benchmarks.wiki / Public workspace
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Omni-MATH / A wealthy king has his blacksmith fashion him a large cup, whose inside is a…
Problem
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problem
A wealthy king has his blacksmith fashion him a large cup, whose inside is a cone of height 9 inches and base diameter 6 inches. At one of his many feasts, he orders the mug to be filled to the brim with cranberry juice. For each positive integer , the king stirs his drink vigorously and takes a sip such that the height of fluid left in his cup after the sip goes down by inches. Shortly afterwards, while the king is distracted, the court jester adds pure Soylent to the cup until it's once again full. The king takes sips precisely every minute, and his first sip is exactly one minute after the feast begins. As time progresses, the amount of juice consumed by the king (in cubic inches) approaches a number . Find .
Plain-text mathematical notation (without MathML)
A wealthy king has his blacksmith fashion him a large cup, whose inside is a cone of height 9 inches and base diameter 6 inches. At one of his many feasts, he orders the mug to be filled to the brim with cranberry juice. For each positive integer n, the king stirs his drink vigorously and takes a sip such that the height of fluid left in his cup after the sip goes down by (1)/(n²) inches. Shortly afterwards, while the king is distracted, the court jester adds pure Soylent to the cup until it's once again full. The king takes sips precisely every minute, and his first sip is exactly one minute after the feast begins. As time progresses, the amount of juice consumed by the king (in cubic inches) approaches a number r. Find r.
Original LaTeX notation
A wealthy king has his blacksmith fashion him a large cup, whose inside is a cone of height 9 inches and base diameter 6 inches. At one of his many feasts, he orders the mug to be filled to the brim with cranberry juice. For each positive integer $n$, the king stirs his drink vigorously and takes a sip such that the height of fluid left in his cup after the sip goes down by $\frac{1}{n^{2}}$ inches. Shortly afterwards, while the king is distracted, the court jester adds pure Soylent to the cup until it's once again full. The king takes sips precisely every minute, and his first sip is exactly one minute after the feast begins. As time progresses, the amount of juice consumed by the king (in cubic inches) approaches a number $r$. Find $r$.Discussion
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Source and history
- task_id
3c5cab04-ae00-50d1-9bcf-7cb40d571990- task_key
test--3c5cab04-ae00-50d1-9bcf-7cb40d571990- task_revision_id
- 1
- config / split
- / test
initial import