Question

A sphere with diameter $50$ cm intersects a plane $14$ cm from the center of the sphere. What is the number of square centimeters in the area of the circle formed?

• A. $49\pi$
• B. $196\pi$
• C. $429\pi$
• D. $576\pi$
• E. $2304\pi$

Answer 1

The correct option is C $429\pi$

Since the radius of the sphere is half of its diameter and it is given that the diameter of the sphere id $50$ cm, therefore, the radius of the sphere is $r=25$ cm.

The distance from the center of the sphere to the intersecting plane lies along the perpendicular.

Use the pythagoras theorem to get:

$r^2+{14}^2={25}^2\Rightarrow r^2+196=625\Rightarrow r^2=625-196\Rightarrow r^2=429$

The area of the circle formed by the plane and sphere is $\pi r^2$, therefore,

$A=\pi r^2=\pi \left(429\right)=429\pi$