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The inside of a rectangular carton is 48 centimeters long, 32 centimeters wide, and 15 centimeters high. The carton is filled to capacity with k identical cylindrical cans of fruit that stand upright in rows and columns, as indicated in the figure above. If the cans are 15 centimeters high, what is the value of k?
(1) Each of the cans has a radius of 4 centimeters.
(2) Six of the cans fit exactly along the length of the carton.
We are given that a rectangular carton has a length of 48 cm, a width of 32 cm, and a height of 15 cm. We are also given that there are k identical cylindrical cans standing upright in this carton, and each can has a height of 15 cm. We need to determine the value of k, or the total number of cans in the carton.
Because we have the dimensions of the carton, if we are able to determine the diameter of each can, then we will be able to determine the value of k.
Statement One Alone:Each of the cans has a radius of 4 centimeters.
Since the diameter is twice the radius we know that the diameter of each can is 8 cm. This is enough information to determine how many cans fit in the carton. Although we do not have to determine the actual value of k (because this is a data sufficiency problem), let’s determine it anyway.
Since the length of the carton is 48 cm we could fit 48/8 = 6 cans along the length of the carton, and since the width of the carton is 32 cm, we could fit 32/8 = 4 cans along the width of the carton.
Thus the carton could hold a total of 6 x 4 = 24 cans. Statement one alone is sufficient to answer the question. We can eliminate answer choices B, C and E.
Statement Two Alone:Six of the cans fit exactly along the length of the carton.
In a similar fashion to the process used in statement one, we can use the information in statement two to determine the diameter of each can.
(Length of carton)/(diameter of each can) = number of cans that fit along the length of the carton
48/d = 6
48 = 6d
d = 8
Since we have the diameter of each can, we have enough information to determine how many total cans fit in the carton. Statement two alone is also sufficient to answer the question.
The answer is D.
Note that because we have the length of the carton as 48 centimeters, statements 1 and 2 are giving us the same information – the length of the diameter or radius of each can. Whenever the two statements give the same information, the answer has to be D or E. It can’t be A or B because if one statement is sufficient (or insufficient), the other is too. It can’t be C because neither statement is adding information to the other. So as soon as we realized that A was sufficient in this question, we would know the answer is D.
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