Answer:
65 cm
Step-by-step explanation:
A sphere is rolled between two parallel blades that cut away two portions from the top and bottom of the sphere, resulting in two parallel circular faces with radii of 39 cm and 60 cm.
The two-dimensional side view of a sphere is a circle with a radius equal to the sphere's radius. Therefore, in terms of the circle representing the side view of the sphere, we have a circle with pieces from the top and bottom of the circle removed. The removal of the top piece leaves a straight edge 78 cm in length, while the removal of the bottom piece leaves a straight edge 120 cm in length. These straight edges are parallel and 77 cm apart.
To find the radius of this circle (which is the radius of the sphere), we can attribute coordinate points to the endpoints of the straight edges and use the equation of the circle to create equations that can be solved to find the center and the radius.
Let the y-axis be the line on which the vertical diameter of the circle lies. Therefore, let the endpoints of the bottom edge be (-60, 0) and (60, 0), since the radius of this circular face is 60 cm. Let the endpoints of the top edge be (-39, 77) and (39, 77) since the radius of this circular face is 39 cm and the distance between these faces is 77 cm.
The equation of a circle with center (h, k) and radius r is:
[tex](x-h)^2+(y-k)^2=r^2[/tex]
Substitute points (-60, 0) and (60, 0) into this equation and solve for h:
[tex](-60-h)^2+(0-k)^2=r^2\\\\3600 + 120h + h^2 + k^2 = r^2[/tex]
[tex](60-h)^2+(0-k)^2=r^2\\\\3600 - 120h + h^2 + k^2 = r^2[/tex]
[tex]3600 + 120h + h^2 + k^2 = 3600 - 120h + h^2 + k^2\\\\120h = - 120h \\\\h = 0[/tex]
Therefore, the equation of the circle now becomes:
[tex](x-0)^2+(y-k)^2=r^2\\\\x^2+(y-k)^2=r^2[/tex]
Substitute point (60, 0) into this equation and solve for k²:
[tex](60)^2+(0-k)^2=r^2\\\\3600 + k^2 = r^2\\\\k^2 = r^2- 3600[/tex]
Substitute point (39, 77) into this equation:
[tex](39)^2+(77-k)^2=r^2\\\\1521+5929-154k+k^2 = r^2\\\\7450-154k+k^2 = r^2[/tex]
Now, plug in k² = r² - 3600 and solve for k:
[tex]7450 - 154k + r^2 - 3600 = r^2\\\\-154k = -3850\\\\k = 25[/tex]
Therefore, the equation of the circle is now:
[tex]x^2+(y-25)^2=r^2[/tex]
Finally, substitute point (60, 0) into the equation and solve for r:
[tex]60^2+(0-25)^2=r^2\\\\60^2+25^2=r^2\\\\3600+625=r^2\\\\4225=r^2\\\\r=\sqrt{4225}\\\\r=65[/tex]
Therefore, the radius of the circle, and thus the radius of the sphere, is 65 cm.
