Determine an appropriate domain of the function. Identify the independent and depen...
Check the final answer first, then review the worked steps.
Problem
Determine an appropriate domain of the function. Identify the independent and dependent variables. A stone is thrown vertically upward from the ground at a speed of 65 m/s at time t = 0. Its distance d (in m) above the ground (neglecting air resistance) is approximated by the function f(t) = 65t - 5t^2.
Answer
Independent Variable: $t$ (time), Dependent Variable: $d$ or $f(t)$ (distance). Domain: $[0, 13]$
Step-by-step solution
1. Identify Independent and Dependent Variables:
The independent variable is the one that can be changed or controlled, and it is typically represented by 't' for time in physics problems. The dependent variable is the one that is measured or observed, and its value depends on the independent variable. In this case, the distance 'd' (or f(t)) depends on the time 't'.
Independent Variable: $t$ (time in seconds)
Dependent Variable: $d$ or $f(t)$ (distance in meters)
2. Analyze the Function for Physical Constraints:
The function given is $f(t) = 65t - 5t^2$. This function describes the height of the stone above the ground as a function of time. We need to determine a realistic domain for this function based on the physical situation.
3. Determine the Lower Bound of the Domain:
Time cannot be negative in this context, as the stone is thrown at $t=0$. Therefore, the minimum value for $t$ is 0.
Lower bound: $t \ge 0$
4. Determine the Upper Bound of the Domain:
The stone is thrown upwards, reaches a maximum height, and then falls back down. The function $f(t)$ represents the distance above the ground. The stone is on the ground when $f(t) = 0$. We need to find the time(s) when the stone is at ground level.
Set $f(t) = 0$:
$$65t - 5t^2 = 0$$
Factor out $5t$:
$$5t(13 - t) = 0$$
This equation has two solutions:
$5t = 0 \implies t = 0$
$13 - t = 0 \implies t = 13$
The solution $t=0$ represents the initial moment when the stone is thrown from the ground. The solution $t=13$ represents the time when the stone returns to the ground.
Since the function describes the height of the stone while it is in the air, the relevant time interval is from when it is thrown until it hits the ground again. Therefore, the time $t$ must be less than or equal to 13 seconds.
Upper bound: $t \le 13$
5. Combine the Bounds to Define the Domain:
Combining the lower and upper bounds, the appropriate domain for the function in this physical context is $0 \le t \le 13$.
Domain: $[0, 13]$