A camera films the motion of a swing in a park - HSC - SSCE Mathematics Advanced - Question 26 - 2023 - Paper 1

To find the expression for $x(t)$, we need to integrate the rate of change of $x$:

$x(t) = \int -1.5\sin\left(\frac{5\pi}{4}t\right) dt$

Using the antiderivative of sine, we have:

$x(t) = -1.5\left(-\frac{4}{5\pi}\cos\left(\frac{5\pi}{4}t\right)\right) + k$

This simplifies to:

$x(t) = \frac{6}{5\pi}\cos\left(\frac{5\pi}{4}t\right) + k$

Now we need to find the constant $k$. When $t = 0$, $x(0) = 11.2$:

$11.2 = \frac{6}{5\pi}\cos(0) + k$
$11.2 = \frac{6}{5\pi} + k$

Thus, $k = 11.2 - \frac{6}{5\pi}$

Putting it all together, we have:

$x(t) = \frac{6}{5\pi}\cos\left(\frac{5\pi}{4}t\right) + 11.2 - \frac{6}{5\pi}$