6.5*(T-39)*3.14*3.14*80*65*15=3465*3465*35*[1+0.0039*(T-20)]

Let's solve this equation for \( T \). The equation is:

\[
6.5 \times (T - 39) \times 3.14 \times 3.14 \times 80 \times 65 \times 15 = 3465 \times 3465 \times 35 \times \left[1 + 0.0039 \times (T - 20)\right]
\]

### Step 1: Simplify the left-hand side (LHS)

First, simplify the constant multiplication on the left-hand side:

\[
LHS = 6.5 \times (T - 39) \times 3.14 \times 3.14 \times 80 \times 65 \times 15
\]

Calculate the constant term:

\[
3.14 \times 3.14 = 9.8596
\]

\[
9.8596 \times 80 \times 65 \times 15 = 76795.92
\]

So, the LHS becomes:

\[
LHS = 6.5 \times 76795.92 \times (T - 39) = 499673.48 \times (T - 39)
\]

### Step 2: Simplify the right-hand side (RHS)

Simplify the right-hand side:

\[
RHS = 3465 \times 3465 \times 35 \times \left[1 + 0.0039 \times (T - 20)\right]
\]

Calculate the constant multiplication:

\[
3465 \times 3465 = 12001525
\]

\[
12001525 \times 35 = 420053375
\]

So, the RHS becomes:

\[
RHS = 420053375 \times \left[1 + 0.0039 \times (T - 20)\right]
\]

### Step 3: Expand and distribute

Distribute on both sides:

LHS:

\[
499673.48 \times (T - 39) = 499673.48T - 19487225.72
\]

RHS:

\[
420053375 \times \left(1 + 0.0039T - 0.078\right) = 420053375 \times (0.922 + 0.0039T)
\]

Simplify:

\[
RHS = 387749608.75 + 1638218.16T
\]

### Step 4: Combine like terms and solve for \( T \)

Set the equations equal to each other:

\[
499673.48T - 19487225.72 = 387749608.75 + 1638218.16T
\]

Move the \( T \) terms to one side and the constants to the other:

\[
499673.48T - 1638218.16T = 387749608.75 + 19487225.72
\]

\[
-1138544.68T = 407236834.47
\]

Solve for \( T \):

\[
T = \frac{407236834.47}{-1138544.68} \approx -357.69
\]

### Final Answer

The solution for \( T \) is approximately:

\[
T \approx -357.69
\]

Would you like to explore this result further, or do you have any additional questions?

---

**Related Questions:**

1. How do you isolate variables in complex equations?
2. What are the steps to solve equations involving multiple operations?
3. Can you explain how to simplify expressions involving constants?
4. How do you solve equations with terms on both sides?
5. What are common mistakes to avoid when solving algebraic equations?

**Tip:** Always recheck your calculations to ensure that the simplifications and arithmetic operations are accurate, especially when dealing with large numbers.

Learn how to solve the complex equation 6.5*(T-39)*3.14*3.14*80*65*15=3465*3465*35*[1+0.0039*(T-20)] step-by-step with detailed calculations. This problem involves arithmetic operations, equation simplification, and algebraic principles. Suitable for advanced math enthusiasts.

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