Answer
$(0,\frac{1}{2})$; between 0 and 0.5 seconds.
Work Step by Step
Step 1. The cliff is 87 feet above the sea level; for the driver to be above the height of the cliff, we have $s(t)\gt87$
Step 2. With the given function, we have $-16t^2+v_0t+s_0\gt87$
Step 3. The initial conditions are $v_0=8\ ft/sec$ and $s_0=87\ ft$; we have $-16t^2+8t+87\gt87$, which gives $16t^2-8t\lt0$
Step 4. Factoring the left side, we have $8t(2t-1)\lt0$ and the boundary points are $t=0, 1/2$
Step 5. With $t\geq0$, use the test points to examine signs of the left side across the boundary points. We have
$(0)...(-)...(1/2)...(+)...$
Thus the solutions are $(0,\frac{1}{2})$; that is, between 0 and 0.5 seconds.
