A geyser erupts, shooting water into the air with an initial upward velocity of $29.4\,\text{meters per second}$ . Therefore, the height of a single water droplet above the ground in meters, $t$ seconds after it comes out of the geyser, can be modeled by the expression $-4.9t^2 + 29.4t$ . This expression can be written in factored form as $-4.9t(t - 6)$ . What does the number $6$ represent in the expression? $\newline$ (A)the height in meters of a water droplet when it comes out of the geyser $\newline$ (B)the height in meters of a water droplet when it reaches its highest point $\newline$ (C)the time in seconds from when a water droplet comes out of the geyser until it hits the ground $\newline$ (D)the time in seconds from when a water droplet comes out of the geyser until it reaches its highest point

Full solution

Q. A geyser erupts, shooting water into the air with an initial upward velocity of

29.4\,\text{meters per second}

. Therefore, the height of a single water droplet above the ground in meters,

t

seconds after it comes out of the geyser, can be modeled by the expression

-4.9t^2 + 29.4t

. This expression can be written in factored form as

-4.9t(t - 6)

. What does the number

6

represent in the expression?

\newline

(A)the height in meters of a water droplet when it comes out of the geyser

\newline

(B)the height in meters of a water droplet when it reaches its highest point

\newline

(C)the time in seconds from when a water droplet comes out of the geyser until it hits the ground

\newline

(D)the time in seconds from when a water droplet comes out of the geyser until it reaches its highest point

Height Expression Factoring: The expression for the height of the water droplet is $-4.9t^2 + 29.4t$ , which can be factored as $-4.9t(t - 6)$ .
Meaning of Number $6$ : To find the meaning of the number $6$ , we look at the factored form $-4.9t(t - 6)$ . The roots of the quadratic equation represent the times when the height is zero, i.e., when the droplet is at the ground level.
Roots of Quadratic Equation: One root is $t = 0$ , which is when the droplet is just coming out of the geyser. The other root is $t = 6$ , which is the other time when the droplet will be at ground level, meaning it's the total time the droplet is in the air.
Highest Point Calculation: Since the droplet starts at ground level and ends at ground level, the highest point will be at half the total time. So, $6$ seconds is the total time, and half of that is $3$ seconds, which is when the droplet reaches its highest point.