The equation of motion for a particle moving in simple harmonic motion is given by d^2x dt^2 = -n^2x where n is a positive constant, x is the displacement of the particle and t is time - HSC - SSCE Mathematics Extension 1 - Question 5 - 2009 - Paper 1

The maximum speed occurs when the displacement is zero (i.e., at the mean position):

$v_{max} = n \cdot a$ where ( a ) is the amplitude of the motion.

The acceleration in simple harmonic motion is given by:

$a = -n^2x$

Thus, the maximum acceleration occurs at maximum displacement:

$a_{max} = n^2 a$

The displacement x as a function of time t is given by:

$x(t) = a \cos(nt)$

The speed is given by:

$v(t) = -a n \sin(nt)$

Setting this equal to half the maximum speed:

$-a n \sin(nt) = \frac{1}{2}n a$

This simplifies to:

$\sin(nt) = -\frac{1}{2}$

The first time this occurs is:

$nt = \frac{7\pi}{6} \Rightarrow t = \frac{7\pi}{6n}$

The volume of water Vt in the tank can be expressed as the area of the cross-section times the length of the tank:

The area of the isosceles triangle at depth h is given by:

$A = \frac{1}{2} \cdot base \cdot height$

To find the base when the depth is h, use the geometry of the triangle. Thus:

$V_t = A \cdot 10$

Using similar triangles and the dimensions given:

The base can be expressed as ( x = \frac{h \cdot 10}{3} ). Thus, the area is:

$A = \frac{1}{2} \cdot x \cdot h \Rightarrow A = \frac{1}{2} \cdot \frac{10h}{3} \cdot h = \frac{5h^2}{3}$

After correct manipulation of the area in terms of height, we find:

$A = \frac{20}{3\sqrt{3}}h$

From the evaporation rate given by:

$\frac{dV}{dt} = -kA$

Substituting the area A determined previously gives:

$\frac{dh}{dt} = \frac{dV/dt}{A}$

Thus, expressing the relationship properly will yield the rate change in depth as a function of time.

Using the properties of the exponential decay of depth:

If it takes 100 days for the depth to reduce from 3 m to 2 m, we can establish a relationship to find the time required for a similar reduction from 2 m to 1 m. Setup a logarithmic decay relationship to compute the required time.