Grade 11 Jar Question
We have 2 glass jars, both of which contain 375 mL of jam.
The base of one is a regular hexagon, the other is a circle.
The bases have the same area.
Which jar requires less glass to make the sides if the
thickness of the glass for both jars is the same?
Since both the volumes and the base areas are equal, the height h of both
jars must be equal also. The solution then is simply to figure out whether
the area of the 6 rectangles (the sides of the hexagonal jar) is greater or less
than the area of the cylinder (the sides of the circular jar).
We know that:
By equating these, we can find the ratio of r (the radius of the circular base) to s (the side
of the hexagon). Then we can decide the relative size of 6sh and , the lateral areas.
Solution 1 -- just variables:
The interior angles in a hexagon = 120°.
When we draw the apothem and the radius,
we get a 30°, 60°, 90° triangle, and we know the ratio of the sides.
If we set s = the side of the hexagon, we know the apothem, . (as shown)
Find the area of the hexagonal base, set it = the area of a circle with radius r.
Since we know this must be the same as , we can find r in terms of s.
Compare the values of 6sh (the area of the 6 rectangles) with , (the area of the cylinder).
Solution 2 -- with numbers:
Since the interior angles in a hexagon = 120°,
when we draw the apothem and the radius,
we get a 30°, 60°, 90° triangle, for which we know the ratio of the sides.
If we set s the side of the hexagon = 100 units we know the apothem. (as shown)
Find the area of the hexagonal base, then find r, the radius of a circle of equal area.
This means that r², (the square of the circle's radius) is 25980.76211 ÷ = 8269.93...
Compare 600h (the area of the 6 rectangles) with , the lateral area of the cylinder.
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