Question

projectile an arrow is launched upwards. the height of the arrow, in meters, after t seconds can be modeled by the function (h(t)= - 4.9t^{2}+100t + 1). after how many seconds will the arrow first reach an altitude of 100 meters? round to the nearest hundredth of a second.

a) 1.04 seconds

b) 2.09 seconds

c) 19.37 seconds

d) 20.42 seconds

Explanation:

Step1: Set up the equation

Set $h(t)=100$, so $- 4.9t^{2}+100t + 1=100$. Rearrange it to the standard - form of a quadratic equation $ax^{2}+bx + c = 0$. We get $-4.9t^{2}+100t - 99 = 0$. Here, $a=-4.9$, $b = 100$, and $c=-99$.

Step2: Use the quadratic formula

The quadratic formula is $t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Substitute the values of $a$, $b$, and $c$ into the formula:
\[

$$\begin{align*} t&=\frac{-100\pm\sqrt{100^{2}-4\times(-4.9)\times(-99)}}{2\times(-4.9)}\\ &=\frac{-100\pm\sqrt{10000 - 1940.4}}{-9.8}\\ &=\frac{-100\pm\sqrt{8059.6}}{-9.8}\\ &=\frac{-100\pm89.775}{-9.8} \end{align*}$$

\]

Step3: Calculate the two solutions

We have two solutions for $t$:
$t_1=\frac{-100 + 89.775}{-9.8}=\frac{-10.225}{-9.8}\approx1.04$
$t_2=\frac{-100 - 89.775}{-9.8}=\frac{-189.775}{-9.8}\approx19.37$
We want the first - time the arrow reaches the altitude, so we take the smaller value of $t$.

Answer:

A. 1.04 seconds