### Video Transcript

The image below shows a box of
lenses and prisms of different sizes. How many concave lenses are there
in the box?

In this question, we are given an
image of a box that contains 11 different lenses and prisms with varying shapes and
sizes. We are asked to look through this
group of lenses and prisms and find out how many concave lenses are in this
group.

Let’s start by recalling what
exactly a concave lens is and how we can identify them. To understand the shape of a
concave lens, imagine we are starting with a cylinder with the circular ends along
the horizontal axis. Now, imagine that we take two
spheres represented by circles in this image and overlap them with our cylinder with
the edges of the circle lining up with the corners of the cylinder like shown. If we remove this overlapping
material from the cylinder, we will get the shape of a concave lens.

Notice that a concave lens is
thickest at the edges of the lens and thinnest in the middle of the lens. So, when we look at the objects in
the box, we are looking for two attributes, curved surfaces and being thicker at the
edges and thinner in the middle. With this in mind, let’s take a
look at these lenses and prisms and find out how many here are concave.

Starting in the top-left corner, we
have our first lens to look at. This lens is thinner along the
middle and thicker on the edges with curved surfaces, which means this is a concave
lens. Moving to the right, we have our
second object. This one is the same thickness
along the whole shape and is distinctly in the shape of an S. So this is not concave. Looking now at object three, we see
that it is thinner on the edge and is made up of straight surfaces. So this is not what we want.

Starting on a new line for the
fourth object, we see that it is not only shaped like a triangle, but it also has no
curved surfaces, so it is not a concave lens. Object five is thinner on the edges
and thicker in the middle, but it has curved surfaces, meaning that this is our
first convex lens of the group. Now, object six has thicker edges,
a thin middle, and the surface is curved. So everything checks out, and we
have our second concave lens. Now to object seven, it is thinner
on the edge and its straight edges make it a distinct diamond. So it is not what we are looking
for.

Going below to the eighth object,
its surfaces are curved, but they are thicker along the edge and thinner in the
middle, meaning this is a convex, not concave, lens. This is the same for object
nine. It is thinner on the edge and
thicker in the middle, meaning it is a convex lens.

Now, moving to object 10, we can
see that it is thicker on the edge and thinner in the middle. So it may at first appear to be
concave, but notice that the surface is made up of straight lines. Since they are not curved, it
cannot be a concave lens. Our final object, number 11, is
thinner on the edge and thicker in the middle and has a curved surface, meaning this
is another convex lens. And that’s all of the objects in
the diagram.

So, out of these 11 lenses and
prisms in the box, there are a total of two concave lenses.