Answer:
A and C similar
Step-by-step explanation:
A: 80/100 = 0.8
B: 100/120 = 0.833...
C: 72/90 = 0.8
Since the ratio of the lengths of the sides of rectangle A and C are equal, the only similar rectangles are A and C.
Answer: A and C similar
Which of the following transformations is non-rigid? A.translationB.dilationC.reflectionD.rotation
From the question the transformations that non-rigid is dilation. The correct answer is B.
what is nonrigid transformation?Any transformation of a geometric object that alters the size but not the shape is referred to as a nonrigid transformation. Examples of non-rigid types of transformation include stretching and dilation. Any action taken on a shape is referred to as a metamorphosis.
Rigid transformation is When a figure undergoes a stiff transformation, its size and shape are left unaltered. Because the image is unchanged, stiff transformations include translations, rotations, and reflections.
As a result, a dilation is a non-rigid transformation because the size of the image is changing.
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How many solutions are there to the inequality x₁ + x₂ + x₃ ≤ 11, where x₁, x₂, and x₃ are nonnegative integers?
There are 66 solutions to the inequality x₁ + x₂ + x₃ ≤ 11, where x₁, x₂, and x₃ are nonnegative integers.
To solve this inequality, we must find all the possible combinations of nonnegative integers x₁, x₂, and x₃ that add up to 11 or less. This can be done by breaking down the problem into a series of nested loops. The outermost loop will iterate through all possible values of x₁, while the two inner loops will iterate through all possible values of x₂ and x₃ that add up to the remaining value of 11 minus x₁. We know that the maximum value of x₁ is 11, so the outermost loop will run from 0 to 11. For each value of x₁, the inner loops will run from 0 to (11 - x₁), incrementing by 1 each time. This will ensure that all possible combinations of x₂ and x₃ that add up to 11 minus x₁ are tested. Once we have all the combinations, we can count them to get the total number of solutions. In this case, there are 66 solutions to the inequality.
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in excel, to construct a residual plot, input of a y range and an x range is needed. aimee is examining the relationship between age and square foot range (sqft) of living space. in the scenario provided, what would be the y and what would be the x range data?
For the given case, in the residual plot, the x range is for the age and the y range must be the square foot
A residual plot appears the difference between the observed response and the fitted response values. The ideal remaining plot, called the null residual plot, appears as a random scatter of focuses shaping a roughly constant width band around the identity line. It is vital to check the fit of the model and presumptions – steady change, normality, and freedom of the errors, utilizing the residual plot, alongside normal, sequence, and lag plot.We are given the information that there is a relationship between age and square foot range (sqft) of living space, so for the residual plot the y range must be for the square foot and the x range must the age
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colin surveyed 12 teachers at his school to determine how much each person budgets for lunch. he recorded his results in the table. a 6-column table with 2 rows. the first row contains entries 10, 5, 8, 10, 12, 6. the second row contains entries 8, 10, 15, 6, 12, 18. what does the relationship between the mean and median reveal about the shape of the data?
The mean x=10 and the median=10 are equal, then the data is a normal distribution and the shape will be symmetrical "bell curve".
In the given question, Colin surveyed 12 teachers at his school to determine how much each person budgets for lunch.
He recorded his results in the table.
A 6-column table with 2 rows
The first row contains entries 10, 5, 8, 10, 12, 6.
The second row contains entries 8, 10, 15, 6, 12, 18.
We have to find the relationship between the mean and median reveal about the shape of the data.
The data elements are given as:
10, 5, 8, 10, 12, 6, 8, 10, 15, 6, 12, 18
Start by sorting the above data elements in ascending order.
5, 6, 6, 8, 8, 10, 10, 10, 12, 12, 15, 18
The mean of the dataset is then calculated as:
X = 5+6+6+8+8+10+10+10+12+12+15+18/12
X = 120/12
X = 10
The median of element is then calculated as:
Median = n/2th
Median = 12/2th
Median = 6th
This means that, the median is of the 6th elements.
So, we have:
Median = 10
When the mean and the median are equal, then
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Question 11
Erica recorded the number of sales she made during each of the first 4 months when she began her new career. Erica plotted the data on a coordinate grid, and she noted that the data correlate to an exponential equation, as shown What is the domain of the function shown in the graph?
The domain of the graph is {1, 2, 3, 4}
What is Domain?"All the values" that are input into a function are referred to as the domain of a function. The set of all potential inputs for a function is its domain.
Given:
As, we know domain of a function is the set of all possible inputs for the function.
from the Graph, we can find the coordinates ( 1, 10), (2, 20), (3, 40) and ( 4, 80)
So, the domain of the graph is {1, 2, 3, 4}
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A ladder is leaning against a vertical wall makes an angle of 20° with the ground. The foot of the ladder is 3 m from the wall. Find the length of ladder (Round to the nearest whole number).
If you choose the 20 degree acute angle, the distance between the foot of the ladder and the wall represents the [Select] leg. Since we are given the [Select] leg and solving for the hypotenuse, we need to use [Select] to solve for the length of the ladder. The length of the ladder when solved is [Select] meters.
Options for blank #1: adjacent, opposite, hypotenuse
Options for blank #2: adjacent, opposite, hypotenuse
Options for blank #3: sine, cosine, tangent
Options for blank #4: 3, 1, 9, 4
The length of the ladder is approximately 3 m. The correct option is the first option 3
Calculating the length of a ladderFrom the question, we are to calculate the length of the ladder.
From the given information,
The ladder makes an angle 20° with the ground
and
The foot of the ladder is 3 m from the wall
This scenario gives a right triangle where the hypotenuse is the ladder and the adjacent is the distance of the foot of the ladder from the wall.
Let the length of the ladder be x
Thus,
Using SOH CAH TOA
We can write that
cos 20° = 3 / x
x = 3/(cos 20°)
x = 3.19
x ≈ 3 m
Hence, the length is approximately 3 m
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Help me please I need help
The percentage of the down payment in the purchase is 15% and the down payment of $67500 will be 45% of the purchase price.
Calculating Percentage:The ratio of the actual worth to the total value multiplied by 100 is the formula for calculating the percentage.
The percentage formula is given by
Percentage, % = (Value × 100) / Total Value.Here we have
A family buys a studio apartment for $ 150,000
The down payment paid by the family is $ 22500
a. Percentage of down payment in the purchase
From the above data,
The percentage of the down payment = [tex][ \frac{22500 }{150,000} \times 100 ][/tex]
= [tex][ \frac{22500 }{1500} ][/tex] = 15
Therefore,
The percentage of the down payment is 15%
b. The percentage of the purchase price would be a $67500 down payment?
Let x be the percentage of the down payment in the purchase amount
=> [tex]\frac{x}{100} \times $ 150,000 = 67500[/tex]
=> [tex]x \times 1500 = 67500[/tex]
=> x = 67500/1500
=> x = 45%
Therefore,
The percentage of the down payment in the purchase is 15% and the down payment of $67500 will be 45% of the purchase price.
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Data that are left-skewed would include:
O values that are unusually low.
O values that are unusually high.
O values that are symmetrically distributed.
O values that are all less than the median.
Data that are left-skewed would include values that are unusually high.
What is meant by symmetry of data?A data is symmetric when it has a mirror image on both sides of the mean. For a symmetric data the mean = mode = median.
What is left-skewed data?If the data is asymmetric, it can either be left-skewed or right-skewed. A left-skewed data has mots of its values greater than the mean. In this case the mean < median < mode. Left-skewed data mostly contains very less amount of values with smaller magnitudes.
Hence, data that are left-skewed would include values that are unusually high.
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how do i solve 4( К-8 ) = -3+4к
Answer:
No solutions, there is nothing that will make the equation true
Step-by-step explanation:
You need to isolate the variable
4(K-8) = -3+4K
First get ride of the parenthesis by distributing
(4 x k) - (4 x 8) =-3+4k
4k-32 = -3+4k
next rearrange the equation to have the variable on only one side
4k-32 = -3+4k (subtract 4k from one side)
-32 = -3
The variables canceled out, leaving an equation. If the equation is true there are infinite equations while if the equation is false there are no solutions
-32 is NOT equal to -3, meaning there are no solutions
if you select a random sample of full-time college students, there is an hance that the sample mean is less than how many hours per week?
The sample mean is less than 26.92 hours per week .
What is standard deviation?
Your dataset's average level of variability is represented by the standard deviation. In general, it reveals how far away from the mean each value is.
A low standard deviation denotes that values are grouped closely to the mean, while a high standard deviation denotes that values are typically spread widely from the mean.
Let X~ Normal (26, 4/√16) distribution ie Normal (26, 1) distribution
We want,
P(X<a) = 0.82
Using standard normal approximation,
P(z< (a-26)/1) = 0.82
Using standard normal distribution tables, we get
(a-26)/1 = 0.915
(Since P(z<0.915) = 0.82)
So, a= 26+0.915= 26.915
So the required value is 26.92 (rounded to two decimals)
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Consider the graph of the line y = 1//2x – 4 and the point (−4, 2). The slope of a line parallel to the given line is . A point on the line parallel to the given line, passing through (−4, 2), is . The slope of a line perpendicular to the given line is . A point on the line perpendicular to the given line, passing through (−4, 2), is ✔ (–2, –2) .
For the given line y = 1/2x - 4,
The slope of the line parallel to the given line is m = 0.The point on the line parallel to the given line, passing through (-4, 2) is (0, 4) and its equation is x - 2y + 8 = 0.The slope of the line perpendicular to the given line is m = -2.The point on the line perpendicular to the given line, passes through (-4, 2) is (-2, -2) and its equation is y = -2x - 6.How to calculate the slopes and equations of parallel and perpendicular lines of a line?Consider a line with the equation ax+by+c=0
The slope of the line is m = -b/a.The slope of the line parallel to the given line is m = 0.The slope of the line perpendicular to the given line is m = a/b or the product of their slopes equal to -1.The equation of the line parallel to the given line and passing through the point (x1, y1) is a(x - x1) + b(y - y1) = 0.The equation of the line perpendicular to the given line and passing through the point (x1, y1) is b(x - x1) - a(y - y1) = 0.Calculation:It is given that, the equation of a line is given as y = 1/2x - 4.
The slope of the given line is m = 1/2; (since it is in the form of y=mx+c)
Then, the slope of the line parallel to the given line is m = 0.
And the equation of the parallel line passing through the point (-4, 2) is
a(x - x1) + b(y - y1) = 0; where a = 1; b = -2 (y = 1/2x-4 ⇒ 2y = x - 8)
⇒ 1(x + 4) - 2(y - 2) = 0
⇒ x + 4 -2y + 4 = 0
⇒ x - 2y +8 = 0
So, the point on the parallel line is (0, 4).
The slope of the line perpendicular to the given line is m = -1/1/2 = -2.
And the equation of the perpendicular line passing through the point (-4, 2) is
b(x - x1) - a(y - y1) = 0; where a = 1; b = -2;
⇒ -2(x + 4) - 1(y - 2) = 0
⇒ -2x - 8 - y +2 = 0
⇒ y = -2x - 6
So, the point on the perpendicular line is (-2, -2).
All of these are shown in the graph below:
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In 1846 the depth of the river was 5 feet deep. In 1847 it dropped to 3.5 feet. This year, 1848, it rose to 5.3 feet.
What is the percentage change
Answer:
From 1846 to 1847 there was a 30% decrease and from 1847 to 1848 it increased by 51.4%
Step-by-step explanation:
How it decreased by 30%
V2-V1/V1x100
V2=3.5 V1=5
=(3.5−5)|5|×100
=−1.55×100
=−0.3×100
=−30%change
=30%decrease
How it increased by 51.4%
V2-V1/V1x100
V2=5.3 V1=3.5
=(5.3−3.5)|3.5|×100
=1.83.5×100
=0.514286×100
=51.4286%change
=51.4%increase
How do you evaluate [(1/(x)−(1/x^(2)+x))] as x approaches 0?
As x approaches 0, the denominator of the expression [(1/(x)−(1/x^(2)+x))] approaches infinity and the numerator approaches 0. Therefore, the expression approaches 0 as x approaches 0.
1. Evaluate the denominator of the expression [(1/(x)−(1/x^(2)+x))] as x approaches 0.
2. The denominator of the expression [(1/(x)−(1/x^(2)+x))] approaches infinity as x approaches 0.
3. Evaluate the numerator of the expression [(1/(x)−(1/x^(2)+x))] as x approaches 0.
4. The numerator of the expression [(1/(x)−(1/x^(2)+x))] approaches 0 as x approaches 0.
5. Therefore, the expression [(1/(x)−(1/x^(2)+x))] approaches 0 as x approaches 0.
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the volume of a rectangular box with a square base remains constant at 1,100 cm3 as the area of the base increases at a rate of 2 cm2/sec. find the rate at which the height of the box is decreasing when each side of the base is 12 cm long. (do not round your answer.)
If the area of the area of the base increases at rate of 2 cm²/sec then the rate at which the height of the box is decreasing is 0.11 cm/sec .
In the question ,
it is given that ,
the volume of the rectangular box having square base is 1100 cm³ ;
the rate of decrease of the area is (dA/dt) is = 2 cm²/sec ;
we know that ; Volume = Area × Height ;
the area of the square base of side 12 cm is = 12 × 12 = 144 ;
So , height = Volume/Area ;
= 1100/144 ;
Now , V = A × h ;
differentiating the above equation w.r.t. "t" ,
we get ;
dV/dt = A×(dh/dt) + h×(dA/dt)
0 = 144×(dh/dt) + (1100/144)×(2)
dh/dt = -2200/(144 × 144)
= -0.106095
≈ -0.11
Therefore , the rate of change of height is = 0.11 cm/sec .
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How do you write 3/4 as a percent?
Answer:
0.75 or 75%
Step-by-step explanation:
I know that 2/4 is 0.5 or 50% and 1/4 is 0.25 or 25% so it goes by 25. One more than 2/4 is 3/4, which is what you are trying to find, so it would be 0.75, or 75%.
data shows that the united states has less intergenerational mobility than germany. that means that:
The data shows that United States has less intergenerational mobility than Germany this data means that the success of the children of United States depends more on the parents income status in comparison to Germany.
Intergeneration mobility is the extent to which the outcomes of the doings of an individual are different in comparison to his or her parents. This basically means the extent to which an individual is gaining something by his own potential.
According to the data United States has less intergeneration mobility in comparison to the intergenerational mobility of Germany.
From this data that outcomes of a children born in United States depends highly on the parents economic and financial status than German kids.
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select all of the following equation(s) that are quadratic in form. x4 – 6x2 – 27 = 0 3x4 = 2x 2(x 5)4 2x2 5 = 0 6(2x 4)2 = (2x 4) 2 6x4 = -x2 5 8x4 2x2 – 4x = 0
An equation is quadratic if the greatest power of the coefficients is 2. Hence, for the given equations, only 6(2x + 4)2 = (2x + 4) + 2 is quadratic in nature.
What in mathematics is a quadratic equation?
A quadratic equation is a second-order polynomial equation with one variable that goes like this: x ax2 + bx + c = 0. with a ≠ 0 .
The fundamental theorem of algebra ensures that it has at least one solution because it is a second-order polynomial problem. The answer could be simple or complicated.
Why it is called quadratic equation?
In mathematics, a quadratic issue is any problem that involves multiplying a variable by itself, often known as "squaring."
This vocabulary is based on the fact that a square's area is equal to the product of its side length and itself. The Latin word quadratum, which means square, is where the word "quadratic" derives.
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The complete question is -
Select all of the following equation(s) that are quadratic in form.
x4 – 6x2 – 27 = 0
3x4 = 2x
2(x + 5)4 + 2x2 + 5 = 0
6(2x + 4)2 = (2x + 4) + 2
6x4 = -x2 + 5
8x4 + 2x2 – 4x = 0
8010005 in international number system
Answer:
8,010,005
Step-by-step explanation:
eight million ten thousand and five
hope this helps
radius of a right circular cylinder is increasing at a rate of 6 inches per minute, and the height is decreasing at a rate of 4 inches per minute. what is the rate of change of the surface area when the radius is 6 inches and the height is 18 inches?
The rate of change of the surface area when the radius is 6 inches and the height is 18 inches is 979.68 in² per minute.
Given:
radius of a right circular cylinder is increasing at a rate of 6 inches per minute, and the height is decreasing at a rate of 4 inches per minute.
we are asked to find the rate of change of the surface area when the radius is 6 inches and the height is 18 inches.
we have dr/dt = 6 in/min and dh/dt = -4 in/min
Volume, V = πr²h
Differentiate the above expression to get:
dV/dt = 2πrh dr/dt + πr² dh/dt
SUbstitute the value we get:
dV/dt = 2π(6)(18)(6) + π(6)²(-4)
= 4069.44 - 452.16
= 3617.28 in³ per minute.
The volume is increasing at a rate of 3617.28 in³ per minute.
Area = 2πr(h+r)
Differentiate the above expression:
dA/dt = 2π {(h+r) dr/dt + r (dh/dt + dr/dt)}
dA/dt = 2π {(18+6)(6) + 6(-4+6)}
= 2π(144 + 12)
= 312π
= 979.68 in² per minute.
The area is increasing at a rate of 979.68 in² per minute.
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The rate of change of the surface area when the radius is 6 inches and the height is 18 inches is 979.68 in² per minute.
Given:
The radius of a right circular cylinder is increasing at a rate of 6 inches per minute, and the height is decreasing at a rate of 4 inches per minute.
We are asked to find the rate of change of the surface area when the radius is 6 inches and the height is 18 inches.
We have dr/dt = 6 in/min and dh/dt = -4 in/min
In geometry, it is defined as a three-dimensional shape having two circular shapes at a distance called the height of the cylinder.
We know the volume of the cylinder is given by:
Volume, V = πr²h
Differentiate the above expression to get:
dV/dt = 2πrh dr/dt + πr² dh/dt
Substitute the value we get:
dV/dt = 2π*6*18*6 + π*6² * -4
= 4069.44 - 452.16
= 3617.28 in³ per minute.
The volume is increasing at a rate of 3617.28 in³ per minute.
Area = 2πr(h+r)
Differentiate the above expression:
dA/dt = 2π {(h+r) dr/dt + r (dh/dt + dr/dt)}
dA/dt = 2π {(18+6)(6) + 6(-4+6)}
= 2π(144 + 12)
= 312π
= 979.68 in² per minute.
The area is increasing at a rate of 979.68 in² per minute.
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( 3x^2 - 5x + 7) (2x^2 + 6x - 4) Try solving this one on your own first! What is the coefficient of the x3 term in the final polynomial? 8 -10 6 18
On solving the provided question, by multiplying the give polynomials, we got to know that - [tex]( 3x^2 - 5x + 7) X (2x^2 + 6x - 4)[/tex] = [tex]6x^4 + 8x^3 - 28x^2 + 62x - 28[/tex], the coefficient of x3 is 8
what are polynomials?An mathematical expression known as a polynomial requires that all of the variables' exponents be integers. Each polynomial variable's exponent must be an integer that is not negative. Although a polynomial is made up of a constant and a variable, it cannot be divided by a variable.
Given polynomials are -
[tex]( 3x^2 - 5x + 7) and (2x^2 + 6x - 4)[/tex]
By multiply the given polynomials, we got -
[tex]( 3x^2 - 5x + 7) X (2x^2 + 6x - 4)[/tex]
= [tex]6x^4 + 8x^3 - 28x^2 + 62x - 28[/tex]
So, the coefficient of x3 is 8
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evaluate the following expression for x=8
a. 82
b.178
c.854
d. 188
Answer:
178
Step-by-step explanation:
SORRY IF ITS WRONG IM BAD AT MATH :(
While training for a marathon, you consume at least 2200 calories a day. For one session of exercise, you consume 400 calories. How many calories do you consume for the rest of the day?
The amount of calories you consume for the rest of the day is 1800
How to determine the amount of calories you consume for the rest of the dayFrom the question, we have the following parameters that can be used in our computation:
Total amount of calories for a day = 2200 calories
Total amount of calories for one session = 400 calories
The above parameters can be represented as
Total amount of calories for one session + Total amount of calories for the end of the day = Total amount of calories for a day
Substitute the known values in the above equation, so, we have the following representation
400 + Total amount of calories for the end of the day = 2200
Evaluate the like terms
Total amount of calories for the end of the day = 2200 - 400
So, we have
Total amount of calories for the end of the day = 1800
Hence, the amount of calories is 1800
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1. After a pay raise, Kate's monthly salary increased from $3500 to $3780. Find the percentage increase in her pay raise.
Answer:
8%
Step-by-step explanation:
1. Subtract the larger number from the smaller:
3780 - 3500 = 280
2. Divide the result from step 1 by the FIRST salary amount:
280 ÷ 3500 = 0.08 NOTE: Order Matters!!!
3. Change the decimal to a percent
Given the following exponential function, identify whether the change represents growth or decay, and determine the percentage rate of increase or decrease. y=99(1.04)^4
The percentage rate of increase or decrease of the function y=99(1.04)^4 is 4%
What is an exponential function?This is a type of function that are of 3 main parts
the starting or initial value
base function
the exponents
Considering the given problem, y=99(1.04)^4
the starting = 99
the base = 1.04
the exponents = x
The base is what determines increase or decrease with a base of 1.04 shows a percentage increase is 4%
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Jerry bulit a table with a square top. the perimeter of the tabletop is 18 feet. He knows that the side lentgh of the table is 3, 3 1/4 4 or 4 1/2 use the equation 18 = 4s where s is the side length of the table to determine the side length explain how you found your answer
Answer:
4 1/2 ft
Step-by-step explanation:
18 = 4s
Divide both sides by 4.
18/4 = 4s/4
9/2 = s
4 1/2 = s
The side of the table is 4 1/2 ft.
Light travels at about 186,000 miles per second.
Which shows an equivalent speed?
(A)
(В)
O
D
1.86x103 miles per second
1.86x104 miles per second
1.86x105 miles per second
1.86x106 miles per second
Answer:
The answer is 1.86x10 6
Step-by-step explanation:
10x6 is 100k and the speed light travels is 186k so to get the answer you need to find the one that gets to 100k and then multiply it by 1.86
the function c gives the temperature, in degrees celsius, that corresponds to a temperature of x degrees fahrenheit. if a temperature increased by 19.8 degrees fahrenheit, how much did the temperature increase in degrees celsius?
When the temperature increased by 19.8 degrees fahrenheit , the temperature in degrees Celsius got increased by 11 degree
The equation C that gives the temperature in degree Celsius is
C = [tex]\frac{5}{9}(x - 32)[/tex]
Where x denotes the degree Fahrenheit
According to the question,
Temperature has increased by 19.8 degree Fahrenheit
So, let the new temperature be " x' "
x' = x + 19.8
Therefore, For calculating temperature in degree Celsius
=> C' = 5/9 (x' - 32)
Substituting the value of x'
=> C' = 5/9 (x + 19.8 - 32)
=> C' = 5/9 ( x - 32) + 5×19.8/9
=> C' = C + 99/9
=> C' = C + 11
Hence , The temperature increased by 11 degree Celsius
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At a particular restaurant, each mini hotdog has 60 calories and each mozzarella stick has 70 calories. A combination meal with mini hotdogs and mozzarella sticks is shown to have 1080 total calories and 5 more mini hotdogs than mozzarella sticks. Write a system of equations that could be used to determine the number of mini hotdogs in the combination meal and the number of mozzarella sticks in the combination meal. Define the variables that you use to write the system.
Answer:
the combination meal contains 18 mini hotdogs and 13 mozzarella sticks.
Step-by-step explanation:
To determine the number of mini hotdogs and mozzarella sticks in the combination meal, we can use the following system of equations:
Let x be the number of mini hotdogs in the combination meal.
Let y be the number of mozzarella sticks in the combination meal.
The first equation represents the total number of calories in the combination meal: 60x + 70y = 1080
The second equation represents the difference between the number of mini hotdogs and mozzarella sticks: x - y = 5
Thus, the system of equations is:
60x + 70y = 1080
x - y = 5
To solve this system, we can use a variety of methods, such as substitution or elimination. For example, we can eliminate x by adding the equations together, which yields:
60x + 70y = 1080
x - y = 5
60x - x + 70y - y = 1080 - 5
59x + 69y = 1075
Dividing both sides by 59, we find that x = 18.
Substituting this value into the second equation, we find that y = 18 - 5 = 13.
Therefore, the combination meal contains 18 mini hotdogs and 13 mozzarella sticks.
Answer:
There are 9 pizza rolls and 12 mini hotdogs in the combination meal.
Step-by-step explanation:
We are given the following in the question.
Let x be the number of pizza rolls sold and y be the number of mini hotdogs in the combination meal.
Total number of items in combination meal = 21
Thus, we can write the equation
Calorie in a pizza roll = 60
Calorie in a mini hotdog = 70
Total calories = 1380
Thus, we can write the equation
Solving the two equations, we get,
Thus, there are 9 pizza rolls and 12 mini hotdogs in the meal.
What is the antiderivative of 1/sinx?
The anti-derivative in other words integration of the inverse of sine function i.e 1/sinx is
x sin⁻¹x + √(1 - x²) + C.
Integration of a function is the inverse process of differentiation. The integral is just the inverse derivative. The inverse integral of sin is written as
x sin⁻¹x + √(1 - x²) + C.
where ∫ --> the sign of integration, dx --> that the integration of sine inverse with respect to x, C --> the constant of integration.
Integral of Sin Reverse proof by integration by parts :
The formula for integration by parts is ∫f(x)g(x)dx = f(x) ∫g(x)dx - ∫[d(f(x) ) /dx×∫g(x) dx] sin⁻¹x can be written as sin⁻¹x = sin⁻¹x.1.f(x) = sin⁻¹x, g(x) = 1
d(sin⁻¹x)/dx = 1/√(1 - x²)
Using these equations and facts,
∫sin⁻¹x dx = ∫sin⁻¹x.1 dx
= sin⁻¹x ∫1dx - ∫[d(sin⁻¹x)/dx × ∫1 dx] dx
= x sin⁻¹x - ∫[1/√(1 - x²) × x] dx
= x sin⁻¹x - ∫x/√(1 - x²) dx
= x sin⁻¹x + (1/2) ∫-2x/√(1 - x²) dx [multiplication and division by 2]
= x sin⁻¹x + (1/2) ∫(-2x)(1 - x²)-1/2 dx
= x sin⁻¹x + (1/2) [(1 - x²)-1/2 + 1/ (-1/2 + 1)] + C {using the formula ∫[f(x)]nf'(x)dx=[f(x)]n +1/(n + 1) +C}
= x sin⁻¹x + (1/2) [(1 - x²)1/2/ (1/2)] + C
= x sin⁻¹x + (1 - x²)1/2 + C
= x sin⁻¹x + √(1 - x²) + C
Therefore the inverse derivative of 1/sinc is x sin⁻¹x + √(1 - x²) + C.
To learn more about Integration , refer:
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Which parent functions have a domain of all real values of x? Select all that
apply.
A. Reciprocal
B. Cubic
C. Quadratic
D. Cube root
E. Linear
F. Absolute value
Answer:
A D F
Step-by-step explanation:
The parent functions with a domain of all real values of x are the linear function, the absolute value function, the square root function, and the cube root function.
The linear function is defined by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. The domain of this function is all real values of x.
The absolute value function is defined by the equation y = |x|, where |x| is the absolute value of x. The domain of this function is all real values of x.
The square root function is defined by the equation y = √x, where √x is the square root of x. The domain of this function is all nonnegative real values of x.
The cube root function is defined by the equation y = ∛x, where ∛x is the cube root of x. The domain of this function is all nonnegative real values of x.
Therefore, the correct answers are A, D, and F. The reciprocal function, the cubic function, and the quadratic function do not have a domain of all real values of x.
