Problem 9
1 liter = 1000 cubic cm
The base of the carton has area 50 cm^2, or 50 square cm.
Let A = 50 to represent the area.
The height h is unknown. It multiplies with the value of A to get the volume
V = A*h
1000 = 50*h
50h = 1000
h = 1000/50
h = 20
Answer: The carton is 20 cm tall
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Problem 10
Part (a)
- Along the 4 cm side of the box, we can fit 4/2 = 2 dice side by side
- Along the 7 cm side of the box, we can fit 7/2 = 3.5 = 3 dice. Note how I rounded down instead of up. Having 4 dice will lead to 4*2 = 8 cm, but 8 is larger than 7.
- Along the 5 cm side of the box we can fit 5/2 = 2.5 = 2 dice
This 3D configuration will allow us to fit 2*3*2 = 12 dice in total
Answer: 12 dice will fit
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Part (b)
The box has volume 5*4*7 = 140 cubic cm
Each die has volume 2*2*2 = 8 cubic cm. There are 12 dice we can fit in, so 12*8 = 96 cubic cm is the amount of volume taken up by the dice
We have 140-96 = 44 cubic cm of empty air left over.
Answer: 44 cubic cm
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Problem 11
1 liter = 1000 cubic cm
8 liters = 8000 cubic cm
So he has 8000 cubic cm of soil
The volume of the cube is
(22.5)^3 = (22.5)*(22.5)*(22.5) = 11,390.625 cubic cm
He won't have enough soil to completely fill the cube shaped planter
We can be able to determine this by recalling that 2^3 = 8, so (20)^3 = 8000. Meaning that a cube shaped planter of sides 20 cm will lead to a volume of 8000 cubic cm. Anything over 20 cm will lead to a larger volume.
So in short, spotting the 22.5 being larger than 20 is a quick way to know that the planter has more volume compared to the amount of soil he has.
Answer: He won't have enough soil to completely fill the planter
