Levi kicks a soccer ball up into the air with an initial upward velocity of $14.7$ meters per second. Therefore, the ball's height above the ground in meters, $t$ seconds after it is kicked, can be modeled by the expression $-4.9t^2 + 14.7t$ . This expression can be written in factored form as $-4.9t(t - 3)$ . $\newline$ What does the number $3$ represent in the expression? $\newline$ (A)the height in meters of the ball when it is kicked $\newline$ (B)the time in seconds from when the ball is kicked until it hits the ground $\newline$ (C)the time in seconds from when the ball is kicked until it reaches its highest point $\newline$ (D)the height in meters of the ball when it reaches its highest point

Full solution

Q. Levi kicks a soccer ball up into the air with an initial upward velocity of

14.7

meters per second. Therefore, the ball's height above the ground in meters,

t

seconds after it is kicked, can be modeled by the expression

-4.9t^2 + 14.7t

. This expression can be written in factored form as

-4.9t(t - 3)

\newline

What does the number

3

represent in the expression?

\newline

(A)the height in meters of the ball when it is kicked

\newline

(B)the time in seconds from when the ball is kicked until it hits the ground

\newline

(C)the time in seconds from when the ball is kicked until it reaches its highest point

\newline

(D)the height in meters of the ball when it reaches its highest point

Expression Analysis: The expression for the ball's height is $-4.9t(t - 3)$ . To understand what the number $3$ represents, let's look at the factored form of the quadratic equation.
Factored Form: The factored form indicates the roots of the equation, which are the values of $t$ where the height is zero. The roots are $t = 0$ and $t = 3$ .
Root Interpretation: The root $t = 0$ represents the time when the ball is kicked. The root $t = 3$ represents another point in time when the height of the ball is zero, which is when the ball hits the ground.
Vertex Calculation: Since the ball starts at ground level and returns to ground level, the highest point will be exactly in the middle of the time interval between the two roots. This is the vertex of the parabola, which occurs at $t = \frac{3}{2}$ or $1.5$ seconds.
Significance of $3$ : However, the question asks what the number $3$ represents. Since $3$ is the time when the ball hits the ground, it is not the time it takes to reach the highest point. The number $3$ represents the total time the ball is in the air before it hits the ground.