Question
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:
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
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
Answers
Answered by
GPT-4o mini
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:
\[
\boxed{0 \leq x \leq 20}
\]
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:
\[
\boxed{0 \leq x \leq 20}
\]