T(w) = 0.1w2 + 2.151w + 20
where T is the temperature in degrees Celsius and w is the power input in watts.
(a) How much power is needed to maintain the temperature at 204°C?

To find the power \( w \) needed to maintain the temperature at \( 204^\circ \text{C} \), we need to solve the given quadratic equation for \( w \).

The equation provided is:

\[
T(w) = 0.1w^2 + 2.151w + 20
\]

We set \( T(w) = 204 \) and solve for \( w \):

\[
204 = 0.1w^2 + 2.151w + 20
\]

First, subtract 204 from both sides:

\[
0 = 0.1w^2 + 2.151w + 20 - 204
\]

\[
0 = 0.1w^2 + 2.151w - 184
\]

Now, we solve this quadratic equation using the quadratic formula:

\[
w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Where:
- \( a = 0.1 \)
- \( b = 2.151 \)
- \( c = -184 \)

Substitute these values into the quadratic formula:

\[
w = \frac{-2.151 \pm \sqrt{(2.151)^2 - 4(0.1)(-184)}}{2(0.1)}
\]

Let's calculate this step by step.

The two possible solutions for the power \( w \) are approximately:

- \( w_1 = 33.47 \) watts
- \( w_2 = -54.98 \) watts

Since power cannot be negative in this context, the physically meaningful solution is:

\[ w \approx 33.47 \text{ watts} \]

Thus, approximately **33.47 watts** of power is needed to maintain the temperature at \( 204^\circ \text{C} \).

Would you like more details or have any further questions?

Here are some related questions to consider:
1. How would the temperature change if the power input were increased to 40 watts?
2. Can you graph the function \( T(w) = 0.1w^2 + 2.151w + 20 \) to visualize its behavior?
3. What happens to the temperature as the power input becomes very large?
4. How would you solve the equation if \( T(w) \) were given in Fahrenheit instead of Celsius?
5. What physical situations might require solving for \( w \) in the context of this equation?

**Tip:** When solving quadratic equations, always check if the discriminant (the value under the square root) is positive, zero, or negative to understand the nature of the roots.

T(w) = 0.1w^2 + 2.151w + 20 where T is the temperature in degrees Celsius and w is the power input in watts. (a) How much power is needed to maintain the temperature at 204°C?

This problem involves solving the quadratic equation T(w) = 0.1w^2 + 2.151w + 20 to find the power needed to maintain a temperature of 204°C. Learn how to apply the quadratic formula to determine the solution, suitable for students in Grades 9-12.

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