The problem involves a river flow modeled by the function $f(t) = 250t$, where $t$ is time in seconds and $f(t)$ is the volume of water (in cubic meters) that has flowed past a point after $t$ seconds. Let's break down and solve the sub-questions:

(a) Find $f(30)$

To find $f(30)$, substitute $t = 30$ into the function:

$f(30) = 250 \cdot 30 = 7500 \, \text{cubic meters.}$

So, $f(30) = 7500 \, \text{cubic meters}$.

(b) What does $f(30)$ mean in this situation?

$f(30)$ represents the volume of water that has flowed from the river after 30 seconds. Based on the calculation:

At $t = 30 \, \text{seconds}$, the volume of water from the river is ( 7500 , \text{cubic meters}.

(c) How long will it take for $100,000 \, \text{cubic meters}$ of water to flow down the river?

To find the time $t$, we use the function $f(t) = 250t$ and solve for $t$ when $f(t) = 100,000$:

$100,000 = 250t$

$t = \frac{100,000}{250} = 400 \, \text{seconds.}$

So, it will take $400 \, \text{seconds}$ for $100,000 \, \text{cubic meters}$ of water to flow down the river.

Do you need further details or clarifications? Let me know!

Related Questions

1. How does the flow rate $250 \, \text{m}^3/\text{s}$ affect the time for large volumes of water to flow?
2. What would $f(t)$ equal after 1 minute?
3. How can the function be modified for a varying flow rate instead of a constant one?
4. If $f(t) = 300t$, how would the time change for $100,000 \, \text{m}^3$?
5. Can this formula predict water volume in scenarios of flooding or decreased flow?

Tip:

Always double-check your substitution and units when working with time and rates to ensure your answers remain consistent!