Question 50986
Step 1.
Rewrite your equation in the form of a quadratic function.
{{{h(t) = -16t^2 + rt}}} This is function for the height, h, of an object propelled upwards as a function of time, t. Here, r is given as 180 ft/sec. This fits the situation of Mark's rocket. You are being asked to find at what time, t, will the height, h, be 464 feet. Because your equation is quadratic, you can expect to get two solutions.

Step 2.  Set h(t) = 464 in the quadratic equation above and solve for t.

{{{464 = -16t^2+180t}}} Put this into standard form by subtracting 464 from both sides.
{{{-16t^2+180t-464 = 0}}}

Step 3. Solve the quadratic equation for t by the most convenient method. You can simplify this equation a bit by dividing through by -4 to get:
{{{4t^2-45t+116 = 0}}} Now you can factor this.
{{{(t-4)(4t-29) = 0}}} Apply the zero product principle:
{{{t-4 = 0}}} and/or {{{4t-29 = 0}}}
If {{{t-4 = 0}}} then t = 4
If {{{4t-29 = 0}}} then {{{t = 29/4}}} or {{{t = 7.25}}}

The two solution are:
The height of the rocket will reach 464 feet in 4 seconds, ascending.
The height of the rocket will be at 464 feet again in 7.25 seconds descending.