course Mth 271
Solve using proportionalities by stating the appropriate proportionality law and finding the proportionality constant: •If a sand pile 2.7 meters high has a mass of 8660.521 kg, then what would we expect to be the mass of a geometrically similar sand pile 6.7 meters high?
• If there are 1.5309 billion grains of sand exposed on the surface of the first sand pile, how many grains of sand we expect to be exposed on the surface of the second?
To find the mass of the second sand pile, I used the equation y = ax^3 and solved for a using the values given for the first sand pile.
y = ax^3
y1 = 2.7
x1 = 8660.521kg
2.7 = a(8660.521kg)^3
2.7/(8660.521^3) = a = 4.156*10^-12
y2 = 6.7
6.7 = (4.156*10^-12)x^3
6.7/(4.156*10^-12) = x^3
(1.612*10^12)^(1/3) = (x^3)^(1/3)
x = 11725.04 kg
For the second part, I used the value 1.5309*10^9 as the x value and the same values of y to find the new value of a, which would represent a constant:
2.7 = a(1.5309*10^9)^3
2.7/(3.5819*10^27) = a = 7.525*10^-27
From here I used the second value of y and value I found for a to find the x of the second sand pile:
6.7 = (7.525*10^-27)x^3
6.7/(7.525*10^-27) = x^3
(8.9033*10^26)^(1/3) = (x^3)^(1/3)
x = 9.62*10^8 grains of sand
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The surface of a sandpile can be covered by tiny squares, which scale only as the second power of the linear dimension. So the proportionality appropriate to this question about the surface would be y = k x^2.
Your procedures are otherwise correct.