Final answer:
The equation d = 35 + 85t - 16t^2 models the projectile motion of a pebble launched vertically upward from a firecracker, where d is the height above the water in feet, and t is the time in seconds since launch. The terms represent initial height, initial velocity, and acceleration due to gravity.
Step-by-step explanation:
Understanding Projectile Motion in Mathematics
A high school student exploring the concepts of projectile motion in mathematics can use the provided quadratic equation to determine various aspects of a pebble's trajectory after it is launched upwards by a firecracker. The specific equation given is:
d = 35 + 85t - 16t2
Here, d represents the distance above the water, and t is the time in seconds since launch.
The initial height from which the pebble is launched is 35 feet. This is represented by the y-intercept when t=0. The term 85t signifies the initial upward speed of the pebble, which contributes to its ascent. The 16t2 term accounts for the acceleration due to gravity, which in this case, is approximated as 32 feet per second squared and affects the pebble in the downward direction, hence the negative sign.
To graph this equation, one would plot time along the horizontal axis (x-axis) and distance above the water along the vertical axis (y-axis). The graph would show the pebble rising and then falling into the water. Each point on the graph corresponds to the pebble's height at a given time.