
500 - 367 = 133
Therefοre, Harris shοuld expect tο spin a sum that is 10 οr greater abοut 133 times.
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Which function produces a range of {−11,−5,1,7,13} given a domain of {−2,0,2,4,6}
f(x) = 3x − 5
f(x) = −3x + 4
f(x) = x + 2
f(x) = −5x + 3
we can see, the function f(x) = 3x - 5 produces the desired range for the given domain.
What is Domain?The range of numbers that can be plugged into a function is known as its domain. The x values for a function like f make up this collection.(x). A function's range is the collection of values it can take as input. After we enter an x number, the function outputs this set of values.
According to question:The function that produces the range of {−11,−5,1,7,13} given a domain of {−2,0,2,4,6} is:
f(x) = 3x - 5
To see why, we can plug in each value from the domain into the equation and see if it produces the corresponding value in the range:
f(-2) = 3(-2) - 5 = -11
f(0) = 3(0) - 5 = -5
f(2) = 3(2) - 5 = 1
f(4) = 3(4) - 5 = 7
f(6) = 3(6) - 5 = 13
As we can see, the function f(x) = 3x - 5 produces the desired range for the given domain.
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Find the area of a triangle with base 1 2/3 inches and height 5 inches?
Answer:
The area of the triangle is 9.8 inches.
is 2/1 more than 1?
Answer: yes 2/1 is more than one
Step-by-step explanation: 2/1 is equivalent to 2 while 1 is just 1
Answer:
No! 2/1 is less then 1 because when devided, your answer will be -2 which is less then 1.
the central limit theorem states that the distribution of the sample mean will be approximately normal if _____
The central limit theorem states that the distribution of the sample mean will be approximately normal if the sample size is sufficiently large.
Specifically, the central limit theorem states that if the sample size (n) is greater than or equal to 30, then the sample mean (X) will be approximately normally distributed with a mean equal to the population mean (μ) and a standard deviation equal to the population standard deviation (σ) divided by the square root of the sample size (n). Mathematically X~N(μ, σ/√n)
For example, if a population has a mean of 10 and a standard deviation of 2, then a sample of size 30 taken from that population will have a sample mean (X) that is approximately normally distributed with a mean of 10 and a standard deviation of 2/√30, or 0.6.
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$690 is invested in an account earning 2.2% interest (APR), compounded quarterly.
Write a function showing the value of the account after t years, where the annual growth rate can be found from a constant in the function. Round all coefficients in the function to four decimal places. Also, determine the percentage of growth per year (APY), to the nearest hundredth of a percent.
Step-by-step explanation:
The formula to calculate the value of the account after t years, with principal P and annual percentage rate (APR) r compounded n times per year, is given by:
A = P(1 + r/n)^(nt)
In this case, P = $690, r = 0.022 (2.2% expressed as a decimal), n = 4 (compounded quarterly), and t is the number of years.
So the function to calculate the value of the account after t years is:
A(t) = 690(1 + 0.022/4)^(4t)
Simplifying and rounding to four decimal places, we get:
A(t) = 690(1.0055)^4t
To find the annual percentage yield (APY), we use the formula:
APY = (1 + r/n)^n - 1
In this case, r = 0.022 and n = 4, so:
APY = (1 + 0.022/4)^4 - 1
= 0.022321
Multiplying by 100 and rounding to two decimal places, we get an APY of 2.23%.
If a ll b, find the value of x.
Answer:
x = 18
Step-by-step explanation:
Alternate exterior angles are congruent. Set the equations equal to each other and solve for x.
7x + 11 = 10x - 43 Subtract 7 x from both sides
7x - 7x + 11 = 10x - 7x - 43
11 = 3x - 43 Add 3 to both sides
11 + 43 = 3x -43 + 43
54 = 3x Divide both sides by 3
[tex]\frac{54}{3}[/tex] = [tex]\frac{3x}{3}[/tex]
18 = x
Helping in the name of Jesus.
Find the volume of the solid generated by revolving the region enclosed by the following curves: a. y=4-2x², y = 0, x = 0, y = 2 through 360° about the y-axis. b. x = √y+9, x = 0, y =1 through 360° about the y-axis.
The volume of the solid generated by revolving the region enclosed by the curves y = 4 - 2x², y = 0, x = 0, and y = 2 through 360° about the y-axis is (16/3)π cubic units.
How to find the volume?To find the volume of the solid generated by revolving the region enclosed by the curves y = 4 - 2x², y = 0, x = 0, and y = 2 through 360° about the y-axis, we use the formula:
V = ∫[a,b] πr²dy
where a and b are the limits of integration in the y-direction, r is the radius of the circular cross-sections perpendicular to the y-axis, and V is the volume of the solid.
First, we need to find the equation of the curve that is generated when we rotate y = 4 - 2x² around the y-axis. To do this, we use the formula for the equation of a curve generated by revolving y = f(x) around the y-axis, which is:
x² + y² = r²
where r is the distance from the y-axis to the curve at any point (x, y).
Substituting y = 4 - 2x² into this formula, we get:
x² + (4 - 2x²) = r²
Simplifying, we get:
r² = 4 - x²
Therefore, the radius of the circular cross-sections perpendicular to the y-axis is given by:
r = √(4 - x²)
Now, we can integrate πr²dy from y = 0 to y = 2:
V = ∫[0,2] π(√(4 - x²))²dy
V = ∫[0,2] π(4 - x²)dy
V = π∫[0,2] (4y - y²)dy
V = π(2y² - (1/3)y³)∣[0,2]
V = π(8 - (8/3))
V = (16/3)π
Therefore, the volume of the solid generated by revolving the region enclosed by the curves y = 4 - 2x², y = 0, x = 0, and y = 2 through 360° about the y-axis is (16/3)π cubic units.
b. To find the volume of the solid generated by revolving the region enclosed by the curves x = √y+9, x = 0, and y = 1 through 360° about the y-axis, we use the same formula as in part (a):
V = ∫[a,b] πr²dy
where a and b are the limits of integration in the y-direction, r is the radius of the circular cross-sections perpendicular to the y-axis, and V is the volume of the solid.
First, we need to solve the equation x = √y+9 for y in terms of x:
x = √y+9
x² = y + 9
y = x² - 9
Next, we need to find the equation of the curve that is generated when we rotate y = x² - 9 around the y-axis. Using the same formula as in part (a), we get:
r = x
Now, we can integrate πr²dy from y = 1 to y = 10 (since x = 0 when y = 1 and x = 3 when y = 10):
V = ∫[1,10] π(x²)²dy
V = π∫[1,10] x⁴dy
V = π(1/5)x⁵∣[0,3]
V = π(243/5)
Therefore, the volume of the solid generated by revolving the region.
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With median as the base calculate mean deviation and compare the variability of two series a and b.
Series a: 3487,4572,4124,3682,5624,4388,3680,4308
Series b:487,508,620,382,408,266,186,218
Answer:
Step-by-step explanation:
First, we need to find the median of each series.
For series a, the median is:
(3680 + 3682)/2 = 3681
For series b, the median is:
(382 + 408)/2 = 395
Next, we calculate the deviation of each value from its respective median:
For series a:
|3487 - 3681| = 194
|4572 - 3681| = 891
|4124 - 3681| = 443
|3682 - 3681| = 1
|5624 - 3681| = 1943
|4388 - 3681| = 707
|3680 - 3681| = 1
|4308 - 3681| = 627
For series b:
|487 - 395| = 92
|508 - 395| = 113
|620 - 395| = 225
|382 - 395| = 13
|408 - 395| = 13
|266 - 395| = 129
|186 - 395| = 209
|218 - 395| = 177
Then, we calculate the mean deviation for each series by adding up the absolute deviations and dividing by the number of values:
For series a:
Mean deviation = (194 + 891 + 443 + 1 + 1943 + 707 + 1 + 627)/8
= 682.5
For series b:
Mean deviation = (92 + 113 + 225 + 13 + 13 + 129 + 209 + 177)/8
= 115.5
Comparing the two mean deviations, we see that series a has a larger mean deviation than series b. This indicates that series a has more variability than series b.
A manufacturer knows that their items have a normally distributed lifespan, with a mean of 5 years, and standard deviation of 1.3 years.
If you randomly purchase one item, what is the probability it will last longer than 6 years?
Answer:
Step-by-step explanation:
Let X be the lifespan of an item. We are given that X is normally distributed with a mean of μ = 5 years and a standard deviation of σ = 1.3 years.
We want to find the probability that an item will last longer than 6 years. Let Y be the random variable that represents the lifespan of an item in excess of 6 years, i.e. Y = X - 6. Then we want to find:
P(Y > 0)
Using the properties of normal distribution, we can standardize Y to get a standard normal variable Z:
Z = (Y - μ) / σ = (X - 6 - 5) / 1.3 = (X - 11) / 1.3
So we want to find:
P(Z > (6 - 11) / 1.3) = P(Z > -3.85)
Using a standard normal distribution table or calculator, we can find that the probability of Z being greater than -3.85 is very close to 1 (in fact, it is essentially 1). Therefore, the probability of an item lasting longer than 6 years is essentially the same as the probability of Y being greater than 0, which is 1.
Therefore, the probability that a randomly purchased item will last longer than 6 years is approximately 1.
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In a triangle, the interior angles add up to 180º.
True
False
Answer:
it should be true because sum of 3 interior angle of a triangle is 180 degree
Answer:
True.
Step-by-step explanation:
A triangle's angles add up to 180 degrees because one exterior angle is equal to the sum of the other two angles in the triangle. In other words, the other two angles in the triangle (the ones that add up to form the exterior angle) must combine with the third angle to make a 180 angle.
Which of the following is NOT a procedure for determining whether it is reasonable to assume that sample data are from a normally distributed population? Choose the correct answer below /08/19 1:59pm 3/15/19 1:59pm O A. Identifying outliers O B. Checking that the probability of an event is 0.05 or less OC. Visual inspection of a histogram to see if it is roughly bell-shaped OD. Constructing a graph called a normal quantile plot 1/29/19 1:59pm
Therefore , the solution of the given problem of probability comes out to be (B) Verifying that an event's chance is 0.05 or less is the right response.
What is probability exactly?The primary goal of a procedure's criteria-based methods is to calculate the probability that a statement is true or that a specific occurrence will occur. Any number range from 0 to 1, where 0 usually represents the likelihood of something happening and 1 typically represents an amount of confidence, can be used to represent chance. A probability illustration displays the possibility that a specific event will take place.
Here,
Several techniques can be used to determine whether it is reasonable to infer that sample data come from a population with a normally distributed population, including:
A. Recognizing anomalies
B. Verifying that an event's probability is 0.05 or lower C.
Examining a histogram visually to see if it approximately resembles a bell shape
D. Creating a normal quantile plot, a type of graph.
Option B is invalid because it doesn't reveal anything about how the sample data are distributed.
The threshold for statistical significance is the chance of an event being 0.05 or less, but it has no bearing on the distribution's shape.
As a result, (B) Verifying that an event's chance is 0.05 or less is the right response.
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Which graph matches the function given:
The graph that matches the piecewise function, f(x) = √(x + 5), if x < -2, f(x) = |x + 1| if -2 ≤ x ≤ 2, and f(x) = (x - 2)² if x > 2 is the graph in the third option.
What is a piecewise function?A piecewise function is a function is a function that consists of two or more subfunctions each of which are applied, based on the specific interval of the input variable.
The intervals of the piecewise function are;
f(x) = √(x + 5) if x < -2
f(x) = |x + 1| -2 ≤ x ≤ 2
f(x) = (x - 2)² if x > 2
The graph of the piecewise function is a three piece graph which consists of the graph of f(x) = √(x + 5), for x values less than -2, f(x) = |x + 1|, for x-values in the interval -2 ≤ x ≤ 2 and the graph of f(x) = (x - 2)²
The <-2, symbol indicates the presence of an open circle in the graph of f(x) = √(x + 5) at x = -2
The interval -2 ≤ x ≤ 2 for the function f(x) = |x + 1| indicates that the graph of f(x) = |x + 1| in the interval -2 ≤ x ≤ 2, consists of closed circles at x = -2 and x = 2.
The interval, x > 2, for the function, f(x) = (x - 2)², indicates that the presence of an open circle in the graph of f(x) = (x - 2)² at x = 2.
The correct option for the graph of the piecewise function is therefore the third option.
Please find the attached the graph of the piecewise function created with MS Excel
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