text: # Thread: Related rates problem
  
  1. ## Related rates problem
  
  Hello, i'm not exactly sure how to do this one.
  
  Water is draining for a conical filter into a cylindrical cup at the rate of 10 cubic centimeters per minute. The conical filter measures 10 cm tall and 10 cm across its top. The diameter of the cylindrical cup is 15 cm.
  
  (a) How fast is the level of water in the filter falling when the water in the filter is 8 cm deep?
  
  (b) At the same time as in part a, how fast is the level of water in the cup rising?
  
  I can somewhat get it, but I need a start and some explanation. Thanks a lot.
  
  2. First you can get by triangle analogies that the radius of the water cone is always half the height. Let h be the heigth of this water cone.
  $V = \frac{\pi h^3}{4}$
  $\frac{dV}{dh} = \frac{3 \pi h^2}{4}$
  $\frac{dh}{dV} = \frac{4}{3 \pi h^2}$
  You are given $\frac{dV}{dt} = 10$
  $\frac{dh}{dV}\frac{dV}{dt} = \frac{dh}{dt} = 10\frac{4}{3 \pi h^2}$
  Replace h by what value you want and be careful that your units are consistent.
  Exercice b is quite easier than exercice a
explanation: The document contains a math problem related to related rates, specifically involving a conical filter and a cylindrical cup. The problem is divided into two parts:
  
  1. Determining how fast the level of water in the filter is falling when the water is 8 cm deep.
  2. Determining how fast the level of water in the cup is rising at the same time.
  
  The document also includes a detailed solution for part (a), providing the necessary steps and formulas to solve the problem.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Thread: Related rates problem
  
  1. ## Related rates problem
  
  Hello, i'm not exactly sure how to do this one.
  
  Water is draining for a conical filter into a cylindrical cup at the rate of 10 cubic centimeters per minute. The conical filter measures 10 cm tall and 10 cm across its top. The diameter of the cylindrical cup is 15 cm.
  
  (a) How fast is the level of water in the filter falling when the water in the filter is 8 cm deep?
  
  (b) At the same time as in part a, how fast is the level of water in the cup rising?
  
  I can somewhat get it, but I need a start and some explanation. Thanks a lot.
  
  2. First you can get by triangle analogies that the radius of the water cone is always half the height. Let h be the heigth of this water cone.
  $V = \frac{\pi h^3}{4}$
  $\frac{dV}{dh} = \frac{3 \pi h^2}{4}$
  $\frac{dh}{dV} = \frac{4}{3 \pi h^2}$
  You are given $\frac{dV}{dt} = 10$
  $\frac{dh}{dV}\frac{dV}{dt} = \frac{dh}{dt} = 10\frac{4}{3 \pi h^2}$
  Replace h by what value you want and be careful that your units are consistent.
  Exercice b is quite easier than exercice a
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math problem related to related rates, specifically involving a conical filter and a cylindrical cup. The problem is divided into two parts:
  
  1. Determining how fast the level of water in the filter is falling when the water is 8 cm deep.
  2. Determining how fast the level of water in the cup is rising at the same time.
  
  The document also includes a detailed solution for part (a), providing the necessary steps and formulas to solve the problem.
  
  Label: question-and-answer