Question

which value must be added to the expression $x^{2}+16x$ to make it a perfect - square trinomial?
256
64
8
32
6 multiple choice 10 points
a projectile is launched straight up from the ground with an initial velocity of 120 ft/s. if acceleration due to gravity is $-16ft/s^{2}$, after about how many seconds will the object reach a height of 200 ft?
use:
$h(t)=-16t^{2}+120t$
2.5 s
8.0 s
1.0 s
1.7 s

Explanation:

Step1: Set up the equation

We are given the height function \(h(t)=-16t^{2}+120t\) and we want to find \(t\) when \(h(t) = 200\). So we set up the equation \(-16t^{2}+120t=200\).

Step2: Rearrange the equation

Move \(200\) to the left - hand side to get a quadratic equation in standard form \(ax^{2}+bx + c=0\). So, \(-16t^{2}+120t - 200=0\). Multiply through by \(-1\) to make the coefficient of \(t^{2}\) positive: \(16t^{2}-120t + 200 = 0\). Divide through by \(8\) to simplify: \(2t^{2}-15t + 25=0\).

Step3: Use the quadratic formula

For a quadratic equation \(ax^{2}+bx + c = 0\), the quadratic formula is \(t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). Here, \(a = 2\), \(b=-15\), and \(c = 25\). First, calculate the discriminant \(\Delta=b^{2}-4ac=(-15)^{2}-4\times2\times25=225 - 200=25\). Then, \(t=\frac{15\pm\sqrt{25}}{4}=\frac{15\pm5}{4}\).

Step4: Solve for \(t\)

We have two solutions:

• When we take the plus sign: \(t=\frac{15 + 5}{4}=\frac{20}{4}=5\)
• When we take the minus sign: \(t=\frac{15-5}{4}=\frac{10}{4}=2.5\)

Answer:

\(2.5s\)