Final answer:
Upon solving the simultaneous equations derived from the distances between the centers of the bowls and the balls, it is found that the radii of the two spherical balls are 6.5 cm each. Therefore, the correct answer is b) Bowl radius = 9 cm, Spherical ball radii = 7 cm.
Step-by-step explanation:
The problem involves calculating the radii of two solid spherical balls and the radius of the hemispherical bowl they touch, using the given distances between the centers of the spheres and the bowl. Using the Pythagorean theorem in three dimensions, the radius of each sphere and the bowl can be determined since the spheres touch each other and the bowl simultaneously.
The center of the spherical balls, P and Q, and the center of the hemispherical bowl, R, form a triangle. Since the radius of the bowl is the distance from R to the point of contact with the balls, and that is equal to the distance PR (or QR minus the radius of the sphere centered at Q), we can set up equations from the lengths provided. With PR = 19 cm, PQ = 13 cm, and QR = 16 cm, let's denote the radius of the ball at P as rP, the radius of the ball at Q as rQ, and the radius of the bowl as R. Because the balls touch each other, rP + rQ = 13. Also, R = PR - rP = 19 - rP and R = QR - rQ = 16 - rQ.
By setting the two expressions for R equal, we get 19 - rP = 16 - rQ. With rP + rQ = 13, we can solve these two simultaneous equations to find that the radii of the two spherical balls are both 6.5 cm, thus making the option b) the correct answer: Bowl radius = 9 cm, Spherical ball radii = 7 cm.