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MMLU-Pro / 8322 / A planning engineer for a new alum plant must present some estimates to his…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

question

A planning engineer for a new alum plant must present some estimates to his company regarding the capacity of a silo designed to contain bauxite ore until it is processed into alum. The ore resembles pink talcum powder and is poured from a conveyor at the top of the silo. The silo is a cylinder 100ft100 \mathrm{ft} high with a radius of 200ft200 \mathrm{ft}. The conveyor carries ore at a rate of $60,000 \pi \mathrm{~ft}^3 / \mathrm{h}$ and the ore maintains a conical shape whose radius is 1.5 times its height. If, at a certain time tt, the pile is 60ft60 \mathrm{ft} high, how long will it take for the pile to reach the top of the silo?
Plain-text mathematical notation (without MathML)
A planning engineer for a new alum plant must present some estimates to his company regarding the capacity of a silo designed to contain bauxite ore until it is processed into alum. The ore resembles pink talcum powder and is poured from a conveyor at the top of the silo. The silo is a cylinder 100ft high with a radius of 200ft. The conveyor carries ore at a rate of $60,000 \pi \mathrm{~ft}^3 / \mathrm{h}$ and the ore maintains a conical shape whose radius is 1.5 times its height. If, at a certain time t, the pile is 60ft high, how long will it take for the pile to reach the top of the silo?
Original LaTeX notation
A planning engineer for a new alum plant must present some estimates to his company regarding the capacity of a silo designed to contain bauxite ore until it is processed into alum. The ore resembles pink talcum powder and is poured from a conveyor at the top of the silo. The silo is a cylinder $100 \mathrm{ft}$ high with a radius of $200 \mathrm{ft}$. The conveyor carries ore at a rate of $60,000 \pi \mathrm{~ft}^3 / \mathrm{h}$ and the ore maintains a conical shape whose radius is 1.5 times its height. If, at a certain time $t$, the pile is $60 \mathrm{ft}$ high, how long will it take for the pile to reach the top of the silo?

category

math

options

  1. 10.5 $\mathrm{h}$
  2. 15.7 $\mathrm{h}$
  3. 16.4 $\mathrm{h}$
  4. 9.8 $\mathrm{h}$
  5. 13.3 $\mathrm{h}$
  6. 14.0 $\mathrm{h}$
  7. 8.5 $\mathrm{h}$
  8. 11.2 $\mathrm{h}$
  9. 12.6 $\mathrm{h}$
  10. 7.2 $\mathrm{h}$

src

scibench-calculus

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Source and history

upstream_id / Original problem ID
8322
task_id
50592853-14a5-53ec-bc6d-cd3c25f2dec3
task_key
default--test--8322
task_revision_id
1
config / split
default / test

Official source

initial import