A high-speed laser scanner rotates on a pivot 8 cm from a straight detector rail. The displacement h h\,h cm of the laser spot from the center of the rail is modelled by the function h(t)=8tan(πt15)\displaystyle h(t) = 8 \tan \left( \frac{\pi t}{15} \right)h(t)=8tan(15πt), where t t\,t is the time in microseconds. A digital cursor moves along the same rail with displacement z z\,z cm.
State the period of h(t)h(t)h(t).
Write down the number of points at which the laser spot and the cursor coincide (the number of roots of h(t)=z(t)h(t) = z(t)h(t)=z(t)) in the following cases:
(i) z(t)=8πt3\displaystyle z(t) = \frac{8\pi t}{3}z(t)=38πt in the interval −15<t<15-15 < t < 15−15<t<15
(ii) z(t)=4πt3\displaystyle z(t) = \frac{4\pi t}{3}z(t)=34πt in the interval −60<t<60-60 < t < 60−60<t<60
(iii) z(t)=4πt3\displaystyle z(t) = \frac{4\pi t}{3}z(t)=34πt in the interval −1500<t<1500-1500 < t < 1500−1500<t<1500
