For f(x) = x5 + x3 + x, the value of f1(3) is 1. Because of the inverse function feature, the value of f(f1(2)) is 2.
What is function?A function is an equation with just one solution for y for every x. A function produces exactly one output for each input of a certain type. Instead of y, it is usual to call a function f(x) or g(x). f(2) indicates that we should discover our function's value when x equals 2. A function is an equation that depicts the connection between an input x and an output y, with precisely one output for each input. Another name for input is domain, while another one for output is range.
Here,
f(x)=x^5 + x^3 + x
f(1)=3
f^-1(3)=1
f[f −1 (x)]=f −1 [f(x)]=x
f(f^(-1)2)=2
The value of f−1(3) is 1 for f(x) = x^5 + x^3 + x. The value of f(f−1(2)) is 2 by property of inverse function.
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a particle is moving along the x-axis with position function x ( t ) = t 3 − 5 t 2 3 t 2 for t ≥ 0 . over which of the following time intervals is the particle speeding up?
The particle is speeding up over the time interval (10/6, ∞).
The particle is speeding up when its acceleration is positive, which occurs when its second derivative is greater than 0. The second derivative of x (t) is 6t - 10, so the particle is speeding up when t > 10/6. Therefore, the particle is speeding up over the time interval (10/6, ∞).
To calculate this, we take the second derivative of x (t):
x''(t) = 6t - 10
We then set this equal to 0 and solve for t
6t - 10 = 0
t = 10/6
Therefore, the particle is speeding up over the time interval (10/6, ∞).
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what are the x-intercepts for the graph below?
Answer:
C. (-5,0) ( -2, 0)
Step-by-step explanation:
The X-intercept is when the y = 0
We see the x-intercept on the graph is located at (-5,0) and (-2,0), so the answer is C.
item 9 una makes $64 a week, plus 15 percent of the cost of meals that she serves. use the equation y = 0.15x 64 to predict what she makes in a week when she serves $355 worth of meals.
The amount of money una earns in a week she serves $355 worth of meals including her base earnings is $177.25
How to solve equation?Amount una makes in a week= $64
Additional amount una earn = 15% of x
Cost of meal she serves = x
Total amount she makes= y
y = 0.15x + 64
If she serves $355 worth of meals.
y = 0.15x + 64
y = 0.15(355) + 64
= 53.25 + 64
y = $177.25
In conclusion, una makes a total of $177.25 in a week.
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Ruby’s model for a Fourth of July party hat is shown. The model is composed of a rectangle and 2 right triangles. What is the area of Ruby’s model for a Fourth of July party hat?
Total area of Ruby’s model for a Fourth of July party hat is 182.75 square centimeter.
What is Area of Rectangle?The area of Rectangle is length times of width.
Let us find the area of the red colour right triangle.
Area of red triangle=1/2×13.5×9
=60.75 square cm.
Area of blue triangle=1/2×8×9
=36 square cm.
Area of rectangle=Length×breadth
=(13.5+8)4
=21.5×4
=86 square cm.
The total area of Ruby’s model for a Fourth of July party hat is
60.75+36+86
182.75 square centimeter.
Hence, total area of Ruby’s model for a Fourth of July party hat is 182.75 square centimeter.
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Please help meeeee !!!!!long division step by step
Answer:
25 r12
Step-by-step explanation:
first, u ask yourself how many times can 30 go in 7, well it can't so you do 30 can go in 76, 2 times because. 30x2 is 60 and it doesn't go over 76.
30x2 is 60 so on the first box u put 2. Then u multiply again 30x2 which is 60 so u put the 60 on the box below the 76. U subtract
Next, you would subtract it which would turn out to be 16. Since 16 can't go in 30 equally, u would drop down the 2 in the 762 now you can do 30 into 162 which is 5 times bc 30x5 is 150 and doesn't go over so u put the 5 in the box next to the 2.
30x5 is 150, u put that in the box below 162. Then u subtract 162-150 which is 12. So you have a remainder of 12.
density dependent population growth in scetion 5.9, we introduced a model for population growth. our model predicts that a population of organisms will grow according to the differential equation:
In nature, there are various constraints on population number and expansion. Some are depending on the density, whereas others are not.
A population's per capita growth rate changes—typically declines—with rising population density due to density-dependent limiting constraints. One illustration is how individuals in a group compete for scarce food.
Independent of population density, several factors impact the per capita growth rate. Natural catastrophes like forest fires are examples.
Diverse limiting variables can combine in intricate ways to generate different patterns of population expansion. Some populations exhibit cyclical oscillations, where the population size varies cycle by cycle in a predictable manner.
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a town's population has been growing linearly. in 2003 the population was 21,000. the population has been growing by 1700 people each year. write an equation for the population, p, years after 2003.
An equation for the population, p, years after 2003 is 45000 + 1700t .
What is an equation for the population?The population of a town has been increasing linearly. The population was 21,000 in 2003. 1700 persons have been added to the population annually.
Locate the necessary equation:
An equation for the population, p, years after 2003.
Given: Population in 2003 was 45,000.
Additionally, annual population growth equals 1,700.
After t years, the population will expand by 1,700t because it has been rising linearly.
Population t years later = 45000 + 1700t, then.
P(t) = 45000 + 1700t is the needed equation as a result.
An equation for the population, p, years after 2003 is 45000 + 1700t .
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6. Work The dollar amount d that Julia earns varies directly as the number of hours !
that she works, and d=$116. 25 when 1 = 15 h. Find t when d = $178. 25.
Work the dollar amount d that Julia earns varies directly as the number of hours is 23 hours is the solution.
What is the time t?According to the information provided, Julia's earnings correspond closely to the amount of hours she puts in at work.
Also, at t = 15 h, d equals $116.25.
The direct variation's formula is d = kt.
where k is the variational constant.
Find k and replace the known values:
116.25 = k * 15
k = 116.25/15 k = 7.75
Find t now that d is equal to 178.25. In the direct variation equation, swap out d and k, then figure out t:
23 hours are equal to 178.25 = 7.75t t.
t = 23 hours is the solution.
Work the dollar amount d that Julia earns varies directly as the number of hours is 23 hours is the solution.
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Julia had a bag filled with gumballs. There were 1 watermelon, 2 lemon-lime, and 3 grape gumballs. What is the correct sample space for the gumballs in her bag?
Sample space = watermelon, lemon-lime, lemon-lime, grape, grape, grape
Sample space = lemon-lime, watermelon, grape
Sample space = 1, 2, 3, 4, 5, 6
Sample space = 1, 2, 3
Option A. is the correct answer.
Sample space = watermelon, lemon-lime, lemon-lime, grape, grape, grape.
What is a sample space?A sample space is a set of potential results from a random experiment. The letter "S" is used to denote the sample space. Events are the subset of possible experiment results. Depending on the experiment, a sample area could contain a variety of results.
here, we have,
Given that, Julia had a bag filled with gumballs. There were 1 watermelon, 2 lemon-lime, and 3 grape gumballs.
now,
Sample space contains,
2 lemon-lime: Elements in sample space are lemon-lime, lemon-lime,
1 watermelon: Elements in sample space are watermelon,
3 grape: Elements in sample space are grape, grape, grape,
In total there are 13 elements in sample space.
Therefore, option A . is the correct answer.
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Answer:
Step-by-step explanation:
Option A. is the correct answer.
Sample space = watermelon, lemon-lime, lemon-lime, grape, grape, grape.
Determine the intervals where the function f(x) = e^(x) Cos(x) (0 less than or equal to X less than equal to 2Pi) is increasing and where it is decreasing ?
If the function is f(x) = eˣ Cos(x) where (0≤ x ≤2π) , then the function will be increasing in interval (0, π/4) U (7π/4, 2π), and decreasing in interval (π/4, 7π/4) .
In order to find the intervals where function f(x) = eˣ Cos(x) is increasing and where it is decreasing, we find the points where the derivative of the function, f'(x), is positive and negative,
The derivative of function : f(x) = eˣ cos(x) is:
⇒ f'(x) = eˣ cos(x) + eˣ (-sin(x))
⇒ eˣ (cos(x) - sin(x))
So , To find the intervals where f'(x) is positive or negative, we can use the sign of the derivative to determine if the function is increasing or decreasing.
we know that ;
⇒ When f'(x) > 0, the function is increasing.
⇒ When f'(x) < 0, the function is decreasing.
For function " eˣ (cos(x) - sin(x)) " ,
we can see that f'(x) is positive when cos(x) > sin(x) and negative when cos(x) < sin(x).
By using a unit circle, we can see that the inequality cos(x) > sin(x) is satisfied for x in the interval (0, π/4) U (7π/4, 2π), and
the inequality cos(x) < sin(x) is satisfied for x in the interval (π/4, 7π/4) .
The given question is incomplete , the complete question is
Determine the intervals where the function f(x) = eˣ Cos(x) where (0≤ x ≤2π) is increasing and where it is decreasing ?
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a shoe sells 200 pairs at $60. for each $1 increase, sales decrease by 2. determine maximum (vertex) PLS HELP I NEED THE ANSWERRRR WILL MARK BRAINLIEST
6.1/3.9 = 2.8x/x+8.1
solve for x
Answer:
0.89
Step-by-step explanation:
6.1/3.9=28/10x/x+81/10
61/39=28x(x)/10+8.1
61/39-81/10=28x(x)/10
610/390_3159/390=28x(x)/10
2469/390=28x(x)/10
2469/390=14/5x(x)
2.51=14/5x
12.55=14x
x=0.89
The graph of the line m is shown.use the similar slope triangles to compare the slope of segment RT and TV
Note that from the graph given above and their slopes, it is correct to state that mRT = mTV.
What is the justification for the above assertion?First, note that the slope or gradient of a line in mathematics is a quantity that specifies both the direction and the steepness of the line.
Here, Slope RT = Δy/Δx = SR/ST = 2/4 = 1/2
Slope TV = Δy/Δx = UT/UV = 1/2
Thus, since the Slope of RT = 1/2 and the Slope of TV = 1/2 it i correct to state that mRT = mTV.
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Let f be the function defined by f(x) = Inx/x. What is the absolute maximum value of f?(A) 1(B) 1/e(C) 0(D) -e(E) f does not have an absolute maximum value
The absolute maximum value of f is 1.
The function f(x) is defined as the natural logarithm of x (lnx) divided by x. This function is undefined when x is equal to 0, but for positive x values, the maximum value of f(x) is 1. To prove this, we need to take the derivative of f(x) and set it equal to 0. We can calculate the derivative of f(x) by using the quotient rule.
The quotient rule states that for two functions u(x) and v(x), the derivative of their quotient can be calculated by taking the derivative of u(x) times v(x) minus the derivative of v(x) times u(x), all divided by v(x)2. After taking the derivative of f(x) = lnx/x, we get f'(x) = (1/x) – (lnx/x2).
Now we set f'(x) = 0 and solve for x. This gives us x = 1, so when x = 1, we get the maximum value of f(x) = ln1/1 = 1. Therefore, the absolute maximum value of f is 1.
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in Mrs. Washington's class, there are 2 students in band for every
3 students in chorus. In Mr. Utley's class there are 5 students in band for every 7 students in chorus. Complete the ratio tables. Which class has a greater ratio of students in band to students in chorus?
Answer:
Here are the completed ratio tables for both classes:
Mrs. Washington's class:
Band: 2
Chorus: 3
Total: 2 + 3 = 5
Band-to-Chorus ratio: 2 : 3
Mr. Utley's class:
Band: 5
Chorus: 7
Total: 5 + 7 = 12
Band-to-Chorus ratio: 5 : 7
Based on the ratios, it appears that Mr. Utley's class has a greater ratio of students in band to students in chorus. The ratio in Mr. Utley's class is 5 : 7, while the ratio in Mrs. Washington's class is 2 : 3.
Step-by-step explanation:
Step-by-step explanation:
radius are a special form of fractions. but they are fractions, and a such all rules of calculation and comparison apply.
mrs. Washington's class has a ratio of 2/3 (or 2:3 or 2÷3).
2/3 = 0.666666666...
mrs. utley's class has a ratio of 5/7.
5/7 = 0.714285714...
so, the class of mrs. Utley had a greater ratio.
to compare the ratios (like any fraction) within calculating their decimal value, we need to bring them to the same denominator :
.../3 and .../7
we need to bring both to .../21 :
2/3 × 7/7 = 14/21
5/7 × 3/3 = 15/21
so, we can clearly see that the ratio of mrs. Utley's class is greater, as 15 is greater than 14.
35 POINTS AND BRAINLIEST
A zoologist recorded the speed of two cheetahs. Cheetah A ran 18 miles in 16 minutes. Cheetah B ran 54 miles in 50 minutes. Which statement is correct?
Cheetah A has a higher ratio of miles per minute than Cheetah B because 18 over 16 is greater than 54 over 50.
Cheetah A has a higher ratio of miles per minute than Cheetah B because 18 over 16 is less than 54 over 50.
Cheetah B has a higher ratio of miles per minute than Cheetah A because 18 over 16 is greater than 54 over 50.
Both cheetahs have the same ratio of miles per minute.
find the horizontal and vertical asymptotes of the curve. y = 3ex ex − 5
The graph of y = 3ex ex − 5 has a horizontal asymptote of y = 3 and a vertical asymptote of x = -3/5.
The equation y = 3ex ex − 5 can be rewritten as y = 3x2ex − 5. In order to find the horizontal and vertical asymptotes of this function, we must first find its degree of the numerator and denominator. The numerator is a second degree polynomial while the denominator is a first degree polynomial.
The horizontal asymptote will be the line y = b/a, where b and a are the coefficients of the highest degree terms in the numerator and denominator, respectively. In this case, b = 3 and a = 1, so the horizontal asymptote is y = 3.
The vertical asymptote of a function is the line x = a/b, where a and b are the coefficients of the highest degree terms in the numerator and denominator, respectively. In this case, a = 3 and b = -5, so the vertical asymptote is x = -3/5.
Therefore, the graph of y = 3ex ex − 5 has a horizontal asymptote of y = 3 and a vertical asymptote of x = -3/5.
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Using the Law of Sines to solve the triangle if ∠ A = 43 ∘ , ∠ C = 75 ∘ , b = 15 : ∠ B is degrees; a = ; c = ; Round to two decimal places if needed. Assume ∠ A is opposite side a , ∠ B is opposite side b , and ∠ C is opposite side c .
The values using Law of Sines are:
∠ B = 62∘; b = 11.58 and c = 16.40.
Explain the Law of Sines?The relationship between a triangle's sides and angles is demonstrated using the sine rule. It is specified as:
A/Sin(A) = B/Sin(B) = C/Sin(C)
Where A, B, and C are indeed the angles and A, B, and C are the angles' opposing sides.
∠ A = 43 ∘ , ∠ C = 75 ∘ , b = 15 ;
Part a:
∠ B = 180 - ∠ A - ∠ C
∠ B = 180 - 43 ∘ - 75∘
∠ B = 62∘
Part b: the value of a.
a/Sin(A) = b/Sin(B)
a = b/Sin(B) * Sin(A)
a = 15/Sin(62) * Sin(43)
b = 11.58
Part c: the value of c
b/Sin(B) = c/Sin(C)
c = b/Sin(B) * Sin(C)
c = 15/Sin(62) * Sin(75)
c = 16.40
Thus, the values using Law of Sines are obtained.
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Baldwin drew a cale drawing of an apartment. In real life, a rug by the door i 4 feet long. It i 5 inche long in the drawing. What i the cale factor of the drawing?
The scale factor is 1/5 inch per foot.
Apartment Rug Scale FactorThe scale factor of the drawing can be found by dividing the length of the rug in real life by the length of the rug in the drawing.
4 feet / (5 inches/12 inches/foot) = 4 * 12 / 5 = 9.6 inches
So the scale factor is 9.6 inches / 4 feet = 1/5 inch per foot.
The method used to find the scale factor is applicable to any drawing or model where the size of an object is known in both the real-life version and the drawing or model. As long as the ratio of the size of the object in the real-life version to the size of the object in the drawing or model is constant, the scale factor can be found using the same method.
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These tiles represent the expression 2x + x + 6.
&
XX
3x + 6
X
Which expression is equivalent to 2x + x + 6?
Submit
1
1
2x + 8
8x
or
Watch a video
8x + 1
You have
16y6/x2 is the corresponding formula.The necessary equivalent expression is therefore 16y6/x2.
An expression is defined?Expressions in mathematics are made up of variables combined with the use of operations and predetermined rules. An equation, some numbers, etc., could be used as examples.
Expressions in mathematics are made up of variables combined with the use of operations and predetermined rules.
Expressions in mathematics are made up of variables combined with the use of operations and predetermined rules. An equation, some numbers, etc., could be used as examples.
There is a phrase that says,
{4xy³/x²}²
Expressed more succinctly,
{4xy³/x²}²
= {4y³/x}²
= 16y⁶/x².
The necessary equivalent expression is therefore 16y6/x2.
The complete question is,
Which of the following expressions is the same as the one given?
(4xy^3/x^2)
^2
A. 16y^6 \sB. 16y
5 x 2, 16 x 2, 5 y, and 16 y.
^6/x^2
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Use the fact that
sin
38
°
≈
0
.
6157
sin
38
°
≈
0
.
6157
to solve for
m
.
m
.
cos
m
°
≈
0
.
6157
Answer:
Below
Step-by-step explanation:
Remember the trig identity sin^2 + cos^2 = 1
.6157^2 + cos ^2 = 1
cos^2 = 1 - .6157^2
cos^2 = .6209
cos = ~ ± .7879
how many ways are there to select 10 countries in the united nations to serve on a council if 3 is selected from a block of 52, 1 are selected from a block of 62 and 6 are selected from the remaining 75 countries?
There are 12,804,280,000 ways to select 10 countries in the United Nations to serve on a council.
UN Council Country SelectionThe number of ways to select 3 countries from a block of 52 is given by the combination formula:
C(52, 3) = 52! / (3! * (52 - 3)!) = 19600.The number of ways to select 1 country from a block of 62 is given by:C(62, 1) = 62.
The number of ways to select 6 countries from the remaining 75 countries is given by:C(75, 6) = 75! / (6! * (75 - 6)!) = 178,365.
Finally, to find the number of ways to select 10 countries in total, we multiply the number of ways to select 3, 1, and 6 countries:
19600 * 62 * 178365 = 12,804,280,000.
So there are 12,804,280,000 ways to select 10 countries in the United Nations to serve on a council.
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Let y (x) be the solution of the differential equation xlogxdydx+y=2xlogx
, x≥1
. Then y (e) is equal to,
(a). e
(b). 0
(c).2
(d). 2e
y = 2 is the solution of differential equation .
What exactly does differential equation mean?
An equation that connects one or more unknown functions and their derivatives is known as a differential equation in mathematics.
Applications typically use functions to describe physical quantities, derivatives to indicate the rates at which those quantities change, and differential equations to define a relationship between the two.
( x logx )dy/dx + y = 2xlogx
dividing by xlogx, we get
dy/dx + y/xlogx = 2
d(IF × Y ) = 2logx dx
IF = e∫1/xlogx dx
= [tex]e^{log(logx)} = logx[/tex]
rightarrow Multiplying by IF, we get
d(IF * Y ) = 2logx dx
By integrating, we get
y logx = ∫2 logx dx
By using Product Rule on RHS, we get
ylogx = 2[logx ∫ 1 - ∫((∫1) d/dx(logx)) dx
ylogx = 2[x(logx - 1 )] + c
Put x = 1, we get
0 = 2[1(0 - 1 )] + c
c = 2
ylogx = 2[x[logx - 1 )] + 2
at x = e
y = 2
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what is the percent change of the data set? if your answer
The percent change of the data set is 10%.
Percentage change is a way to describe the difference between an old value and a new value, expressed as a ratio and then multiplied by 100.
If the price of a product was $5 last month and it increased to $5.50 this month, we can calculate the percentage change as
=> (5.50 - 5) / 5 x 100 = 10%.
This means that the price of the product has increased by 10% compared to last month. If the price had gone down, we would have a negative percentage change.
Percentage change is a useful tool to compare data sets, especially when dealing with large numbers.
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Office Bargains sells 150 pens for $7.50. What is the price per pen? $ per pen
Answer:
5 cents each
Step-by-step explanation:
to find out how much each pen costs, you would divide the total price(7.50) by the number of pens(150) to get to the price for each pen ($0.05)
Answer:
To find this, you just want to divide 7.50 by 150.
Take note that since were looking for how much each individual pen costs, 7.50 with come first in the equation, as shown:
7.50/150 = 0.05
Each Pen Costs 5 cents each!
Step-by-step explanation:
Hope it helps! =D
question in image
answer choices
y= -3x + 2
y= -1/3x + 15
y = -1/3x + 2
y= -3x + 15
The correct line of best fit is y = -1/3x + 2. Option C
What is the line of best fit?A line of best fit is a line that is used to approximate the underlying pattern or relationship between two variables in a scatterplot. The line of best fit is drawn so that the number of points above the line and below the line are approximately equal (and the line passes through as many points as possible).
The line of best fit is used to make predictions about the response variable based on the predictor variable, and to understand the strength and direction of the relationship between two variables.
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What are the types of Discontinuities?
Three different types of discontinuity exist.
Discontinuous Jump.Continuous Infinity.Displacing Discontinuity.What is Discontinuities?Discontinuous functions are those parts of the graph that are not connected to one another. If both the left-hand limit and the right-hand limit of a function f(x) exist but are not equal, the function is said to have a first-kind discontinuity at x = a.You need to be familiar with the four different sorts of discontinuities: essential, removable, jump, and point. Factoring the function's numerator and denominator first. When a number is zero in both the numerator and denominator, this is known as a point of discontinuity. There is a point of discontinuity there there is a zero for the numerator and denominator.To learn more about Discontinuities, refer to:
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To graph the function f, we plot the points x, f(x) Correct: Your answer is correct. in a coordinate plane. To graph f(x) = x2 − 5, we plot the following points. (x, 0) (x, x − 3) (x, 2x) (x, 1) (x, x2 − 5) So the point 3, is on the graph of f. The height of the graph of f above the x-axis when x = 3 is Complete the table. x f(x) = x2 − 5 (x, y) −2 −1 0 1 2 0
The complete table for the function f(x)=x^2-2 is given below.
What is a function?
A function is defined as a relation between a set of inputs having one output each. In simple words, a function is a relationship between inputs where each input is related to exactly one output. Every function has a domain and co-domain or range. A function is generally denoted by f(x) where x is the input. The general representation of a function is y = f(x).
There are several types of functions in math. Some important types are:
Injective function or One to one function: When there is mapping for a range for each domain between two sets.
Surjective functions or Onto function: When there is more than one element mapped from domain to range.
Polynomial function: The function which consists of polynomials.
Inverse Functions: The function which can invert another function.
Now,
Given
f(x)=x^2-2
The table will be filled as
x f(x) =x^2-2 (x,y)
-2 2 (-2,2)
-1 -1 (-1,-1)
0 -2 (0,-2)
1 -1 (1,-1)
2 2 (2,2)
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Right question is in image.
A wire of length 66cm i bent into the hape of a circle. Find it area. If it i bent into the hape of a quare,what will be it area ?which figure encloe _________take π=22\7
The area of the circle is 346.5 cm² while the area of the square is 272.25 cm². The figure that encloses more area is the circle.
If a wire of length 66cm is bent into the shape of a circle and square, then this length will be its circumference and perimeter, respectively.
C = 66cm = 2πr
r = 10.5 cm
P = 4s = 66cm
s = 16.5 cm
The area of the circle is given by the formula
A = πr²
A = π(10.5 cm)²
A = 346.5 cm²
So, the area of the circle is 346.5 cm².
The area of the square is given by the formula
A = s²
A = (16.5 cm)²
A = 272.25 cm²
So, the area of the square is 272.25 cm².
Therefore, the figure that encloses the greater area is the circle.
The problem seems unreadable, it must have been...
"a wire of length 66 cm is bent into the shape of a circle. find its area. if the same wire is bent into the shape of a square, what will be its area? which figure enclose more area, the circle or the square? take π = 22/7."
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Let R be the region bounded by the parabola y=x^2 and the line y=4
(a) What is the volume of the solid generated when R is rotated about the line y=4 ?
V=
(b) What is the volume of the solid generated when R is rotated about the line y=5 ?
V=
So, when R is rotated along the line y = 4, the volume of the solid formed is roughly 12.57 cubic units. So, when R is rotated about the line y = 5, the volume of the solid formed is roughly 81.34 cubic units.
What is volume?Every three-dimensional item requires some amount of space. The volume of this space is measured. Volume is defined as the space occupied by an item inside the confines of three-dimensional space. It is also known as the object's capacity.
Here,
The volume of the solid generated when R is rotated about the line y = 4 is given by the formula:
V = π * ∫[a,b] (x² - 4)² dx
where a and b are the x-coordinates of the points where the parabola y = x² intersects the line y = 4. To find these points, we set y = x² and y = 4 and solve for x:
x²= 4
x = ±2
So the points of intersection are (2, 4) and (-2, 4). The region R is the part of the parabola between these points, so a = -2 and b = 2.
Substituting these values into the formula for V, we have:
V = π * ∫[-2,2] (x² - 4)² dx
Evaluating the definite integral, we obtain:
V = π * [x⁵/5 - 4x³/3 + 16x] evaluated from -2 to 2
= π * (16/5 - 16/3 + 16 - (-16/5 + 16/3 - 16))
= π * (32/5)
= 12.57 ≈ 12.57 (rounded to 4 decimal places)
For the volume of the solid generated when R is rotated about the line y = 5, we can use the same formula:
V = π * ∫[a,b] (x² - 5)² dx
where a and b are the x-coordinates of the points where the parabola y = x² intersects the line y = 5. These can be found in the same way as before:
x² = 5
x = ±√5
So the points of intersection are (√5, 5) and (-√5, 5). The region R is the part of the parabola between these points, so a = -√5 and b = √5.
Substituting these values into the formula for V, we have:
V = π * ∫[-√5,√5] (x² - 5)² dx
Evaluating the definite integral, we obtain:
V = π * [x⁵/5 - 5x³/3 + 25x] evaluated from -√5 to √5
= π * (25/5 - 25/3 + 25√5/2 - (-25/5 + 25/3 - 25√5/2))
= π * (50/3 + 25√5)
= 50.97 + 25√5 ≈ 50.97 + 25 * 2.236 = 81.34 (rounded to 2 decimal places)
So the volume of the solid generated when R is rotated about the line y = 4 is approximately 12.57 cubic units. So the volume of the solid generated when R is rotated about the line y = 5 is approximately 81.34 cubic units.
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