Solve the problem.
-A coin is tossed upward from a balcony 242 feet high with an initial velocity of 16 feet per second, then its height after $t$ seconds is given by the equation $h ( t ) = - 16 t ^ { 2 } + 16 t + 242$. During what interval of time will the coin be at a height of at least $50 \mathrm { ft }$ ?
A) $4 \leq t \leq 8$
B) $3 \leq t \leq 4$
C) $0 \leq t \leq 1$
D) $0 \leq t \leq 4$

Question

Quizplus · Accepted Answer

To find the interval of time during which the coin will be at a height of at least 50 ft, we need to solve the inequality $ -16t^2 + 16t + 242 \geq 50 $. Subtracting 50 from both sides gives $ -16t^2 + 16t + 192 \geq 0 $. Factoring out -16 gives $ -16(t^2 - t - 12) \geq 0 $, which simplifies to $ t^2 - t - 12 \leq 0 $ (reversing the inequality sign because we divided by a negative number). Factoring the quadratic equation gives $ (t-4)(t+3) \leq 0 $. By the zero product property, the solutions are $ t = 4 $ and $ t = -3 $. Since time cannot be negative, we disregard $ t = -3 $. Testing intervals around $ t = 4 $, we find that the inequality holds for $ 0 \leq t \leq 4 $, indicating the coin is at least 50 ft high during this time interval.

Solve the Problem ttt Seconds Is Given by the Equation h(t)=−16t2+16t+242h ( t ) = - 16 t ^ { 2 } + 16 t + 242h(t)=−16t2+16t+242

Solve the Problem $t$ Seconds Is Given by the Equation $h ( t ) = - 16 t ^ { 2 } + 16 t + 242$