To solve this problem, we need to calculate the deflection of a bimetallic strip used in a temperature control device. The bimetallic strip bends due to the difference in thermal expansion coefficients of the two metals.

We'll break down the steps based on the information provided and the hint about using a small-angle approximation (sin(θ/2) ≈ θ/2 or using the nonlinear sine equation as needed).

Problem Breakdown:

1. Known Parameters:

   • Nominal Temperature: 200°F
   • Temperature Deviation: ±1°F (i.e., we're considering two cases: 199°F and 201°F).
   • Length of the Strip: 4 inches

2. Deflection Formula: The deflection $\delta$ at the end of the strip can be calculated using a relation derived from the bending of bimetallic strips, which involves thermal expansion and material properties. In general, the deflection is related to the temperature change, $\Delta T$, by the equation:

   $\delta = L \cdot \theta$

   where:

   • $\delta$ = deflection at the end of the strip,
   • $L$ = length of the strip (4 inches),
   • $\theta$ = angle of curvature resulting from the temperature change.

3. Curvature Angle $\theta$: The curvature angle $\theta$ depends on the temperature change and material properties of the bimetallic strip. The relationship for small deflections (small angles) is typically linearized, but it can also be expressed in terms of the sine function for more accuracy.

   Given the tip to either linearize or use the sine function as-is, we assume:

   $\sin\left(\frac{\theta}{2}\right) \approx \frac{\theta}{2} \quad \text{for small angles}$

   We can use this approximation because the temperature deviation (±1°F) is small relative to the nominal temperature.

4. Temperature Change $\Delta T$: For the deviations of ±1°F from the nominal temperature, the temperature change $\Delta T$ is either $+1°F$ or $-1°F$. We will use these changes to compute the corresponding deflection.

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Solution Process:

1. Find the angle of curvature $\theta$: The angle of curvature $\theta$ is related to the temperature change $\Delta T$ by a proportionality constant that depends on the properties of the materials making up the bimetallic strip. This constant, often denoted by $\alpha$, depends on the difference in thermal expansion coefficients between the two metals.

   In this case, $\theta$ is proportional to $\Delta T$, so:

   $\theta = C \cdot \Delta T$

   where $C$ is a constant related to the material properties of the strip. Since $\Delta T$ is small, we can use the linearized equation for small deflections.

2. Compute the deflection $\delta$: Once we know $\theta$, the deflection $\delta$ can be calculated using the equation:

   $\delta = L \cdot \theta$

   We'll now calculate the deflection for both $\Delta T = +1°F$ and $\Delta T = -1°F$.

Let me know if you'd like further clarification or specific values for the material constant $C$, as this would typically be provided in the textbook or relevant references.