A geyser erupts, shooting water into the air with an initial upward velocity of 29.4meters 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 −4.9t2+29.4t. This expression can be written in factored form as −4.9t(t−6). What does the number 6 represent in the expression?(A)the height in meters of a water droplet when it comes out of the geyser(B)the height in meters of a water droplet when it reaches its highest point(C)the time in seconds from when a water droplet comes out of the geyser until it hits the ground(D)the time in seconds from when a water droplet comes out of the geyser until it reaches its highest point
Q. A geyser erupts, shooting water into the air with an initial upward velocity of 29.4meters 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 −4.9t2+29.4t. This expression can be written in factored form as −4.9t(t−6). What does the number 6 represent in the expression?(A)the height in meters of a water droplet when it comes out of the geyser(B)the height in meters of a water droplet when it reaches its highest point(C)the time in seconds from when a water droplet comes out of the geyser until it hits the ground(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 −4.9t2+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.
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