Problem of the Week Archive
Do You Want to Build a Snowman? – December 7, 2015
Problems & Solutions
Braden and Venita went outside to play in the snow on their day off of school. They
decided to build a snowman. They rolled three tightly packed snowballs of different
sizes and stacked them, one on top of the other. The three snowballs were
perfectly spherical and their diameters were in the ratio 2 to 3 to 4. If the
smallest snowball had a diameter of 12 inches, what was the height of
the snowman in inches?
There are three snowballs, one small, one medium and one large, whose diameters are in ratio
2 to 3 to 4, which is equivalent to saying 1 to 3/2 to 2. Since we are given the diameter of the
small snowball, we can find the diameters of the medium and large snowballs by multiplying the
diameter of the small snowball by 3/2 and by 2, respectively. So the diameters of the three
snowballs are 12 in, 3/2 × 12 in = 18 in and 2 × 12 in = 24 in. Since the snowballs are
stacked one on top of the other to create the snowman, the height of the snowman will be
12 in + 18 in + 24 in = 54 in.
How many cubic feet of snow were in Braden and Venita’s snowman? Express your answer as a
decimal to the nearest tenth.
The formula for volume of a sphere uses the radius of the sphere and the problem asks for the answer in cubic feet so
first, we find the radius of each snowball in feet. The radii in feet are 12 in ÷ 12 in/ft ÷ 2 = 0.5 ft,
18 in ÷ 12 in/ft ÷ 2 = 0.75 ft and 24 in ÷ 12 in/ft ÷ 2 = 1 ft. The total volume of the snow is the sum of the volumes of
the three spherical snowballs: 4/3 × π × (0.5 ft)3 + 4/3 × π × (0.75 ft)3 + 4/3 × π × (1 ft)3 ≈ 6.5 cubic feet.
Hector and Davena decided they wanted to build a snowman too. They made their snowman twice
the height of Braden and Venita’s snowman. They kept the same proportions when creating the new
snowman. The number of cubic feet of snow in Hector and Davena’s snowman was how many times
that in Braden and Venita’s snowman?
If Hector and Davena doubled the height of the snowman, that means they doubled the radius of each snowball. The ratio
of the volume of a sphere to the volume of a sphere with double the radius is r3 to (2r)3 or r3 to 23r3. Using this ratio we
know the volume of Hector and Davena’s snowman will be 23 = 8 times the volume of Braden and Venita’s snowman.