Question

1. projectile. an arrow is launched upwards. the height of the arrow, in meters, after t sec - onds can be modeled by the function h(t)= - 4.9t² + 100t + 1. after how many seconds will the arrow first reach an altitude of 100 meters? round to the nearest hundredth of a second. 1.04 seconds 2.09 seconds 19.37 seconds 20.42 seconds

Explanation:

Step1: Set up the equation

We are given the height - function $h(t)=-4.9t^{2}+100t + 1$. We want to find $t$ when $h(t) = 100$. So, we set up the equation $-4.9t^{2}+100t + 1=100$.

Step2: Rearrange the equation

Rearrange it to the standard quadratic - form $ax^{2}+bx + c = 0$. We get $-4.9t^{2}+100t+1 - 100 = 0$, which simplifies to $-4.9t^{2}+100t - 99 = 0$. Multiply through by - 1 to get $4.9t^{2}-100t + 99 = 0$. Here, $a = 4.9$, $b=-100$, and $c = 99$.

Step3: 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:

$$
LATEXBLOCK0
$$

Step4: Calculate the two values of t

First, $t_1=\frac{100 + 89.775}{9.8}=\frac{189.775}{9.8}\approx19.37$. Second, $t_2=\frac{100 - 89.775}{9.8}=\frac{10.225}{9.8}\approx1.04$. We are looking for the first time the arrow reaches the height, so we take the smaller value.

Answer:

1.04 seconds