course Mth 271
If a sand pile 4 meters high has a mass of 148000 kg, then what would we expect to be the mass of a geometrically similar sand pile 18 meters high? Using the differential, estimate the mass of sand required to increase the height of the pile from 4 meters to 4.06 meters.
The given formula for the sand pile (taken from Class Notes 6) is y =ax^3.
y represents the volume
x represents the diameter
To find the value of a, I substituted the values for y and x into the equation: y = 4, x = 148000
y = ax^3
4 = a(148000)^3
a = 4/(148000^3) = 1.124*10^-15
I used this value of a and the given value of y to find the value of x:
y = ax^3
18 = (1.124*10^-15)x^3
18/(1.124*10^-15) = x^3
(1.456*10^16)^(1/3) =( x^3)^(1/3)
x = 244185.8299 kg
To find the estimate of mass of sand, I took the derivative of the original problem:
y = ax^3
‘dy/dx = 3x^2’dx
I then used x = 4 and ‘dx as 4.06 – 4 = .06:
‘dy/dx = (3(4)^2)(.06)
‘dy = 2.88
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Nicely done.