In Figure 2 OAB is a sector of a circle, radius 5 m - Edexcel - A-Level Maths Pure - Question 7 - 2005 - Paper 2

To find the angle ∠OAB in radians, we use the inverse cosine function:

$$\angle OAB = \cos^{-1}\left(\frac{7}{25}\right).$$

Calculating this gives:

$$\angle OAB \approx 1.247\ ext{ radians (to 3 decimal places)}.$$

The area of a sector can be calculated using the formula:

$$\text{Area} = \frac{1}{2} r^2 \theta,$$

where r is the radius and (\theta) is the angle in radians. Substituting the values:

$$\text{Area} = \frac{1}{2} \cdot 5^2 \cdot 1.247 = \frac{1}{2} \cdot 25 \cdot 1.247 \approx 15.588\ m^2.$$

To find the shaded area, we need to subtract the area of triangle OAB from the area of the sector OAB. The area of triangle OAB can be calculated using:

$$\text{Area} = \frac{1}{2} \times base \times height.$$

In triangle OAB, the height corresponds to the line from point O to the midpoint of chord AB. Using the Pythagorean theorem:

• Let h be the height and x be half of AB (3 m).

$$h^2 + 3^2 = 5^2\ h^2 + 9 = 25\ h^2 = 16\ h = 4.$$

Thus, the area of triangle OAB is:

$$\text{Area} = \frac{1}{2} \cdot 6 \cdot 4 = 12 m^2.$$

The shaded area is calculated as:

$$\text{Shaded Area} = \text{Area of Sector} - \text{Area of Triangle} \approx 15.588 - 12 = 3.588 m^2.$$