A volunteer tutor wants to know HOW to solve this problem. It is not
necessary to give the answer if you have explained it so that a
student can get it...
A sphere rests upon a plane. At a certain time of day, its shadow
extends 10m from the point where the sphere touches the plane. The
sphere is instantly replaced by a vertical meter stick resting upon
the same spot. If the stick casts a 2m shadow, what is the radius of
the sphere?

This answer is incorrect.

A sphere will cast a longer shadow than a stick of the same height.

Here is how to solve this (Disclaimer: it has been a long time since I
took geometry, so this may not be entirely clear or correct -
hopefully it will be somewhat helpful though).

First, solve for the angles of the triangle made up of the meter stick
and its shadow.

Next, imagine the triangle made up of the sphere and its shadow.  The
vertical side of this triangle extends above the top of the sphere and
the hypotenuse touches the sphere somewhere on its side.  Using the
angles you obtained from the first step, you can determine that the
vertical side of this triangle is 5m.

Next, imagine the triangle that connects the center of the sphere, the
point where the hypotenuse of the last triangle touches the sphere,
and the top of the last triangle.  One of the sides of this triangle
is equal to the radius of the sphere and the hypotenuse is equal to
the radius of the sphere plus the distance between the top of the
sphere and the top of the last triangle.  Solve for the ratio between
the hypotenuse and the radius of the circle.  As the hypotenuse plus
the radius of the sphere is equal to 5m, you can use this ratio to
solve for the radius.

The answer I obtained was 1.545m.

badge, I agree with you that the answer given by hibiscus is
incorrect. I also agree with your proper technique to solving the
problem. However, being too lazy to solve the triangles, I constructed
the problem using CAD. The radius I get (which I believe to be
correct) is 2.36m. Do you agree?

10*tan(.5*atan(.5))=2.3606797749978972

1. Calculate the angle of the triangle formed by the meter stick's
shadow: atan(.5) This is triangle 1.

2a. Draw a triangle: There's a new triangle formed by the sphere's
shadow. This is triangle 2a--it is similar to triangle 1.
2b. Draw a line inside: The radius of the sphere touches a point on
the hypotenuse of this triangle. Draw a line from this point to the
center of the sphere.
2c. Draw another line inside: Draw a line from the end of the shadow
on the plane (not part that touches the sphere) the to the center of
the sphere.

3. There should now be 3 triangles inside the triangle formed by the
sphere's shadow.
3a. The triangle formed by line (2b) is a similar to this triangle.
This is triangle 3a.
3bc. There are two triangles formed by line (2c). They are mirrors of
each other. The shortest side of each of these triangles is the radius
of the sphere. The top triangle is 3b. The bottom triangle is 3c.

4a. Consider the vertical line along the vector from the ground
touching the sphere and its center. The angle this line forms with
triangle 3a at the center of the sphere is atan(.5) by similar
triangles (1->2a->3a).
4b. Triangles 3b and 3c occupy the rest of the hemisphere: Pi-atan(.5)
4c. Each triangle occupies exactly half this angle (mirror triangles):
.5*(Pi-atan(.5))
4d. Solve for the other angle in triangle 3c:
2*Pi = .5*Pi + .5*(Pi-atan(.5)) + angle
angle = Pi - .5*Pi - .5*(Pi-atan(.5))
angle = Pi - .5*Pi - .5*Pi - .5*atan(.5)
angle = Pi - Pi - .5*atan(.5)
angle = .5*atan(.5)

5. Solve for the height of triangle 3c:
tan(.5*atan(.5)) = y/10
y = 10*tan(.5*atan(.5))

The height is approximately 2.36 meters.

Oops made a mistake... The correction should be:

4d. Solve for the other angle in triangle 3c:
Pi = .5*Pi + .5*(Pi-atan(.5)) + angle

There are only Pi interior angles in a triangle (not 2Pi).

Proper job synk.