A uniform rectangular plate of length B = 32 cm and height A = 15 cm has a rectangular corner cut out of it of length D = 9 cm and height C = 10 cm. The plate is made of a material of area mass density σ. For this problem we set the origin at the lower left corner of the plate with the x-axis horizontal pointing right and the y-axis vertical pointing up.

(a) Calculate the value of the y-coordinate, in centimeters, for the center of mass of the plate.
(b) Calculate the value of the x-coordinate, in centimeters, for the center of mass of the plate.

To calculate the value of the y-coordinate for the center of mass, we can begin by finding the y-coordinates for the center of masses for both rectangles.

Also, let the small rectangle be 'Rectangle 1', and the bigger rectangle 'Rectangle 2'.

Smaller rectangle:

Since D = 9 cm and B = 32 cm, the remaining width of the small rectangle is equal to B - D = 32 - 9 = 23 cm. The midpoint of this width is:
[tex]w_{1m} = \frac{23}{2} = 11.5 cm[/tex]

Now, the height of this rectangle is 10 cm. The midpoint of this height is 10/2 = 5 cm. However, this is not the actual y-coordinate. Since the height of the block below is 5 cm, we must add the two because this rectangle is on top of the other.

[tex]h_{1m}= 5 + 5 = 10 cm[/tex]

Larger rectangle:
We can simply take the midpoints of its dimensions to solve for its center of mass.

[tex]w_{2m} = \frac{B}{2} = \frac{32}{2} = 16 cm[/tex]

[tex]h_{2m} = \frac{A-C}{2} = \frac{15-10}{2} = 2.5 cm[/tex]

Now, take the averages of the coordinates for both rectangles to solve.

y-coordinate:
[tex]y_{cm} = \frac{10 + 2.5}{2} = \frac{12.5}{2} = \boxed{6.25 cm}[/tex]

x-coordinate:
[tex]x_{cm} = \frac{11.5 + 16}{2} = \frac{27.5}{2} = \boxed{13.75 cm}[/tex]