1. **Problem (a): Find the size of \(\angle BAC\) to 1 decimal place.** Since the problem does not provide explicit coordinates or side lengths, we assume \(\triangle ABC\) is given or can be solved with provided data (not shown here). Typically, use the Law of Cosines or vector dot product to find \(\angle BAC\). 2. **Problem (b): Find the exact value of the area of \(\triangle ABC\).** Use the formula for the area of a triangle, such as \(\frac{1}{2}ab\sin C\) or Heron's formula if side lengths are known. --- 3. **Problem 2: Show that the internal area \(A\) of the running track is given by \(A = 300r - \pi r^2\).** - The track consists of a rectangle and two semi-circles. - Let the length of the rectangle be \(L\) and the width be \(2r\) (since radius \(r\) is half the width). - The total perimeter (track length) is given as 300 m. The perimeter consists of: - Two lengths \(L\) - Two semicircular arcs, which together form a full circle of circumference \(2\pi r\) So, $$ 2L + 2\pi r = 300 $$ Solving for \(L\): $$ 2L = 300 - 2\pi r \implies L = 150 - \pi r $$ The internal area \(A\) is the area of the rectangle plus the area of the two semicircles (which make a full circle): $$ A = L \times 2r + \pi r^2 = 2r(150 - \pi r) + \pi r^2 = 300r - 2\pi r^2 + \pi r^2 = 300r - \pi r^2 $$ 4. **Find the maximum value of \(A\) in terms of \(\pi\).** - Treat \(A = 300r - \pi r^2\) as a quadratic function in \(r\). - The maximum occurs at the vertex of the parabola. The vertex \(r\) is at: $$ r = -\frac{b}{2a} = -\frac{300}{2(-\pi)} = \frac{300}{2\pi} = \frac{150}{\pi} $$ Substitute back into \(A\): $$ A_{max} = 300 \times \frac{150}{\pi} - \pi \left(\frac{150}{\pi}\right)^2 = \frac{45000}{\pi} - \pi \frac{22500}{\pi^2} = \frac{45000}{\pi} - \frac{22500}{\pi} = \frac{22500}{\pi} $$ **Final answers:** - (a) \(\angle BAC\) depends on given data (not provided). - (b) Area of \(\triangle ABC\) depends on given data (not provided). - Internal area formula: \(A = 300r - \pi r^2\). - Maximum internal area: \(\frac{22500}{\pi}\) square meters.