A metal rod has a length of 100cm at 200°c. At what temperature will it's length be 99.4 cm if linear expansivity of the material of the rod is 0.00002k-1?

We can use the formula for linear expansion:

ΔL = αLΔT

Where:
ΔL = change in length (final length - initial length)
α = linear expansivity (0.00002 K^-1)
L = initial length (100 cm)
ΔT = change in temperature

We are given that the final length is 99.4 cm. Therefore, we can rewrite the formula as:

99.4 - 100 = 0.00002 * 100 * ΔT

-0.6 = 0.002 * ΔT

ΔT = -0.6 / 0.002
ΔT = -300 K

Therefore, the temperature at which the length of the rod will be 99.4 cm is 200°C - 300°C = -100°C.

Step 1: Define the variables

- L0 = Initial length = 100 cm
- Lf = Final length = 99.4 cm
- α = Linear expansivity = 0.00002 K-1
- Δt = change in temperature = ?

Step 2: Use the formula for linear expansion

The formula for linear expansion is:

```
Lf = L0 + α - L0 - Δt
```

Step 3: Rearrange the formula to solve for Δt

```
Δt = (Lf - L0) / (α - L0)
```

Step 4: Substitute the values into the formula

```
Δt = (99.4 cm - 100 cm) / (0.00002 K-1 - 100 cm)
```

Step 5: Calculate Δt

```
Δt = -0.6 K
```

Step 6: Convert Δt to degrees Celsius

```
Δt (°C) = -0.6 K - (1 °C / 1 K)
```

Δt (°C) = -0.6 °C

Therefore, the temperature must decrease by 0.6 °C for the metal rod to have a length of 99.4 cm.