Question

david is piloting a drone to film a short scene for a movie. the altitude of the drone in feet at t seconds is modeled by the quadratic function $h(t) = -2t^2 + 16t + 10$ for the time interval $t = 0$ to $t = 8$. using this model, which statement is true about the drone’s flight? a. the maximum altitude reached by the drone is 4 feet. b. the altitude of the drone is increasing for the entire duration of the flight. c. the maximum altitude reached by the drone is 42 feet. d. the initial altitude of the drone is 16 feet.

Explanation:

Step1: Analyze the quadratic function

The quadratic function is \( h(t) = -2t^2 + 16t + 10 \). For a quadratic function \( ax^2+bx + c \), the vertex (which gives the maximum or minimum) occurs at \( t = -\frac{b}{2a} \). Here, \( a=-2 \), \( b = 16 \), so \( t=-\frac{16}{2\times(-2)}=\frac{- 16}{-4} = 4 \) seconds.

Step2: Find the maximum altitude

Substitute \( t = 4 \) into \( h(t) \): \( h(4)=-2(4)^2+16(4)+10=-2\times16 + 64+10=-32 + 64+10=42 \) feet.

Step3: Analyze option B

Since the coefficient of \( t^2 \) is negative (\( a=-2<0 \)), the parabola opens downwards. So the function increases from \( t = 0 \) to \( t = 4 \) and decreases from \( t = 4 \) to \( t = 8 \). So the altitude is not increasing for the entire duration.

Step4: Analyze option D

Initial altitude is at \( t = 0 \), \( h(0)=-2(0)^2+16(0)+10 = 10 \) feet, not 16.

Step5: Analyze option A

From step 2, maximum altitude is 42, not 4.

Answer:

C. The maximum altitude reached by the drone is 42 feet.