Let's solve the given problem step-by-step.

Problem Overview

The flow rate of water is $f(t) = 250t$, where:

• $t$: time in seconds
• $f(t)$: volume of water in cubic meters at time $t$

Part (a): Find $f(30)$

We substitute $t = 30$ into the formula $f(t) = 250t$: $f(30) = 250 \times 30 = 7500 \, \text{cubic meters.}$

So, the volume of water after 30 seconds is 7500 cubic meters.

Part (b): What does $f(30)$ mean?

The function $f(t) = 250t$ calculates the total volume of water flowing in cubic meters after $t$ seconds. Therefore, $f(30)$ represents: $\text{At 30 seconds, the total volume of water flowing from the river is 7500 cubic meters.}$

Part (c): Time to Flow 100,000 Cubic Meters

To find how long it will take for 100,000 cubic meters of water to flow, set $f(t) = 100,000$: $f(t) = 250t \implies 100,000 = 250t$ Solve for $t$: $t = \frac{100,000}{250} = 400 \, \text{seconds.}$

It will take 400 seconds for 100,000 cubic meters of water to flow.

Final Answers

(a) $f(30) = 7500 \, \text{cubic meters}$
(b) $f(30)$ means: At 30 seconds, the total volume of water flowing from the river is 7500 cubic meters.
(c) Time to flow 100,000 cubic meters: $t = 400 \, \text{seconds.}$

Do you want further clarification on any part?

5 Related Questions:

1. How is the flow rate of a river typically calculated in real-life situations?
2. What does a linear relationship between time and volume indicate about the flow?
3. If the flow rate changes to $f(t) = 300t$, how does that affect the calculations?
4. What would $t = 0$ imply in the context of this problem?
5. How can you visualize this function as a graph?

Tip:

Always double-check units in mathematical problems—mixing units (e.g., seconds and minutes) can lead to errors!