Final answer:
To calculate the cross-sectional area of the ball where the plane is 15 cm from the center, use the Pythagorean theorem to find the radius of the circle formed by the cross-section, then apply the circle area formula, yielding a cross-sectional area of 64π cm².
Step-by-step explanation:
To find the cross-sectional area of a ball when it is cut by a plane that is located at a certain distance from its center, you can use the Pythagorean theorem on the resulting circle formed from the cross-section. Since the radius of the ball is 17 cm and the plane is 15 cm from the center, we can imagine a right triangle with one leg as the radius of the new circle (we'll call this r), the other leg as the distance from the plane to the center of the sphere (15 cm), and the hypotenuse as the radius of the sphere (17 cm).
Using the Pythagorean theorem, we can find the length of the leg representing the radius of the new circle (r):
152 + r2 = 172
r2 = 172 - 152
r2 = 289 - 225
r2 = 64
r = 8 cm
Now that we have the radius of the circle (r = 8 cm), we can find the cross-sectional area (A) by using the formula A = πr2:
A = π(8 cm)2
A = π(64 cm2)
A = 64π cm2
Therefore, the cross-sectional area of the ball at the plane 15 cm from the center is 64π cm2.