What is the range of A(t)?
Understand the Problem
The question is asking for the range of the function A(t), which represents the altitude of a crane's beam over a duration of time, given that it can lift 30 meters at a steady rate of 6 meters per minute for a total of 5 minutes. We need to determine the possible values that A(t) can take during this process.
Answer
The range of $A(t)$ is all real numbers from $0$ to $30$.
Answer for screen readers
The range of $A(t)$ is all real numbers from $0$ to $30$.
Steps to Solve
- Identify the time range for A(t)
The crane lifts the beam over a period of 5 minutes. Therefore, the variable $t$ ranges from $0$ to $5$.
- Calculate the maximum altitude of the beam
The crane lifts the beam at a steady rate of $6$ meters per minute for $5$ minutes. The total height lifted is: $$ 30 = 6 \times t $$ When $t = 5$, the maximum height is $30$ meters.
- Determine the altitude A(t) at different time points
During the lifting, the altitude at any time $t$ can be represented as: $$ A(t) = 6t $$ When $t = 0$, $A(t) = 0$; when $t = 5$, $A(t) = 30$.
- Determine the range of A(t)
Since $t$ varies continuously from $0$ to $5$, the function $A(t)$ moves continuously from $0$ meters to $30$ meters. Therefore, the range is all possible values between $0$ and $30$, including all decimal values.
The range of $A(t)$ is all real numbers from $0$ to $30$.
More Information
The crane's lifting process creates a linear relationship between $t$ and $A(t)$. This means as time increases, the altitude increases consistently until it reaches the maximum height of $30$ meters.
Tips
- Assuming the range is limited to only whole numbers or specific multiples. Since $A(t)$ can take on any value between $0$ and $30$, it's important to consider all real numbers in that interval.
AI-generated content may contain errors. Please verify critical information
