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Alpha Optics and Beta Vision buy high-precision optical lenses from the same manufacturer. The price of a lens is inversely proportional to the cube of its diameter. Alpha Optics buys 30 lenses, each with a diameter of 6 mm. Beta Vision buys 90 lenses in total: 30 lenses with a diameter of 3 mm, 30 lenses with a diameter of 2 mm, and 30 lenses with a diameter of 1 mm. How many times as much does Beta Vision pay as Alpha Optics?
A. 80
B. 120
C. 180
D. 251
E. 300
A. 80
B. 120
C. 180
D. 251
E. 300
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Official Solution:
Alpha Optics and Beta Vision buy high-precision optical lenses from the same manufacturer. The price of a lens is inversely proportional to the cube of its diameter. Alpha Optics buys 30 lenses, each with a diameter of 6 mm. Beta Vision buys 90 lenses in total: 30 lenses with a diameter of 3 mm, 30 lenses with a diameter of 2 mm, and 30 lenses with a diameter of 1 mm. How many times as much does Beta Vision pay as Alpha Optics?
A. 80
B. 120
C. 180
D. 251
E. 300
Since the price of a lens is inversely proportional to the cube of its diameter, the price of a lens with diameter \(d\) mm can be written as \(\frac{k}{d^3}\), where \(k\) is a constant and \(d\) is the diameter in mm.
Alpha’s total cost is:
\(30 * \frac{k}{6^3} = 30 * \frac{k}{216}\)
Beta’s total cost is:
\(30 * \frac{k}{3^3} + 30 * \frac{k}{2^3} + 30 * \frac{k}{1^3}=\)
\(= 30(\frac{k}{27} + \frac{k}{8} + k)=\)
\(= 30 * \frac{251k}{216}\)
Therefore, \(\frac{Beta}{Alpha} = \frac{30 * \frac{251k}{216}}{30 * \frac{k}{216}} = 251\).
Answer: D
Alpha Optics and Beta Vision buy high-precision optical lenses from the same manufacturer. The price of a lens is inversely proportional to the cube of its diameter. Alpha Optics buys 30 lenses, each with a diameter of 6 mm. Beta Vision buys 90 lenses in total: 30 lenses with a diameter of 3 mm, 30 lenses with a diameter of 2 mm, and 30 lenses with a diameter of 1 mm. How many times as much does Beta Vision pay as Alpha Optics?
A. 80
B. 120
C. 180
D. 251
E. 300
Since the price of a lens is inversely proportional to the cube of its diameter, the price of a lens with diameter \(d\) mm can be written as \(\frac{k}{d^3}\), where \(k\) is a constant and \(d\) is the diameter in mm.
Alpha’s total cost is:
\(30 * \frac{k}{6^3} = 30 * \frac{k}{216}\)
Beta’s total cost is:
\(30 * \frac{k}{3^3} + 30 * \frac{k}{2^3} + 30 * \frac{k}{1^3}=\)
\(= 30(\frac{k}{27} + \frac{k}{8} + k)=\)
\(= 30 * \frac{251k}{216}\)
Therefore, \(\frac{Beta}{Alpha} = \frac{30 * \frac{251k}{216}}{30 * \frac{k}{216}} = 251\).
Answer: D
