Calculus

It would be so helpful if someone could check my answers!

A machine starts dumping sand at the rate of 20 m3/min, forming a pile in the shape of a cone. The height of the pile is always twice the length of the base diameter. The volume formula for a right circular cone is V= 1/3pi*r2.*h

A. After 5 minutes, what is the height of the pile?
h = 2d or 4r
v = π/3 r^2 h
v = π/3 (h/4)^2 h
v = π/48 h^3

After 5 minutes, 100m^3 of sand would have been dumped. Therefore:
π/48 h^3 = 100
So 100 = π/48 h^3
48 (100) = (π/48 h^3) 48
4800 = π*h^3
4800/π = h^3
∛(4800/π) = h
approximately 11.51m

B. After 5 minutes, how fast is the height increasing?
dv/dt = π/4 h^2 dh/dt
20 = π/4 ∛(4800/π)^2 dh/dt
dh/dt = 20 * 4/π * ∛(π/4800)^2
(I was having some trouble simplifying this one and the two below it)

C. After 5 minutes, how fast is the base radius increasing?
since h = 4r, dh/dt = 4 dr/dt
4(dr/dt) = 20 * 4/π * ∛(π/4800)^2
dr/dt = (20 * 4/π * ∛(π/4800)^2)/4

D. After 5 minutes, how fast is the area of the base increasing?
base area: A=πr^2,
dA/dt = 2πr dr/dt
2π * (∛(4800/π))/4 * dr/dt
2π ∛(4800/4π) * dr/dt
2π ∛(4800/4π) * (20 * 4/π * ∛(π/4800)^2)/4

  1. 👍 1
  2. 👎 0
  3. 👁 396
  1. I see you fixed my error where I said h=2r.
    (A) looks good
    (B) 20 * 4/π * ∛(π/4800)^2 = 1/∛(45π) ≈ 0.19196
    (C) as you say, dr/dt = 1/4 of dh/dt ≈ 0.04799
    (D) Oops . why did you drag the 4 under the radical?
    2π * (∛(4800/π))/4 * dr/dt
    = π/2 * (∛(4800/π)) * 1/4 * 1/∛(45π) = ∛(5π/24) = 0.8682

    1. 👍 0
    2. 👎 0
  2. Thank you so much for your time and help!

    1. 👍 0
    2. 👎 0

Respond to this Question

First Name

Your Response

Similar Questions

  1. Calculus rate of change

    A machine starts dumping sand at the rate of 20 m3/min, forming a pile in the shape of a cone. The height of the pile is always twice the length of the base diameter. After 5 minutes, how fast is the base area of the base

    asked by Nicole on July 18, 2016
  2. Calculus

    How would I find out what the height of the pile would be after 5 minutes? A machine starts dumping sand at the rate of 20 m3/min, forming a pile in the shape of a cone. The height of the pile is always twice the length of the

    asked by Peter on December 31, 2014
  3. calculus

    A truck starts dumping sand at the rate of 17 ft^3/ min, forming a pile in the shape of a cone. The height of the pile is always 2/3 the base diameter.

    asked by stephanie on January 12, 2020
  4. physics

    43. A block of mass 2.00 kg rests on a ramp tilted at 25.0°. Coefficients of friction for the block/ramp: μs = 0.300, μk = 0.100. A string passing over a high quality pulley connects the block with a bucket as shown below. The

    asked by Devin on February 17, 2015
  5. Ap physics

    fine sand moves through pipes just like fluid. a sandblasting machine moves fine sand through a pipe of inner diameter 2.8 cm at 75L/minutes. the sand comes out from two identical jets at speed 45 m/s. calculated the diameter of

    asked by jay on December 6, 2015
  1. Calculus

    1.Consider the function f(x)= (3/4)x4 - x3 - 3x2 +6x A. Find the relative extrema for f(x); be sure to label each as a maximum or minimum. You do not need to find function values; just find the x-values. B. Determine the

    asked by Lata on December 14, 2018
  2. Math( help please ! )

    a) f(x) = 4.1x +92.16 b) f(x) = 17.9x + 19.36 c) f(x) = 8.8x + 55.4 ANSWERS: a) 4.8 b) 0.93 c) 2.2, i just have no clue how they got these answers EXTRA INFO: The demand function for a new product is p(x)= - 4x +42.5, where x is

    asked by Poppy on October 20, 2011
  3. Math

    a) f(x) = 4.1x +92.16 b) f(x) = 17.9x + 19.36 c) f(x) = 8.8x + 55.4 ANSWERS: a) 4.8 b) 0.93 c) 2.2, i just have no clue how they got these answers EXTRA INFO: The demand function for a new product is p(x)= - 4x +42.5, where x is

    asked by Nancy on October 20, 2011
  4. Math

    a) f(x) = 4.1x +92.16 b) f(x) = 17.9x + 19.36 c) f(x) = 8.8x + 55.4 ANSWERS: a) 4.8 b) 0.93 c) 2.2, i just have no clue how they got these answers EXTRA INFO: The demand function for a new product is p(x)= - 4x +42.5, where x is

    asked by Nancy on October 20, 2011
  5. English/Lit

    I posted this earlier and someone suggested google-I did research various sites but I was wondering if anyone could perhaps check my answers and see if they would be a correct analysis-thank you I have two questions I'm working on

    asked by Sarah on September 18, 2011

More Similar Questions