Question

Help i need this for rn :( 100 points

Two water balloons were launched into the air at different moments and collided. The water balloons were modeled by the quadratic functions: y = −3x2 + 24x + 3 and y = −4x2 + 25x + 15, where y represents the height in meters and x represents the time in seconds after the launch. What is the time, in seconds, that the balloons collided at the highest point?

3.25 seconds
3.50 seconds
3.75 seconds
4.00 seconds

Answer 1

The collision means that y is equal

• y=-3x²+24x+3
• y=-4x²+25x+15

So

• -3x²+24x+3=-4x²+25x+15
• -3x²+4x²+24x-25x=15-3
• x²-x=12
• x²-x-12=0
• (x-4)(x+3)=0
• x=4,-3

As it's time so must be positive

• x=4s

Answer 2

Explanation:

When the balloons collide, they have the same x and y:

-3x^2 + 24x + 3 = y = -4x^2 + 25x + 15

Rearranging like terms together:

(4x^2 - 3x^2) + (24x - 25x) + (3 - 15) = 0

x^2 - x - 12 = 0

(x - 4)(x +3) = 0

x = 4 or -3

As x represents time, it cannot be negative.

The time when the balloons collide is at 4.00 seconds.