Question

laurie throws a tennis ball toward her dog from a height of 4.5 ft. the initial vertical velocity of the ball is 18 ft/s. at the same time as laurie throws the ball, her dog jumps with an initial vertical velocity of 21 ft/s. when the dog jumps, its mouth is 1.5 ft above the ground. projectile motion formula: $h=-16t^{2}+vt + h_{0}$ $t$ = time, in seconds, since the ball was thrown $h$ = height, in feet, above the ground which system models the height of the tennis ball and the height of the dogs mouth over time? $h = 4.5t + 18$ and $h=-16t^{2}+1.5t + 21$ $h = 18t + 4.5$ and $h=-16t^{2}+21t + 1.5$ $h=-16t^{2}+4.5t + 18$ and $h=-16t^{2}+1.5t + 14$ $h=-16t^{2}+18t + 4.5$ and $h=-16t^{2}+21t + 1.5$ complete what does the $t$-coordinate of the solution to this system represent? the time at which the ball and the dog hit the ground the time at which the ball and the dog begin moving

Explanation:

Step1: Analyze the formula for the tennis ball

The formula for projectile motion is \(h = - 16t^{2}+vt + h_{0}\). For the tennis ball, \(v = 18\) (initial vertical velocity) and \(h_{0}=4.5\) (initial height). So the equation for the tennis ball's height is \(h=-16t^{2}+18t + 4.5\).

Step2: Analyze the formula for the dog's mouth

For the dog's mouth, \(v = 21\) (initial vertical velocity of the jump) and \(h_{0}=1.5\) (initial height of the mouth). So the equation for the dog's mouth height is \(h=-16t^{2}+21t + 1.5\).

Step3: Analyze the meaning of the \(t\) - coordinate of the solution

When we solve the system of equations \(h=-16t^{2}+18t + 4.5\) and \(h=-16t^{2}+21t + 1.5\), we are finding the time \(t\) when the height of the tennis ball (\(h\)) is equal to the height of the dog's mouth (\(h\)). It is not the time when they hit the ground (which would be when \(h = 0\)) or when they begin moving (\(t = 0\)).

Answer:

The system of equations is \(h=-16t^{2}+18t + 4.5\) and \(h=-16t^{2}+21t + 1.5\). The \(t\) - coordinate of the solution represents the time when the height of the tennis ball is equal to the height of the dog's mouth.