(C) When will the​ projectile's height above ground be less than or equal to 480 ft?

Question

If an object is given an initial velocity straight upward of v₀ feet per second from a height of s₀ ​feet, then its altitude s after t seconds is given by the formula s=−16t2+v₀t+s₀. A projectile is launched straight up from ground level with an initial velocity of 272 ​ft/sec.

(C) When will the​ projectile's height above ground be less than or equal to 480 ft?

Answer 1

We are given the formula for the altitude of the projectile as a function of time:

s = -16t2 + v0 t+s₀

where v0 is the initial velocity and s0 is the initial height.

In this case, v0 = 272 ft/sec and s0 = 0 ft (since the projectile is launched from ground level).

We want to find when the projectile's height above ground is less than or equal to 480 ft. In other words, we want to find when s <= 480.

Substituting the values of v0 and s0, we get:

s = -16t^2 + 272t

Now we can set up the inequality:

-16t^2 + 272t <= 480

Simplifying and dividing both sides by -16 (remembering to flip the inequality since we are dividing by a negative number), we get:

t^2 - 17t + 30 >= 0

Now we can factor the quadratic expression:

(t - 2)(t - 15) >= 0

This inequality is true when either both factors are non-negative (i.e. t <= 2 or t >= 15), or both factors are negative (i.e. 2 <= t <= 15).

So the projectile's height above ground will be less than or equal to 480 ft either immediately after launch (when t = 0) or during the time interval [2 sec, 15 sec].

Answer 2

Let the ground be represented by s₀ = 0. Given V₀=272, we need to solve for t. The question asks "when the height is less than or equal to 480 feet", which correlates to when s<=480. Plugging our values in we have the following inequality: -16t²+272t<=480.

Bring all the terms to one side: 16t²-272t+480>=0. Divide by 16 to both sides: t²-17t+30>=0.

Factoring the LHS: (t-15)(t-2)>=0. Solving this inequality, we can see that when t<=2 and t>=15, this inequality holds.

Thus, the projectile's height is less than or equal to 480 ft when t<=2 and t>=15.