The temperature in a greenhouse, recorded over a period of 10 minutes, is represented by the equation: T = 0.5(t + 3)(t + 4)(5 – t) where t is time in minutes and T is temperature in degree Celsius Sketch the curve. Find the domain and range of T. After how many minutes is the temperature first below 40 °C? Use your graph to work out an approximate answer. At 50 °C, plants suffer from thermal death. Do you think this greenhouse is an effective one? Give reasons for your answer

To tackle this problem, let's break it down into steps:

### 1. Sketch the Curve
The given temperature function is:

\[ T(t) = 0.5(t + 3)(t + 4)(5 - t) \]

This is a cubic polynomial, and we need to sketch the curve.

#### Step 1: Find the roots (where \( T(t) = 0 \))

Set \( T(t) = 0 \):

\[
0.5(t + 3)(t + 4)(5 - t) = 0
\]

This equation is zero when any of the factors is zero:

\[
t + 3 = 0 \quad \Rightarrow \quad t = -3
\]
\[
t + 4 = 0 \quad \Rightarrow \quad t = -4
\]
\[
5 - t = 0 \quad \Rightarrow \quad t = 5
\]

So, the roots are \( t = -4 \), \( t = -3 \), and \( t = 5 \).

#### Step 2: Determine the sign of the function between the roots

To sketch the curve, we should examine the behavior of the function between these roots:

- For \( t < -4 \), all factors are negative, so \( T(t) \) is positive.
- For \( -4 < t < -3 \), \( (t + 4) \) is positive, but \( (t + 3) \) and \( (5 - t) \) are negative, so \( T(t) \) is negative.
- For \( -3 < t < 5 \), \( (t + 3) \) and \( (t + 4) \) are positive, but \( (5 - t) \) is negative, so \( T(t) \) is positive.
- For \( t > 5 \), \( (t + 3) \), \( (t + 4) \), and \( (5 - t) \) are all positive, so \( T(t) \) is positive.

#### Step 3: Sketch the Graph

The curve will start above the x-axis when \( t < -4 \), dip below the x-axis between \( -4 \) and \( -3 \), rise above the x-axis between \( -3 \) and \( 5 \), and then rise further as \( t \) increases beyond \( 5 \).

### 2. Domain and Range of \( T(t) \)
- **Domain:** Since \( t \) represents time in minutes over a 10-minute period, \( t \) is likely restricted to \( [0, 10] \). However, the function is defined for all real \( t \), so mathematically, the domain is \( t \in \mathbb{R} \).
- **Range:** To find the range, we need to determine the maximum and minimum values of \( T(t) \). However, we can estimate it from the roots and behavior of the polynomial. We'll do a rough analysis with the graph sketch.

### 3. Find the time when the temperature first drops below 40°C
To find when \( T(t) \) first drops below 40°C, we need to solve:

\[ 0.5(t + 3)(t + 4)(5 - t) = 40 \]

This is a cubic equation, and solving it exactly is complex, but you can approximate it using the graph.

### 4. Analysis at 50°C (Thermal Death)
If plants suffer thermal death at 50°C, we'll need to see if the temperature \( T(t) \) exceeds 50°C at any point. If the graph shows \( T(t) \) exceeding 50°C, the greenhouse would not be effective.

### Graph and Analysis
Given the above steps, you would sketch the graph, identify where \( T(t) \) first falls below 40°C, and check if the curve crosses 50°C. 

### Conclusion
- If \( T(t) \) frequently exceeds 50°C, then the greenhouse is not effective as it would harm the plants.
- If the temperature remains well below 50°C, then the greenhouse is effective.

### Do you want more details or have any questions?

### Related Questions:
1. How do you find the critical points of \( T(t) = 0.5(t + 3)(t + 4)(5 - t) \)?
2. How can you determine the exact time when the temperature first drops below 40°C?
3. Can we find an exact solution for \( T(t) = 50 \) using algebraic methods?
4. How would the graph change if the coefficient 0.5 were different?
5. What would happen if the roots of the polynomial were closer together?

### Tip:
For more accurate results, consider using a graphing calculator or software to visualize the polynomial and precisely find where it meets certain temperature thresholds.

Explore how to sketch and analyze the temperature polynomial T(t) = 0.5(t + 3)(t + 4)(5 - t), focusing on domain, range, and critical points. Learn when the temperature first drops below 40°C and assess the effectiveness of the greenhouse in plant survival.

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