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A designer writes the equation of a spherical tank in metres as x² + y² + z² + 8x - 10y + 2z + 6 =…

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Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

A designer writes the equation of a spherical tank in metres as $x^2 + y^2 + z^2 + 8x - 10y + 2z + 6 = 0$ , with the origin at a reference point and the $xy$ -plane as a level of the ground. Find the three coordinates of the centre of the tank and its radius. Then find the radius of the circle in which the ground plane $z = 0$ cuts the sphere, to two decimal places.

Need a reminder? The key ideaShow

A sphere with centre $(a, b, c)$ and radius $r$ is $(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2$ . In the general form $x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0$ the centre is $(-u, -v, -w)$ and $r^2 = u^2 + v^2 + w^2 - d$ . The plane $z = h$ cuts a circle of radius $\sqrt{r^2 - (h - c)^2}$ .

The worked solution

Have you tried it?

Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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