Question 3 Marks The scale diagram of the aerial view of Warragamba dam is shown below and four triangles have been created to assist with finding the total actual area of the dam. Using the scale 1:7000, your knowledge of trigonometry and the scale diagram above, calculate the actual area of each triangle. Leave your answer correct to the nearest square metre.

Step 1 ：First, we need to find the lengths of the sides of each triangle in the scale diagram. Let's label the triangles as follows: Triangle 1 (top left), Triangle 2 (top right), Triangle 3 (bottom left), and Triangle 4 (bottom right).

Step 2 ：For Triangle 1, we can use the Pythagorean theorem to find the length of the hypotenuse: \(a^2 + b^2 = c^2\), where \(a = 3\), \(b = 4\), and \(c\) is the hypotenuse. So, \(3^2 + 4^2 = c^2\), which gives us \(c = 5\).

Step 3 ：For Triangle 2, we can use the Pythagorean theorem again with \(a = 4\), \(b = 3\), and \(c\) as the hypotenuse. So, \(4^2 + 3^2 = c^2\), which gives us \(c = 5\).

Step 4 ：For Triangle 3, we can use the Pythagorean theorem with \(a = 3\), \(b = 4\), and \(c\) as the hypotenuse. So, \(3^2 + 4^2 = c^2\), which gives us \(c = 5\).

Step 5 ：For Triangle 4, we can use the Pythagorean theorem with \(a = 4\), \(b = 3\), and \(c\) as the hypotenuse. So, \(4^2 + 3^2 = c^2\), which gives us \(c = 5\).

Step 6 ：Now that we have the lengths of the sides of each triangle in the scale diagram, we can find the actual lengths by multiplying each side by the scale factor of 7000. For example, the actual length of a side with a length of 3 in the scale diagram is \(3 \times 7000 = 21000\) meters.

Step 7 ：Using the actual lengths, we can now find the area of each triangle using the formula \(A = \frac{1}{2}ab\sin{C}\), where \(a\) and \(b\) are the lengths of two sides and \(C\) is the angle between them.

Step 8 ：For Triangle 1, we have \(a = 21000\), \(b = 28000\), and \(C = 90^\circ\). So, \(A_1 = \frac{1}{2}(21000)(28000)\sin{90^\circ} = \frac{1}{2}(21000)(28000)(1) = 294000000\) square meters.

Step 9 ：For Triangle 2, we have \(a = 28000\), \(b = 21000\), and \(C = 90^\circ\). So, \(A_2 = \frac{1}{2}(28000)(21000)\sin{90^\circ} = \frac{1}{2}(28000)(21000)(1) = 294000000\) square meters.

Step 10 ：For Triangle 3, we have \(a = 21000\), \(b = 28000\), and \(C = 90^\circ\). So, \(A_3 = \frac{1}{2}(21000)(28000)\sin{90^\circ} = \frac{1}{2}(21000)(28000)(1) = 294000000\) square meters.

Step 11 ：For Triangle 4, we have \(a = 28000\), \(b = 21000\), and \(C = 90^\circ\). So, \(A_4 = \frac{1}{2}(28000)(21000)\sin{90^\circ} = \frac{1}{2}(28000)(21000)(1) = 294000000\) square meters.

Step 12 ：Finally, we can find the total actual area of the dam by adding the areas of each triangle: \(A_{total} = A_1 + A_2 + A_3 + A_4 = 294000000 + 294000000 + 294000000 + 294000000 = 1176000000\) square meters. So, the total actual area of the dam is approximately \(\boxed{1176000000}\) square meters.