Video Transcript
The following combinations of
shapes all have the same total volume. Which has the greatest total
surface area?
In chemistry, the surface area of
reactants impacts the rate of reaction. We can compare the same amount of
reactant in two different forms: one where the reactant is in a large piece, so the
amount of particles on the surface is relatively small. In the other, the reactant is
ground up, perhaps into a fine powder, which increases the amount of particles
exposed to the surface. The reactant in one piece will have
a slower rate of reaction because only the particles on the surface can immediately
react. When the reactant is ground up into
smaller pieces, more particles are available to react.
Based on these pictures, we can
guess that answer choice (B) will have the greatest surface area because there is
the greatest number of shapes. But we can do some math to confirm
our answer. Let’s start off by calculating the
surface area of this cube. The cube has sides of 10
centimeters. So one face of the cube has an area
of 100 centimeters squared. A cube has six faces. So the total surface area will be
six times the area of one face. That gives us 600 centimeters
squared.
Answer choice (C) has two shapes
instead of one. Let’s see if we can figure out the
pattern here by calculating the surface area for these shapes. There are two different size faces
in the shape. One size face has sides of 10
centimeters and 10 centimeters. The area of these faces is 100
centimeters squared. The other faces have sides of five
centimeters and 10 centimeters. So these faces have an area of 50
centimeters squared.
Now let’s calculate the total
surface area for these shapes. Each shape has two faces that have
an area of 100 centimeters squared. And each shape has four faces that
have an area of 50 centimeters squared. So the surface area of one of the
shapes is 400 centimeters squared. There are two of these shapes. So we need to multiply the surface
area we just calculated by two to get the total surface area for both of the
shapes. This means the total surface area
for the two shapes is 800 centimeters squared.
If we were to calculate the surface
area of the remaining shapes, we would find that the surface area for the eight
cubes in answer choice (B) is 1200 centimeters squared. The total surface area for the
shapes in answer choice (D) is 1000 centimeters squared. So we can see the trend here is
that the surface area increases when there is a larger number of shapes.
So we’ve confirmed our answer. The combination of shapes that has
the greatest total surface area is the one with the greatest number of shapes, which
is the eight cubes shown in answer choice (B).
