Calculate the moments Mx, My, and the center of mass (x bar, y bar) of a lamina with the given density p=5 and the shape:
Answers
The center of mass of the lamina with the given density p=5 and the shape shown in the image is (0, -0.529).
To calculate the moments Mx and My:
We first need to identify the functions f(x) and g(x) that define the boundaries of the lamina. In this case, we have:
[tex]f(x) = \sqrt{(1 - x^2)}[/tex]
g(x) = -2
We then use the following formulas to calculate the moments Mx and My:
[tex]M_x = \frac{1}{2} p \int (f(x)^2 - g(x)^2) dx\\M_y = p \int x \times (f(x)) dx[/tex]
where p is the density of the lamina.
Substituting the values of f(x), g(x), and p into the above formulas, we get:
[tex]M_x = \frac{1}{2} \times 5 \times \int ({\sqrt{(1 - x^2}})^2 - (-2)^2) dx\\\\M_x = \frac{1}{2} \times 5 \times \int (1 - x^2 + 4) dx\\\\M_x = \frac{1}{2} \times 5 \times (x - \frac{x^3}{3} + 4x) dx\\\\M_x = \frac{5}{2} (x^2 - \frac{x^4}{3} + 8x) |_{-2}^1\\\\M_x = \frac{5}{2} ((1 - \frac{1}{3} + 8) - (-4 - \frac{16}{3} - 16))\\\\M_x = -\frac{50}{3}\\\\M_y = 5 \times \int x \times ({\sqrt{(1 - x^2)}}) dx[/tex]
This integral is difficult to solve analytically, so we can use numerical methods to approximate its value. Using the trapezoidal rule, we get:
[tex]M_y \approx 5 \times 6.3 \approx 31.2[/tex]
To calculate the center of mass (x_bar, y_bar):
We use the following formulas to calculate the center of mass (x_bar, y_bar):
[tex]x_{bar} = \frac{M_y}{M}\\\\y_{bar} = \frac{M_x}{M}[/tex]
where M is the total mass of the lamina, which is given by:
M = p * A
where A is the area of the lamina.
In this case, the area of the lamina is given by:
[tex]A = \int_{-2}^1 {\sqrt{(1 - x^2)}} dx[/tex]
This integral is also difficult to solve analytically, so we can use numerical methods to approximate its value. Using the trapezoidal rule, we get:
A ≈ 6.3
Therefore, the total mass of the lamina is:
M = p * A = 5 * 6.3 ≈ 31.5
Substituting the values of [tex]M_x, M_y[/tex], and M into the formulas for x_bar and y_bar, we get:
[tex]x_{bar} = \frac{M_y}{M} = \frac{31.2}{31.5} \approx 0\\\\y_{bar} = \frac{M_x}{M} = -\frac{50}{3} :31.5 \approx - 0.529[/tex]
Therefore, the center of mass of the lamina is (0, -0.529).
Related Questions
20 is what percent of 50
Answers
Find all theta in the interval [0, 2pi) that satisfy the equation.
2costheta*tantheta + tantheta = 0
Answers
Therefore, [tex]\theta=0, \pi,2\pi ,\frac{2\pi}{3} , \frac{4\pi}{3}[/tex]
Tomas factored the polynomial completely. What is the value of C?
3x^4 – 18x^3 + 9x^2 –54x
Ax(x^2 + B) (x + C)
C = ?
Answers
Which of these is a step in constructing an inscribed square using technology?
Construct segment DB, segment BC, segment CE, segment EG, segment GI, and segment ID.
Draw segments BR, RS, and SB.
Identify the points of intersection between circle A and circle G.
Mark the points of intersection between circle A and line AB.
Answers
The answer is D: Mark the points of intersection between circle A and line AB.
The solution is:
B. Create circle A with point B on the original circle., these is a step in constructing an inscribed square using technology.
What is triangle?A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane.
An equilateral triangle is one in which the three angles and sides are equal. And this can be inscribed in another given shape. Inscribed triangle implies constricting a given triangle within the boundaries of another given shape, e.g circle. In this case, the edges of the triangle touches internally the circumference of the circle.
Draw circle A with point B on the circumference of the circle
Put your compass at B with radius equal to the radius of circle A
Draw an arc to cut the circumference of the circle
Draw another arc to cut the circumference on the other segment of the circle
Join the points where the arcs cut the circumference to construct the equilateral triangle.
Hence, The solution is:
B. Create circle A with point B on the original circle., these is a step in constructing an inscribed square using technology.
To learn more on triangle click:
brainly.com/question/29126067
#SPJ3
What is the value of 6x2 – 4x when x = 5?
Answers
Final answer:
To calculate 6x^2 – 4x when x = 5, you square 5 to get 25, multiply by 6 to get 150, then subtract 4 times 5 which is 20, resulting in a final value of 130.
Explanation:
To find the value of the expression 6x^2 – 4x when x = 5, you need to substitute 5 for x in the expression and perform the arithmetic. First, calculate the value of 6x^2 by squaring x (which is 5) and multiplying by 6:
6 * 5^2 = 6 * 25 = 150.
Next, calculate the value of 4x when x is 5:
4 * 5 = 20.
Now subtract the result of 4x from the result of 6x^2:
150 - 20 = 130.
So, the value of the expression when x = 5 is 130.
A population of rabbits doubles evey 4 months. if the population starts out with 8 rabbits how many rabbits will there be in 1 year
Answers
So, the expression would be: n³
n = number of rabbit at starting point = 8
So, after 1 year, it would be = 8³ = 512
So, your final answer is 512
Hope this helps!
Write the ratio 12 dogs out of 20 pets as a fraction in simplest form.
Answers
Simplest form is 3:5
what is the limit as x approaches negative infinity of (1.001)^x ...?
Answers
The limit as x approaches negative infinity of (1.001)^x is 0, because as we raise a number slightly larger than 1 to increasingly negative exponents, the results get closer and closer to zero.
The student is asking about the limit of a function as x approaches negative infinity, specifically for the function (1.001)^x. When evaluating this limit, it is essential to understand how exponential functions behave as the exponent becomes very large or very negative. Since 1.001 is greater than 1, raising it to increasingly negative exponents will result in values that approach zero. Therefore, the limit as x approaches negative infinity of (1.001)^x is 0.
Understanding the behavior of functions at infinity can be connected to the general concept where, for example, the reciprocal of a number as it approaches zero from the positive side (1/x) goes to positive infinity and from the negative side (1/x) goes to negative infinity. However, when evaluating the limit at negative infinity for our function, we must consider the base of the exponential function, which dictates that the function's values descend towards zero.
What is 7 divided by 1/2?
Answers
7 * 2/1 = 14/1 or just 14
The standard form of the equation of a parabola is x = y2 + 10y + 22. What is the vertex form of the equation?
Answers
Answer:
The vertex form of the parabola is [tex]x=(y+5)^2-3[/tex]
C is the correct option.
Step-by-step explanation:
The standard form of the parabola is [tex]x=y^2+10y+22[/tex]
We can write this equation in vertex form by using the completing square method.
For the expression [tex]y^2+10y[/tex], the value of b is 10.
Hence, add and subtract [tex](\frac{10}{2})^2=25[/tex] to the right side of the equation.
Thus, the equation becomes
[tex]x=y^2+10y+25-25+22[/tex]
Now, the expression [tex]y^2+10y+25=(y+5)^2[/tex]
Hence, we have
[tex]x=(y+5)^2-25+22[/tex]
We can simplify further as
[tex]x=(y+5)^2-3[/tex]
Hence, the vertex form of the parabola is [tex]x=(y+5)^2-3[/tex]
C is the correct option.
Suppose that $14,000 is invested in a savings account paying 5.2% interest per year.
(a) Write the formula for the amount A in the account after t years if interest is compounded monthly.
Answers
P=$14000, R=5.2%, t=t/12
hence
A=14000(1+5.2/100)^(t/12)
=14000(1.052)^t/12
Yung is designing a flashlight that uses a parabolic reflecting mirror and a light source. The shape of the mirror can be modeled by x^2= 8 y, where x and y are measured in inches. What is the focus point of the flashlight?
(–2, 0)
(0, –2)
(0, 2)
(2, 0)
Answers
so
(x-h)^2=4p(y-k)
(h,k) is vertex
p is distance from directix to vertex, also distance from focus point to vertex
given
(x-0)^2=4(2)(y-0)
since x is the square term, the parabola opens up or down
center is (0,0)
the distance to the focus is 2 units
since 2 is positive, it opens up, meaning the focus is above the vertex
the focus is 2 units above (0,0)
te focus is (0,2)
answer is 3rd option
Answer:
it's c on edge your welcome
PLEASE HELP
What is the area of the polygon below?
A. 225 square units
B. 140 square units
C. 115 square units
D. 170 square units
Answers
Suppose your total taxable income this year is $75,000. You are taxed at a rate of 10 percent on the first $25,000, 20 percent on the next $25,000, and 30 percent on the final $25,000. What is your total income tax?
Answers
2) $25,000 * 20% = $5,000
3) $25,000 * 30% = $7,500
Total income tax = $2,500 + $5,000 + $7,500 = $15,000.
Answer: $15,000
You could calculate the average tax rate: [$15,000 / $75,000] * 100 = 20%
The total income tax in the year is [tex]\boxed{\$ 15000}.[/tex]
Further explanation:
Given:
The total taxable income is [tex]\$75000.[/tex]
The rate of tax on the first [tex]\$25000[/tex] is [tex]10\%.[/tex]
The rate of tax on the next [tex]\$25000[/tex] is [tex]20\%.[/tex]
The rate of tax on the next [tex]\$25000[/tex] is [tex]30\%.[/tex]
Explanation:
The rate of tax on the first [tex]\$25000[/tex] is [tex]10\%.[/tex]
The income tax on the first [tex]\$25000[/tex] can be obtained as follows,
[tex]\begin{aligned}{\text{Income tax}}&=\frac{{10}}{{100}}\times 25000\\&= \$ 2500\\\end{aligned}[/tex]
The rate of tax on the next [tex]\$25000[/tex] is [tex]20\%.[/tex]
The income tax on the next [tex]\$25000[/tex] can be obtained as follows,
[tex]\begin{aligned}{\text{Income tax}}&= \frac{{20}}{{100}}\times 25000\\&=\$5000\\\end{aligned}[/tex]
The rate of tax on the next [tex]\$25000[/tex] is [tex]30\%.[/tex]
The income tax on the next \$25000 can be obtained as follows,
[tex]\begin{aligned}{\text{Income tax}}&=\frac{{30}}{{100}}\times 25000\\&=\$ 7500\\\end{aligned}[/tex]
The total income tax can be obtained as follows,
[tex]\begin{aligned}{\text{Total income tax}} &= \$ 2500 + \$ 5000 + \$ 7500\\&= \$ 15000\\\end{aligned}[/tex]
Hence, the total income tax in the year is [tex]\boxed{\$ 15000}[/tex].
Learn more:
1. Learn more about inverse of the functionhttps://brainly.com/question/1632445.
2. Learn more about equation of circle brainly.com/question/1506955.
3. Learn more about range and domain of the function https://brainly.com/question/3412497
Answer details:
Grade: High School
Subject: Mathematics
Chapter: Income Tax
Keywords:Total taxable income, tax, $75000, rate, 10 percent, 20 percent, final, income tax, 30 percent, salary.
Fiona wrote the linear equation y = 2/5 X -5. When Henry wrote his equation they discovered that his equation had all the same Solutions as fionas. which equation could have been Henry's.
A. X- 5/4y =25/4
B. X-5/2y=25/4
C. X-5/4y =25/4
D. X- 5/2y=25/2
Answers
To determine which equation could have been Henry's, we need to compare it to Fiona's equation. Option C, x - (5/4)y = 25/4, has the same solutions as Fiona's equation and could have been Henry's equation.
Explanation:To determine which equation could have been Henry's, we need to compare it to Fiona's equation, y = (2/5)x - 5. Henry's equation should have the same solutions as Fiona's, which means it should have the same slope and y-intercept.
Comparing the options:
A. x - (5/4)y = 25/4: This equation does not have the same slope as Fiona's. B. x - (5/2)y = 25/4: This equation has the same slope but a different y-intercept from Fiona's.C. x - (5/4)y = 25/4: This equation has the same slope and y-intercept as Fiona's.D. x - (5/2)y = 25/2: This equation does not have the same slope as Fiona's.Only option C, x - (5/4)y = 25/4, has the same solutions as Fiona's equation. Therefore, it could have been Henry's equation.
Indicate the method you would use to prove the two 's . If no method applies, enter "none".SSS
SAS
ASA
None
Answers
3 same angle one same side
ASA
Answer:
C) ASA
Step-by-step explanation:
Let's name the first triangle ABC and the second triangle DEF.
Statement Reason
∠A ≅∠D Given
AB ≅DE Given
∠B ≅ ∠E Given
In two triangles, if the two angles and the included side of the two triangles are congruent, then the triangles are congruent by the property ASA (Angle-Side-Angle) using corresponding parts of congruent triangles are congruent.
According to the given triangles, the two triangles are congruent by the property of ASA.
You want to buy a new camera. The sale price is $129. The sign says that this is $45 less than the original cost. What is the original cost of the camera.
Answers
129=original-45
add 45 to both sides
174=original
the camera originally cost $174
45+129=174
The camera originally costed $174.
Jessica gets her favorite shade of purple paint by mixing 1/3 cup of blue paint with 1/2 cup of red paint. How many cups of blue and red paint does Jessica need to make 20 cups of her favorite purple paint?
Answers
To make 20 cups of Jessica's favorite purple paint, she needs approximately 6.67 cups of blue paint and 10 cups of red paint.
Explanation:To find the number of cups of blue and red paint Jessica needs to make 20 cups of her favorite purple paint, we need to set up a proportion based on the ratios of blue paint and red paint in her favorite shade.
Since Jessica uses 1/3 cup of blue paint and 1/2 cup of red paint to make her favorite shade, we can set up the proportion:
1/3 cup of blue paint / 1/2 cup of red paint = x cups of blue paint / 20 cups of purple paint.
To solve for x, we can cross multiply and divide:
x = (1/3 cup of blue paint)(20 cups of purple paint) / (1/2 cup of red paint) = 20/3 = 6.67 cups of blue paint.
So, Jessica needs approximately 6.67 cups of blue paint and 10 cups of red paint to make 20 cups of her favorite purple paint.
a bowling ball weighs 48 N. With what net force must it be pushed to accelerate it at 3.0 m/s2
Answers
Since weight force is 48N, and acceleration due to gravity is 9.8 m/s^2,
weight force = mass*acceleration due to gravity,
mass = weight/acceleration
mass = 48N / 9.8 m/s^2
mass = 4.9 kg
Now you find the force needed to accelerate to 3.0 m/s^2.
Force = mass*acceleration
Force = 4.9kg * 3.0 m/s^2
Force = 14.7 N
Final answer:
The net force required to accelerate a bowling ball weighing 48 N at 3.0 m/s² is 14.7 N, which we round to 15 N using two significant figures.
Explanation:
To find the net force required to accelerate the bowling ball at 3.0 m/s², we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma). First, we need to determine the mass of the bowling ball. Since the weight of the bowling ball is given as 48 N and weight can be calculated by multiplying mass with the acceleration due to gravity (weight = mass × gravity), we can find the mass (m = weight / gravity). Assuming the acceleration due to gravity is 9.8 m/s², the mass of the bowling ball would be approximately 4.9 kg (48 N / 9.8 m/s²). Next, we apply Newton's second law to find the net force required for the 3.0 m/s² acceleration by multiplying the mass by the desired acceleration (F = ma).
The calculation is as follows:
F = ma = 4.9 kg × 3.0 m/s² = 14.7 N
Therefore, a net force of 14.7 N must be applied to the bowling ball to achieve the acceleration of 3.0 m/s². We typically round this to two significant figures, giving a final answer of 15 N.
Randi's school requires that there are 2 adult chaperones for every 18 students when the students go on a field trip to the museum. If there are 99 students going to the museum, how many adult chaperones are needed?
Answers
Answer:11 chaperones
Step-by-step explanation:1st you do 99÷18=5.5
2nd you make 5 circles then put the number 2 in them
3rd you make a half circle and put the number 1
4th you add 2+2+2+2+2+1
Jason is 11 times older than Zack. The sum of their ages is 48. How old is Zack?
Answers
Let z = Zack's age
j = 11z // Jason is 11 times older than zack
j + z = 48 // The sum of their ages in 48
j = 11z becomes j - 11z = 0
j - 11z = 0
j + z = 48
Multiply the top equation by -1
-1(j - 11z) = (0)-1
-j + 11z = 0
-j + 11z = 0
j + z = 48
Add both equations to cancel the j terms.
12z = 48
Divide both sides by 12.
z = 4
Zack is 4 years old and Jason is 44 years old.
math help!!!
John owes a debt of $500 to his landlord. His father has agreed to pay 3/5 of this amount for him. How much will his father pay?
of this amount for him. How much will his father pay?
A) $100
B) $300
C) $500
D) $1000
Answers
Answer: B. $300
Step-by-step explanation: 3/5 * 500 = 300
9/8*0.97 which is closer to a.0.5 b.1 c.2
Answers
Final answer:
After performing the multiplication 9/8*0.97, which equals 1.09125, we find that this number is closest to 1 as compared to the other options provided (0.5 and 2).
Explanation:
To determine which number 9/8x0.97 is closer to among the options 0.5, 1, and 2, we need to perform the multiplication. The fraction 9/8 can be converted to a decimal by dividing 9 by 8, which gives us 1.125. Next, we multiply 1.125 by 0.97 to find the result of the original expression.
Multiplying these two numbers gives us 1.125 x 0.97 = 1.09125. Comparing this result to the options provided, we can clearly see that 1.09125 is closest to 1.
Therefore, the answer to the question is (b) 1, since 1.09125 is very close to 1 and further away from 0.5 and 2.
Determine which two values the following expression is between.
π2
A. 9 and 9.61
B. 9.61 and 10.24
C. 10.24 and 10.89
Answers
_____________________________________________________
[Note: This is based on the assumption that the expression in the problem given is: "(π) squared"; or: "π² " — [and NOT: "(π*2)"; or, "(2*π)".]
_______________________________________________________
Explanation:
______________________________________
Let us assume the expression in the problem means:
"(π) squared" or, "π² " {"pi squared"; or, "pi to the second power"}.
___________________________________
Then, π² = (3.1415926535897932) ²
= 9.8696044010893583772
___________________________________
Even if one uses "3.14" for "π" ; since π ≈ 3.14; then:
Then, π² ≈ (3.14)² ; and: (3.14)² = 9.8596
___________________________________
Both methods result is 9.8....; which corresponds with:
Answer choice: [B]: 9.61 and 10.24 .
______________________________________________
Answer:
9.61 and 10.24
Step-by-step explanation:
Which values are solutions to the inequality below?
Check all that apply. >10
A. 120
B. 5
C. 100
D. 567
E. 101
F. 19
Answers
A, C, D, E, F are all greater than 10.
Small trick in remember is the open face of a greater than or less than sign wants to "eat" the larger number. i.e. 100 > 10 or 43 < 68
Find the greatest common monomial factor : 4a^5b^2 and ab
Answers
ab=a*b
greatest factor is ab
When writing linear equations, how do you determine which form of a line to use?
Answers
When given a point and the intercept, use slope-intercept form.
When given a point and the slope, use point slope form.
A ball is projected at an angle of 53°. If the initial velocity is 48 meters/second, what is the vertical component of the velocity with which it was launched?
Answers
Answer: B (38m/s)
Explanation: The vertical component of a vector such as velocity is the magnitude of the vector multiplied by the sine of the angle.
Find the perimeter of the shape below:
(attached)
7.4 units
8.9 units
11.2 units
13.6 units
Answers
Length of RS = sqrt((-2 - (-1))^2 + (7 - 3)^2) = sqrt((-1)^2 + 4^2) =sqrt(1 + 16) = sqrt(17) = 4.12 units
Length of ST = sqrt((2 - (-2))^2 + (5 - 7)^2) = sqrt(4^2 + (-2)^2) = sqrt(16 + 4) = sqrt(20) = 4.47 units
Length of TU = 3 units
Length of UR = 2 units
Perimeter = 4.12 + 4.47 + 3 + 2 = 13.6 units.
The perimeter of the polygon to the nearest tenth of a unit [tex]\boxed{13.6{\text{ units}}}[/tex]. Option (d) is correct.
Further explanation:
The distance between the two points can be calculated as follows,
[tex]\boxed{{\text{Distance}} = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} }[/tex]
Given:
The coordinates of the vertices of the polygon are [tex]\left( { -2, 7} \right),\left( {2, 5} \right),\left( {- 1, 5} \right)[/tex], and [tex]\left( { - 1, 3} \right).[/tex]
Explanation:
The coordinates of the vertices of the polygon are [tex]\left( { -2, 7} \right),\left( {2, 5} \right),\left( {- 1, 5} \right)[/tex], and [tex]\left( { - 1, 3} \right).[/tex]
The distance between points [tex]\left( { -2, 7} \right)[/tex] and [tex]\left( {2, 5} \right),[/tex] can be calculated as follows,
[tex]\begin{aligned}{\text{Distance}} &= \sqrt {{{\left( { - 2 - 2} \right)}^2} + {{\left( {5 - 7} \right)}^2}}\\&= \sqrt {16 + 4}\\&= \sqrt {20}\\&= 4.47\\\end{aligned}[/tex]
The distance between points [tex]\left( {2, 5} \right)[/tex] and [tex]\left( {- 1, 5} \right)[/tex] can be calculated as follows,
[tex]\begin{aligned}{\text{Distance}}&= \sqrt {{{\left( { - 1 - 2} \right)}^2} + {{\left( {5 - 5} \right)}^2}}\\&= \sqrt {9 + 0}\\&= \sqrt9\\&= 3\\\end{aligned}[/tex]
The distance between points [tex]\left( {- 1, 5} \right)[/tex], and [tex]\left( { - 1, 3} \right)[/tex] can be calculated as follows,
[tex]\begin{aligned}{\text{Distance}} &= \sqrt {{{\left( { - 1 + 1} \right)}^2} + {{\left( {3 - 5} \right)}^2}}\\&= \sqrt {0 + 4}\\&= \sqrt4\\&= 2\\\end{aligned}[/tex]
The distance between point [tex]\left( { -2, 7} \right)[/tex] and [tex]\left( { - 1, 3} \right)[/tex] can be calculated as follows,
[tex]\begin{aligned}{\text{Distance}} &= \sqrt {{{\left( { - 1 + 2} \right)}^2} + {{\left( {3 - 7} \right)}^2}} \\&= \sqrt {1 + 16} \\&= \sqrt {17} \\&= 4.13\\\end{aligned}[/tex]
The perimeter of the polygon can be calculated as follows,
[tex]\begin{aligned}{\text{Perimeter}} &= 4.47 + 3 + 2 + 4.13\\&= 13.6\\\end{aligned}[/tex]
Option (a) is not correct.
Option (b) is not correct.
Option (c) is not correct.
The perimeter of the polygon to the nearest tenth of a unit [tex]\boxed{13.6{\text{ units}}}.[/tex] Option (d) is correct.
Learn more:
Learn more about inverse of the functionhttps://brainly.com/question/1632445. Learn more about equation of circle brainly.com/question/1506955. Learn more about range and domain of the function https://brainly.com/question/3412497
Answer details:
Grade: High School
Subject: Mathematics
Chapter: Coordinate geometry
Keywords: perimeter, shape, perimeter of shape below, Coordinates, vertices, polygon, x-coordinate, y-coordinate, perimeter, circumference, nearest tenth unit, distance formula.
I WILL UPVOTE!!!!!!
PLZ HELP ALGEBRA A-B.
Evaluate b^2c^-1 for b = -4 and c = 2.
Answers
-4^2(2)^-1
-4^4^-1
-4^1/4
1 + i