A sector of a circular metal sheet with radius 17 cm has an area of 427 cm2. Find the angle of the sector, rounded to three significant figures. Use pi = 3.142.

Here is the solution to the problem:

Given:

• Area of sector $= 427 \text{ cm}^2$
• Radius of circular metal sheet ($R$) $= 17 \text{ cm}$ (This becomes the slant height $l$ of the cone)
• $\pi = 3.142$
• All answers to be rounded to three significant figures.

a) The angle of the sector
Let $\theta$ be the angle of the sector in degrees.
The formula for the area of a sector is:

$$\text{Area} = \frac{\theta}{360^\circ} \pi R^2$$

Substitute the given values:

$$427 = \frac{\theta}{360} \times 3.142 \times (17)^2$$ $$427 = \frac{\theta}{360} \times 3.142 \times 289$$ $$427 = \frac{\theta \times 907.998}{360}$$

Now, solve for $\theta$:

$$\theta = \frac{427 \times 360}{907.998}$$ $$\theta = \frac{153720}{907.998}$$ $$\theta \approx 169.299^\circ$$

Rounding to three significant figures:

$$\theta \approx 169^\circ$$

The angle of the sector is $\boxed{169^\circ