The given bounded region is the set
[tex]R = \left\{(x,y) ~:~ 0 \le x \le 1 \text{ and } x^4 \le y \le x^{1/4} \right\}[/tex]
assuming the curves are [tex]y=x^4[/tex] and [tex]x=y^4\implies y=x^{1/4}[/tex] (since [tex]x>0[/tex] in the first quadrant).
The coordinates of the centroid are [tex](\bar x, \bar y)[/tex] where [tex]\bar x[/tex] and [tex]\bar y[/tex] are the average values of [tex]x[/tex] and [tex]y[/tex], respectively, over the region [tex]R[/tex]. These are given by the ratios
[tex]\bar x = \dfrac{\displaystyle \iint_R x \, dA}{\displaystyle \iint_R dA} \text{ and } \bar y = \dfrac{\displaystyle \iint_R y\,dA}{\displaystyle \iint_R dA}[/tex]
Compute the area of [tex]R[/tex].
[tex]\displaystyle \iint_R dA = \int_0^1 \int_{x^4}^{x^{1/4}} dy \, dx \\\\ ~~~~~~~~ = \int_0^1 \left(x^{1/4} - x^4\right) \, dx \\\\ ~~~~~~~~ = \frac45 - \frac15 = \frac35[/tex]
Integrate [tex]x[/tex] and [tex]y[/tex] over [tex]R[/tex].
[tex]\displaystyle \iint_R x \, dA = \int_0^1 \int_{x^4}^{x^{1/4}} x \, dy \, dx \\\\ ~~~~~~~~ = \int_0^1 x \left(x^{1/4} - x^4\right) \, dx \\\\ ~~~~~~~~ = \int_0^1 \left(x^{5/4} - x^5\right) \, dx \\\\ ~~~~~~~~ = \frac49 - \frac16 = \frac5{18}[/tex]
[tex]\displaystyle \iint_R y \, dA = \int_0^1 \int_{x^4}^{x^{1/4}} y \, dy \, dx \\\\ ~~~~~~~~ = \frac12 \int_0^1 \left((x^{1/4})^2 - (x^4)^2\right) \, dx \\\\ ~~~~~~~~ = \frac12 \int_0^1 \left(x^{1/2} - x^8\right) \, dx \\\\ ~~~~~~~~ = \frac12 \left(\frac23 - \frac19\right) = \frac5{18}[/tex]
Then the centroid's coordinates are
[tex]\bar x = \dfrac{\frac5{18}}{\frac35} = \boxed{\frac{25}{54}} \approx 0.463[/tex]
[tex]\bar y = \dfrac{\frac5{18}}{\frac35} = \boxed{\frac{25}{54}}[/tex]
how many solutions does this equation have ? 2q-3q=-q
Equation have infinity solution
Here the equation is 2q-3q =-q.
After solving this we have
-q = -q
As Left-hand side is equal to the right-hand side.
Therefore whatever the value we will put for the q, the left and right hand side remains the same.
Therefore this equation has an infinity solution.
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The function can be used to determine the side length, x, of a cube given the surface area of the cube, a. the function v(x) = x3 can be used to find the volume of a cube. how can these functions be combined?
The composite function or combined functions of x(A) and V(x) is V(x(A)) = (√A/6)³ option (C) is correct.
It is defined as a special type of relationship, and they have a predefined domain and range according to the function every value in the domain is related to exactly one value in the range.
A function relates an input to an output.
x2 (squaring) is a function
x3+1 is also a function
Sine, Cosine, and Tangent are functions used in trigonometry.
Although part of the question is missing the question is The function x(A)= √A/6 can be used to determine the side length, x, of a cube given the surface area of the cube, A. The function V(x)= x³ can be used to find the volume of a cube. how can these functions be combined?
a. x(v(A))= sqrt x^3/6
b. V(x(A)) = sqrt x3/6
c. V(x(A))=(sqrt A/6)^3
d. x(v(A))= (sqrt A/6)^3
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if p=(3,1) and Q=(-3,-7), find the equation of the circle that has segment PQ as the diameter (x-{?})^2+(y-{?})^2={?}
Answer:
x² + (y + 3)² = 25
Step-by-step explanation:
the centre (C) of the circle is at the midpoint of the diameter.
using the midpoint formula
midpoint = ( [tex]\frac{x_{1}+x_{2} }{2}[/tex] , [tex]\frac{y_{1}+y_{2} }{2}[/tex] )
with (x₁, y₁ ) = P (3, 1 ) and (x₂, y₂ ) = Q (- 3, - 7 )
C = ( [tex]\frac{3-3}{2}[/tex] , [tex]\frac{1-7}{2}[/tex] ) = ( [tex]\frac{0}{2}[/tex] , [tex]\frac{-6}{2}[/tex] ) = (0, - 3 )
the radius r is the distance from the centre to either P or Q
using the distance formula
r = [tex]\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2 }[/tex]
with (x₁, y₁ ) = C (0, - 3 ) and (x₂, y₂ ) = P (3, 1 )
r = [tex]\sqrt{(3-0)^2+(1-(-3)^2}[/tex]
= [tex]\sqrt{3^2+(1+3)^2}[/tex]
= [tex]\sqrt{3^2+4^2}[/tex]
= [tex]\sqrt{9+16}[/tex]
= [tex]\sqrt{25}[/tex]
= 5
the equation of a circle in standard form is
(x - h)² + (y - k)² = r²
where (h, k ) are the coordinates of the centre and r is the radius
here (h, k ) = (0, - 3 ) and r = 5 , then
(x - 0 )² + (y - (- 3) )² = 5² , that is
x² + (y + 3)² = 25
What is the solution set for | s+4 | = 6?
Answer:
s=2,−10
Step-by-step explanation:
Find all solutions for s by breaking the absolute value into the positive and negative components.
(-2,-4) and (3,-9)?
What we answering with these 2 points ?
Where's the question?
Value of x
Pls show steps
Answer:
x = 18
Step-by-step explanation:
Vertical angles are congruent.
4x + 3 = 75 Subtract 3 from both sides of the equation
4x = 72 Divide both sides by 4
x = 18
It took 5 hours to drive to Durham to Birmingham. The average speed was 48mph. What was the distance from Durham to Birmingham?
I need the explanation
Answer:
240 miles
Step-by-step explanation:
Because it took 5 hours going 48 miles per hour, therefore 5 x 48 =240
what is this i need help ASAP
Answer:
The first and fourth statement.
Step-by-step explanation:
1. - 3 - 12/3 + 5
= - 3 - 4 + 5
= - 7 + 5
= - 2
- 2 < 0
4. 4 - 2 - 3/2
= 2 - 3 / 2
= 1 / 2
1 / 2 > 0
Hope this helped! ^^
Sidenote: If you need me to explain for 2, 3 and 5, comment below this and I'll explain.
Which probability value indicates that there is statistical significance? 0.0345 0.4595 0.9534 0.9967
The probability value indicates that there is statistical significance in 0.0345 option (a) is correct.
What is probability?It is defined as the ratio of the number of favorable outcomes to the total number of outcomes, in other words, the probability is the number that shows the happening of the event.
As we know,
Statistical significance is commonly defined as a probability value of 0.05 or even less. P-value can be used in replacement of or in addition to preset levels of confidence when testing the claim.
The value 0.0345 < 0.05
The probability value of 0.0345 indicates that there is statistical significance
Thus, the probability value indicates that there is statistical significance in 0.0345 option (a) is correct.
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Write the name of the metric system prefix associated with each mathematical factor. Do not use any hyphens in your answers and do not abbreviate the prefixes. Spelling counts.
The name of the metric system prefix are : 10^-6: micro , 1/100: centi , 0.001: milli, 1000: kilo
In mathematical exponents, prefixes are referred to as the alphabetic names related to the mathematical factors, such as 10⁻⁹: it nano ,10⁻¹²: Pico and so on.In order to identify the prefixes of the metric system, we first convert from factor to exponent form.
such :
10^-6 is micro
1/100 = 0.01x 10^2: centi
0.001 = 1x 10 ^-3 : milli
1000 = 1x 10^3 : kilo
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Write the name of the metric system prefix associated with each mathematical factor. Do not use any hyphens in your answers and do not abbreviate the prefixes. Spelling counts.
10^-6 ______________
1/100 ______________
0.001 _____________
1000 ______________
A county has 225 acres of vacant land.
Residents voted to turn some of the vacant land.
into parks. Now, there are 36% less acres of
vacant land. How many acres of vacant land
does the county have now?
The acres of vacant land that the county have now will be 144 acres.
How to calculate the value?Since 36% has been used, the remaining percentage will be:
= 1 - 36%
= 64%
Therefore, the acres of vacant land that the county have now will be:
= 64% × 225
= 64/100 × 225
= 0.64 × 225
= 144
Therefore, the acres of vacant land that the county have now will be 144 acres.
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5x - 8x - 7=-3x+5-7 won’t pull an answer to check my answer
Find the output (y) of the function y
20x + 38 if the input (2) is -9.
A function assigns the values. The output(y) of the function y=20x + 38 if the input is -9 is -142.
What is a Function?A function assigns the value of each element of one set to the other specific element of another set.
Given the function y=20x+38, therefore, if the input of the function is -9, then the output can be calculated by substituting the value of x as -9. Therefore, we can write the function as,
y = 20x + 38
y = 20(-9) + 38
y = -180 + 38
y = -142
Hence, the output(y) of the function y=20x + 38 if the input is -9 is -142.
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The population of pine bluff is 654 and is increasing at the rate of 7 per year
If the population of pine bluff is 654 and is increasing at the rate of 7 per year, then the equation will be y=654+7x and the population after 3 years will be 675.
Given that the population of pine bluff is 654 and is increasing at the rate of 7 per year.
We are required to find the equation which shows the population and the population after 3 years.
Equation is basically the relationship between two or more variables that are expressed in equal to form and can be solved to find the value of variables. It may be linear equation, quadratic equation, cubic equation and many more depending on the power of the variables that are present in the equation.
Suppose the number of years be x.
The equation which shows the population after x months be:
y=654+7x
To find the population after 3 years, we can use x=3 in the equation.
y=654+7*3
y=654+21
y=675
Hence if the population of pine bluff is 654 and is increasing at the rate of 7 per year, then the equation will be y=654+7x and the population after 3 years will be 675.
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What is 12 1/2% of 300
Answer:
37.5
Step-by-step explanation:
Answer: 37.5
Step-by-step explanation:
12.5% of 300=37.5
Find the indicated measure(part 3)
I will give brainliest to the right answer with solution
Answer:
VU = 120° , WX = 100°
Step-by-step explanation:
(7)
An inscribed angle is half the measure of its intercepted arc , or
the intercepted arc is twice its inscribed angle, then
arc TV = 2 × 30° = 60°
the diameter is TU then arc TU = 180°
the sum of the arcs in the circle = 360° , so
VU + TV + TU = 360°
VU + 60° + 180° = 360°
VU + 240° = 360° ( subtract 240° from both sides )
VU = 120°
(8)
WY = 2 × 75° = 150°
WX + WY + YX = 360°
WX + 150° + 110° = 360°
WX + 260° = 360° ( subtract 260° from both sides )
WX = 100°
Jill sold half of her comic books
and then bought seventeen
more. She now has 31. With how
many did she begin?
well, she really began with "x" total.
then she sold half of that, so hmmm 1/2 of "x" is just x/2 and she's left with half, so she sold x/2 and she was left only with x/2, and then she bought 17 more, we also know that she has 31 now, so
[tex]\stackrel{\textit{leftover}}{\cfrac{1}{2}x}~~ + ~~\stackrel{\textit{more bought}}{17}~~ = ~~\stackrel{\textit{now she has}}{31}\implies \\\\\\ \stackrel{\textit{multiplying both sides by }\stackrel{LCD}{2}}{2\left(\cfrac{x}{2}~~ + ~~17 \right)~~ = ~~2(31)}\implies x+34=62\implies \LARGE \begin{array}{llll} x=28 \end{array}[/tex]
Which of the following sentences contains a dangling modifier?
A. Exhausted and thirsty, Jack ran off the football field for a drink of water.
B. Embarrassed and humiliated, the singer ran off stage after forgetting the lyrics.
C. Heroically, the firefighter rescued the small child from the burning building.
D. Searching frantically, the homework was finally found and turned in.
Answer:
I'm pretty sure the answer is D
Step-by-step explanation:
I took the test
The sentence that contains a dangling modifier is D. Searching frantically, the homework was finally found and turned in.
What is a dangling modifier?A word or phrase that modifies a term that is not explicitly expressed in the sentence is known as a dangling modifier. A modifier provides a concept's definition, clarification, or more information.
A grammatical modifier that could be mistakenly understood as being connected to a word other than the one intended is known as a hanging modifier. A hanging modifier, which usually refers to a participle, has no subject.
Therefore, the sentence that contains a dangling modifier is Searching frantically, the homework was finally found and turned in.
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Simply 4/16 to the lowest term and find equivalent fraction that has a denominator of 32
Answer:
8
Step-by-step explanation:
this is because If you multiply the denominator by 2 you will get 32
so if you likewise multiply the numerator by 2 you will get 8
Triangle ABC is a right triangle and cos(22.6o)=StartFraction b Over 13 EndFraction. Solve for b and round to the nearest whole number.
Triangle A B C is shown. Angle A C B is 90 degrees and angle C A B is 22.6 degrees. The length of hypotenuse A B is 13 centimeters, the length of A C is b, and the length of C B is a.
Which equation correctly uses the value of b to solve for a?
tan(22.6o) = StartFraction a Over 13 EndFraction
tan(22.6o) = StartFraction 13 Over a EndFraction
tan(22.6o) = StartFraction a Over 12 EndFraction
tan(22.6o) = StartFraction 12 Over a EndFraction
In the given triangle ABC right angled at C and [tex]AC = b[/tex] and [tex]CB = a[/tex], the equation using value of b used for solving for a will be [tex]tan(22.6degrees) = \frac{a}{12}[/tex].
According to the question statement, Triangle A B C is shown. Angle A C B is [tex]90[/tex] degrees and angle C A B is [tex]22.6[/tex] degrees. The length of hypotenuse A B is [tex]13[/tex] centimeters, the length of A C is b, and the length of C B is a.
We are supposed to find the equation using value of b used for solving for a.
Solution: Given,
[tex]AB = 13cm\\AC = b\\BC = a\\Angle BAC = 22.6degrees[/tex]
If hypotenuse is 13cm then value of b is either 5 or 12 as per Pythagorean Triplet. As there is no value such as 5 in the given options we take the value of [tex]b=12 cm[/tex].
[tex]cos(22.6degrees) = \frac{b}{13} =\frac{12}{13} \\sin(22.6degrees) = \frac{a}{13}\\[/tex]
Using the above two equations,
[tex]tan(22.6degrees)=\frac{a}{b} \\b=12\\tan(22.6degrees)=\frac{a}{12}[/tex]
Therefore option three "tan(22.6o) = StartFraction a Over 12 EndFraction" is the correct answer.
Pythagorean Triplet: If a, b and c are three sides of a right angled triangle, c being the hypotenuse then [tex]a^{2} +b^{2} =c^{2}[/tex] and a and b being either of the base and perpendicular.To learn more about Pythagoras theorem, click on the link given below:
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Answer: C tan (22.6)=a/12
Step-by-step explanation: edge 2023
Find the midpoint of the given line segment with the given endpoints
(-3, 8) and (2, -4)
The midpoint of the given line segment with the given endpoints
(-3, 8) and (2, -4) is (-0.5, 2)
Midpoint of coordinatesThe formula for calculating the midpoint of coordinate is expressed as;
M(x, y) = (x₁+x₂/2, y₁+y₋/2 )
Given the coordinate points( -3, 8) and (2, -4)
Substitute
M(x, y) = (-3+2/2, 8-4/2)
M(x, y)= (-1/2, 4/2)
M(x, y) =(-0.5, 2)
Hence the midpoint of the given line segment with the given endpoints
(-3, 8) and (2, -4) is (-0.5, 2)
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The cost of having a package delivered has a base fee of $9.10. every pound over 5 lbs costs an additional $0.42 per pound. write an equation that expresses the cost c in terms of w, the total weight in pounds of a package weighing over 5 lbs.
The expression c = 9.10 + 0.42(w-5) ; w > 5 will represent the cost c in terms of w, the total weight in pounds of a package weighing over 5 lbs.
How to form an equation?Determine the known quantities and designate the unknown quantity as a variable while trying to set up or construct a linear equation to fit a real-world application.
In other words, an equation is a set of variables that are constrained through a situation or case.
Total cost below 5 lbs = base fee
Total cost over 5 lbs = base fee + additional 0.42/pound
Let's say the total cost is c and the total pounds is w
Then,
c = 9.10 + 0.42(w-5) ; w > 5
Here for w = 6 it will gives w - 5 = 6 - 5 = 1 lbs.
Hence "The expression c = 9.10 + 0.42(w) ; w > 5 will represent the cost c in terms of w".
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(PLS HURRY ASAP 100 POINTS)
(Theoretical Probability MC)
The tree diagram represents the sample space for the repeated experiment of a coin being tossed 4 times.
Determine P(at least 3 tails).
A: 0.025
B: 0.0625
C: 0.125
D: 0.3125
Answer:
D) 0.3125====================
Total number of outcomes is 16, count of all outcomes.
Number of outcomes with 3 or more tiles is 5:
HTTT, THTT, TTHT, TTTH, TTTTThe probability is:
P = favorable outcomes / total outcomesP(at least 3 tails) = 5/16 = 0.3125Correct choice is D.
What is an equation of the line that passes through the points (-6, 6) and (-3, 1)?
Answer:
5x + 3y = 18
Step-by-step explanation:
First find slope. m = (y2 - y1)/(x2 - x1) = (6 -1)/(-6 --3) = 5/-3
then. use. y - y1 =. m (x - x1). which gives. y - 1 = -5/3(x - 3)
then. y - 1 = -5/3x + 5
then y = -5/3 x + 6. or. 5x + 3y = 18
30 POINTS
Write and solve an absolute value equation for the minimum and maximum times for an object moving at the given speed to travel the given distance. Absolute value 3 ft. 10 ft/s for 30 ft
Answer:
Step-by-step explanation:
12
Please help! Need answers fast
The common difference of an arithmetic series is 5 and its tenth term is 98,find its first term
Answer:
First term = 53,
Step-by-step explanation:
a10 = a1 + (10 - 1)d so:
98 = a1 + 9*5
a1 = 98 - 45
= 53,
In Exercises 19-24, solve the equation. Determine
whether the equation has one solution, no solution, or
infinitely many solutions. (See Example 3.)
19. 3t+ 4 = 12 + 3t
20. 6d+ 8 = 14 + 3d
21. 2(h+ 1) = 5h - 7
22. 12y + 6 = 6(2y + 1)
23. 3(4g + 6) = 2(6g + 9)
24.
5(1+2m)=(8 + 20m)
Step-by-step explanation:
Given is a linear equation in t

If we want to simplify we find that 3t gets cancelled and we get

This is an impossible situation.
This implies that original equation has no solution at all
Hence answer is
the given has no solution
please help
find the length of AD in the diagram
Answer:
The length of AD is 5 units
Step-by-step explanation:
28x-2=100 solve for x
Answer:
3.64
Step-by-step explanation:
collect like terms
divide both sides by x coefficient
Answer:
x=51/14
Step-by-step explanation:
28x-2=100
+2. +2
28x=98
28x/28=102/28
×=51/14