Mathematics

Mathematics

Problem

A solid metal ball has its centre at C(3,−2,1)C(3, -2, 1) and diameter 10 cm10\text{ cm}. (i) Write the Cartesian equation of the surface of the ball. (ii) Verify, by direct calculation, whether the point P(7,1,1)P(7, 1, 1) lies inside, on, or outside the surface. (iii) Find the distance of the point Q(0,−2,1)Q(0, -2, 1) from the centre CC and state the same relation (inside/on/outside) for QQ.
Intuitive explanation

A sphere is defined by all points equidistant from its center—that distance is the radius. The key insight is that the Cartesian equation captures this geometric constraint algebraically, and then you can test any point by comparing its distance from the center to the radius. Once you have the equation, checking whether a point is inside, on, or outside is just arithmetic: compute the distance and compare it to the radius value.

Intuitive explanation

A sphere is defined by all points equidistant from its center—that distance is the radius. The key insight is that the Cartesian equation captures this geometric constraint algebraically, and then you can test any point by comparing its distance from the center to the radius. Once you have the equation, checking whether a point is inside, on, or outside is just arithmetic: compute the distance and compare it to the radius value.

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