Question

Your friend in the neighboring high-rise-apartment complex in NYC is running low on toilet paper. To stay in compliance with the social distancing rules, you both decide to meet on the roof-top's of your respective apartment building's (exactly 240 feet above the ground). You toss the toilet paper upward towards your friend with an initial velocity of 32 feet-per-second. Unfortuantly, the toilet paper fell to ground. Use the quadratic model h (t) = -16t2 + 32t + 240 where h (t) represents the height at a particular time." In the given scenario, a quadratic model is used to represent the height of a tossed object over time. Which of the following statements is correct about the model h(t) = -16t^2 + 32t + 240?

A) The initial velocity of the tossed object is 240 feet per second.
B) The object reaches its highest point at t = 1 second.
C) The object's maximum height is 240 feet.
D) The object never returns to the ground.

Answer 1

The correct statement about the projectile motion model h(t) = -16t^2 + 32t + 240 is B) The object reaches its highest point at t = 1 second. The initial velocity is 32 feet per second, not 240, and the toilet paper does return to the ground.

Step-by-step explanation:

The provided equation h(t) = -16t^2 + 32t + 240 describes the height of the toilet paper roll at any time t, where h(t) represents the height above ground level in feet and t represents time in seconds. The equation is in the format of a quadratic equation used to model projectile motion where the acceleration due to gravity is represented by the term -16t^2, indicating that gravity is pulling the object down at 32 feet per second squared (since in physics, gravity is typically -32 ft/s^2 and we use half of that in the equation).

The initial velocity is given by the coefficient of the t term, which is 32 feet per second, not 240 feet per second as suggested by statement A. Therefore, statement A is incorrect. The initial position above the ground is the constant term, 240 feet, which is the starting height of the toss, not the maximum height as suggested by statement C. Using calculus or algebra, we can determine the time when the toilet paper reaches its highest point to be at t = 1 second, which is when the derivative of the height with respect to time is zero. This confirms that statement B is correct. Lastly, the toilet paper definitely returns to the ground as the negative quadratic term ensures the parabola opens downwards, suggesting statement D is incorrect.