f(t) = {48 + t^2 - (t^3/12) for 0 <= t < 6, for 6 <= t<= 12

g(x)

t(hours) 6 8 10 12

g(t) cuic meters 306 376 428 474

is f continuous for 0<= t <= 12? Justify your answer

To determine whether the function \( f(t) \) is continuous on the interval \( 0 \leq t \leq 12 \), we need to verify the following:

1. **Piecewise Continuity**: We are given \( f(t) \) in two different forms for different intervals:
   - For \( 0 \leq t < 6 \): \( f(t) = 48 + t^2 - \frac{t^3}{12} \)
   - For \( 6 \leq t \leq 12 \): The function expression for this interval is not provided, but we'll check continuity by ensuring the function values match at \( t = 6 \).

2. **Continuity at \( t = 6 \)**: Since \( f(t) \) is defined differently on the intervals \( [0, 6) \) and \( [6, 12] \), we must check if the left-hand limit and right-hand limit of \( f(t) \) at \( t = 6 \) are equal, and if they equal \( f(6) \).

### Step 1: Evaluate the left-hand limit as \( t \to 6^- \):
For \( t < 6 \), we have \( f(t) = 48 + t^2 - \frac{t^3}{12} \).
Let's find the left-hand limit \( \lim_{t \to 6^-} f(t) \):

\[
\lim_{t \to 6^-} f(t) = 48 + 6^2 - \frac{6^3}{12}
= 48 + 36 - \frac{216}{12}
= 48 + 36 - 18
= 66
\]

So, \( \lim_{t \to 6^-} f(t) = 66 \).

### Step 2: Evaluate the right-hand limit as \( t \to 6^+ \):
We need to know the expression for \( f(t) \) for \( t \geq 6 \). Without this expression, we can't fully check the continuity at \( t = 6 \).

### Step 3: Continuity on each interval:
- The function \( f(t) = 48 + t^2 - \frac{t^3}{12} \) is a polynomial, and polynomials are continuous everywhere. Therefore, \( f(t) \) is continuous on \( [0, 6) \).
- We assume the function for \( [6, 12] \) is continuous as well, but without knowing the exact form, we can't conclusively determine continuity at \( t = 6 \).

### Conclusion:
For \( f(t) \) to be continuous on \( [0, 12] \), we need the function values and limits to match at \( t = 6 \). Since the function for \( t \geq 6 \) is not provided, we cannot definitively confirm continuity at \( t = 6 \) without further information.

Would you like further clarification or help with another part of the question?

Here are some related questions to consider:
1. What is the form of \( f(t) \) for \( 6 \leq t \leq 12 \)?
2. How do you determine limits for piecewise functions?
3. What conditions must be met for a function to be continuous at a point?
4. How do polynomials behave in terms of continuity?
5. Can you provide a graph of \( f(t) \) to visualize continuity?

**Tip**: When checking continuity at a point, always compare the left-hand and right-hand limits with the actual value of the function at that point.

Is f(t) = {48 + t^2 - (t^3/12) for 0 <= t < 6, for 6 <= t <= 12} continuous for 0 <= t <= 12? Justify your answer.

Explore the continuity of the piecewise function f(t) on the interval 0 <= t <= 12. This problem involves evaluating polynomial behavior, limits, and piecewise continuity, with a detailed solution examining the left-hand and right-hand limits at t = 6.

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