A food packet is dropped from a helicopter and is modeled by the function f(x) = −15x2 + 6000. The graph below shows the height f(x), in feet, of the food packet at different times x, in seconds:

Question

graph of function f of x equals 6000 minus 15 multiplied by x squared

Use the graph to determine the domain of f(x) for all viable x values based on the context.

x ≤ 6000
0 ≤ x ≤ 20
−20 ≤ x ≤ 20
All real numbers

Answer 1

To determine the domain of the function \( f(x) = -15x^2 + 6000 \) in the context of the food packet being dropped from a helicopter, we need to consider the physical situation.

1. Context: The function models the height of the food packet over time after it is dropped. The food packet starts from a height of 6000 feet.

2. Time: Since time cannot be negative in this context (the packet cannot be dropped before time zero), we have \( x \geq 0 \).

3. Height: The packet will eventually reach the ground (height = 0). We can find the time when the height will be 0 by solving \( -15x^2 + 6000 = 0 \).

   \[ -15x^2 + 6000 = 0 \] \[ 15x^2 = 6000 \] \[ x^2 = 400 \] \[ x = 20 \quad (\text{since time can't be negative}) \]

Thus, the packet drops from a height of 6000 feet and will reach the ground at \( x = 20 \) seconds.

Putting these two pieces of information together, the packet is in the air from \( x = 0 \) to \( x = 20 \) seconds.

Therefore, the domain of \( f(x) \) based on the context is:

\[ \boxed{0 \leq x \leq 20} \]