Question

“Count Every Particle” is a game conducted by the teachers and students to generate funds for the victims of Typhoon. The game was, the participant will buy a ticket so that they have chance to guess the exact number of particles of the sand that is placed in a vase. The following hints were given:
(1) It is assumed that the volume of every particle is 6.4π x 10^-12 cm^3.
(2) The height of the vase is 32 cm; and
(3) The equation of the curve that the vase made of is given by y=x^2 - 6 wit domain x≥ -2.
To win this game, calculate the exact number of particles that can be placed in the vase.

Answer 1

(1) $V_1=6.4\pi\times10^{-12}$ cm³

(2) $h=32$ cm

(3) $y=x^2-6$

$$y(-2)=(-2)^2-6=-2$$

$$-2+32=30$$

$$x^2-6=30=>x=\pm6$$

$$x=-\sqrt{y+6}$$

$$V=\pi\displaystyle\int_{-2}^{30}(-\sqrt{y+6})^2dy=\pi\bigg[\dfrac{y^2}{2}+6y\bigg]\begin{matrix} 30 \\ -2 \end{matrix}=640\pi(cm^3)$$

$$N=\dfrac{V}{V_1}=\dfrac{640\pi\ cm^3}{6.4\pi\times10^{-12}\ cm^3}=10^{14}$$

The exact number of particles that can be placed in the vase is $10^{14}.$