Question

A region r consists of a square bounded by the lines x = −5, x = 5, y = 0, and y = −10 and a half disk bounded by the semicircle y = 25 − x2 and the line y = 0. find the center of gravity, (x, y), of r.

Answer 1

Exactly (0, -100/(24 + 3*pi)) 
Approximately (0, -2.991792499)  
I am assuming that there's a typo in the formula for the semicircle. The equation given is: 
y = 25 - x^2 
which does NOT describe a semicircle. I assume that the correct equation is: 
y = sqrt(25 - x^2) 
which would describe a semi-circle with a radius of 5.  
First, determine the mass and centroid of the square and the semicircle. For the square, the centroid is the exact center of the square and the mass is the area. So 
Mass = (5 - -5)*(0 - -10) = 10*10 = 100 
Centroid = ((-5 + 5)/2, (0+ -10)/2) = (0/2, -10/2) = (0,-5)  
The mass for the semi-circle is also its area, and the centroid of a semicircle who's center is at the origin (0,0) is:
 (0, 4r/(3*pi) ) where r = radius of semicircle. So
 (0, 4*5/(3*pi))
 (0, 20/(3pi))
 Mass = (1/2)pi*r^2 = (1/2)pi*5^2 = (1/2)pi*25 = 12.5*pi  
 Since the centroid of both the square and semi-circle have a x-coordinate of 0, the x-coordinate of the center of gravity will also be 0. For the y coordinate, we need to use the following formula:
 (m1 + m2 + m3 + ... + mn)*Xcm = m1x1+m2x2+m3x3+...+mn*xn
 where
 m1,m2,m3,...,mn = masses of objects.
 x1,x2,x3,...,xn = distance from datum of objects.
 Xcm = distance from datum of center of mass. 
 I'll use the origin as the datum. So
 (m1 + m2)*Xcm = m1x1+m2x2
 Xcm = (m1x1+m2x2)/(m1 + m2)
 Xcm = (100(-5)+(12.5*pi)(20/(3pi)))/(100 + 12.5*pi)
 Xcm = (-500+250/3)/(100 + 12.5*pi)
 Xcm = (-1500/3+250/3)/(100 + 12.5*pi)
 Xcm = (-1250/3)/(100 + 12.5*pi)
 Xcm = -1250/(300 + 37.5*pi)
 Xcm = -2500/(600 + 75*pi)
 Xcm = -100/(24 + 3*pi) 
 So the center of gravity of r = (0, -100/(24 + 3*pi)) exactly, or approximately (0, -2.991792499)

Answer 2

In this exercise we have to use the knowledge of limits to calculate a part of a semi-circle, like this:

$A=(0, -100/(24 + 3*\pi))$

Knowing that the formula for the circumference is given by:

$y = 25 - x^2 \\y = \sqrt{(25 - x^2)$

Now calculating the mass and centroid we find that:

$Mass = (5 - -5)*(0 - -10) = 10*10 = 100 \\Centroid = ((-5 + 5)/2, (0+ -10)/2) = (0/2, -10/2) = (0,-5)$

The mass for the semi-circle is also its area is:

$Mass = (1/2)\pi*r^2 = (1/2)\pi*5^2 = (1/2)\pi*25 = 12.5*\pi$

So the formula will be:

$(m_1 + m_2 + m_3 + ... + m_n)*Xcm = m_1x_1+m_2x_2+m_3x_3+...+m_n*x_n$

Where:

• m1,m2,m3,...,mn = masses of objects.

• x1,x2,x3,...,xn = distance from datum of objects.

• Xcm = distance from datum of center of mass.

using the given formula and putting the known values ​​we find that:

$(m_1 + m_2)*X = m_1*x_1+m_2*x_2\\ Xcm = (m_1x_1+m_2x_2)/(m_1 + m_2)\\ Xcm = (100(-5)+(12.5*\pi)(20/(3\pi)))/(100 + 12.5*\pi)\\ Xcm = (-500+250/3)/(100 + 12.5*\pi)\\ Xcm = (-1500/3+250/3)/(100 + 12.5*\pi)\\ Xcm = (-1250/3)/(100 + 12.5*\pi)\\ Xcm = -1250/(300 + 37.5*\pi)\\ Xcm = -2500/(600 + 75*\pi)\\ Xcm = -100/(24 + 3*\pi)$

See more about functions at brainly.com/question/5245372