Solve for x: 1000 = (35)(0.02)(x − 293.15) + (0.6)(5.67e−8)(0.02)(x⁴ − 293.15⁴).

Step-by-step solution

1. 1

   Collapse the constants into two coefficients. Group the convection and radiation factors:

   $35 \times 0.02 = 0.7, \qquad 0.6 \times 5.67\times10^{-8} \times 0.02 = 6.804\times10^{-10}$

   so the equation is

   $1000 = 0.7\,(x - 293.15) + 6.804\times10^{-10}\left(x^4 - 293.15^4\right)$

2. 2

   Expand into a quartic in standard form. With $293.15^4 = 7.38515\times10^{9}$, the two constant contributions are $0.7 \times 293.15 = 205.205$ and $6.804\times10^{-10} \times 7.38515\times10^{9} = 5.0249$:

   $6.804\times10^{-10}\,x^4 + 0.7x - 1210.23 = 0$

3. 3

   Do not stop at the linear estimate — this is the trap. Ignoring the quartic term gives

   $x \approx \frac{1210.23}{0.7} \approx 1728.9$

   but substituting that back shows why it fails: $6.804\times10^{-10}(1728.9)^4 \approx 6079\ \text{W}$, six times the entire heat load. The radiation term is dominant at that temperature, not negligible, so $1728.9\ \text{K}$ is far too high.

4. 4

   Solve the quartic numerically instead. The left-hand side is strictly increasing in $x$ for $x > 0$ (both $0.7x$ and $x^4$ increase), so there is exactly one physical root and bisection converges reliably. Bracketing between $300$ and $5000$ and bisecting gives

   $x \approx 947.04\ \text{K}$

5. 5

   Check the root by splitting the two loss mechanisms. At $x = 947.04\ \text{K}$:

   $\text{convection} = 0.7(947.04 - 293.15) = 457.7\ \text{W}$

   $\text{radiation} = 6.804\times10^{-10}\left(947.04^4 - 293.15^4\right) = 542.3\ \text{W}$

   and $457.7 + 542.3 = 1000.0\ \text{W}$, matching the required heat loss. Radiation carries $54\%$ of the load — confirming it could never have been dropped.

6. 6

   State the answer with its physical reading.

   $x \approx 947\ \text{K} \ (\approx 674\ ^\circ\text{C})$

   The general lesson: whenever a balance mixes a linear convection term with a $T^4$ radiation term, the quartic must be solved, because radiation grows so steeply that it overtakes convection well before a few hundred kelvin of overtemperature.