Question

question 1
two docks are 168 m apart. from one dock dave swims out to an island and then swims back to the other dock. the angles between the line joining each dock to the island, and the line joining the docks are 64° and 70°. how far must dave swim? answer to the nearest metre.

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the answer is:

Explanation:

Step1: Find the third angle of the triangle

In a triangle, the sum of interior angles is \(180^\circ\). Let the two given angles be \(A = 64^\circ\) and \(B=70^\circ\), then the third angle \(C=180^\circ-(64^\circ + 70^\circ)=46^\circ\). Let the distance between the docks be \(c = 168\) m, the distance from the first dock to the island be \(a\), and from the island to the second dock be \(b\).

Step2: Apply the Law of Sines to find \(a\)

The Law of Sines states that \(\frac{a}{\sin A}=\frac{c}{\sin C}\). Substituting \(A = 64^\circ\), \(C = 46^\circ\), and \(c = 168\) m, we get \(a=\frac{168\times\sin64^\circ}{\sin46^\circ}\). Calculating \(\sin64^\circ\approx0.8988\) and \(\sin46^\circ\approx0.7193\), then \(a=\frac{168\times0.8988}{0.7193}\approx\frac{151.0}{0.7193}\approx209.9\) m.

Step3: Apply the Law of Sines to find \(b\)

Using the Law of Sines \(\frac{b}{\sin B}=\frac{c}{\sin C}\). Substituting \(B = 70^\circ\), \(C = 46^\circ\), and \(c = 168\) m, we get \(b=\frac{168\times\sin70^\circ}{\sin46^\circ}\). Calculating \(\sin70^\circ\approx0.9397\), then \(b=\frac{168\times0.9397}{0.7193}\approx\frac{157.9}{0.7193}\approx219.5\) m.

Step4: Find the total distance Dave swims

The total distance is \(a + b\). So \(a + b\approx209.9+219.5 = 429.4\approx429\) m (rounded to the nearest metre). Wait, wait, let's recalculate more accurately.

Wait, let's do the calculations with more precision.

First, \(\sin64^\circ\approx0.8987940463\), \(\sin70^\circ\approx0.9396926208\), \(\sin46^\circ\approx0.7193398003\)

For \(a\): \(\frac{168\times0.8987940463}{0.7193398003}=\frac{168\times0.8987940463}{0.7193398003}\)

\(168\times0.8987940463 = 168\times0.898794\approx151.0\) (exact: \(168\times0.8987940463=168\times0.8987940463 = 151.0003997\))

Then \(a=\frac{151.0003997}{0.7193398003}\approx209.91\) m

For \(b\): \(\frac{168\times0.9396926208}{0.7193398003}\)

\(168\times0.9396926208 = 168\times0.9396926208=157.8683603\)

\(b=\frac{157.8683603}{0.7193398003}\approx219.46\) m

Total distance: \(209.91 + 219.46=429.37\approx429\) m? Wait, but maybe I made a mistake in angle assignment. Wait, the two docks are 168 m apart, so the side between the two docks is \(c = 168\), and the angles at the docks are \(64^\circ\) and \(70^\circ\), so the triangle has sides: dock1 - island (\(a\)), island - dock2 (\(b\)), dock1 - dock2 (\(c = 168\)). Angles at dock1: \(64^\circ\), at dock2: \(70^\circ\), so angle at island: \(180 - 64 - 70 = 46^\circ\). So Law of Sines: \(\frac{a}{\sin70^\circ}=\frac{b}{\sin64^\circ}=\frac{168}{\sin46^\circ}\). Oh! Wait, I think I mixed up the angles. The angle at dock1 is between dock1 - dock2 and dock1 - island, so that angle is \(64^\circ\), so the side opposite to \(64^\circ\) is \(b\) (island - dock2), and the angle at dock2 is \(70^\circ\), so the side opposite to \(70^\circ\) is \(a\) (dock1 - island). So correct Law of Sines: \(\frac{a}{\sin70^\circ}=\frac{b}{\sin64^\circ}=\frac{168}{\sin46^\circ}\)

Let's recalculate:

First, \(\sin46^\circ\approx0.7193\), \(\sin70^\circ\approx0.9397\), \(\sin64^\circ\approx0.8988\)

So \(a=\frac{168\times\sin70^\circ}{\sin46^\circ}=\frac{168\times0.9397}{0.7193}\approx\frac{157.87}{0.7193}\approx219.5\) m

\(b=\frac{168\times\sin64^\circ}{\sin46^\circ}=\frac{168\times0.8988}{0.7193}\approx\frac{151.0}{0.7193}\approx209.9\) m

Total distance: \(219.5 + 209.9 = 429.4\approx429\) m? Wait, but let's check with more precise calculation.

\(\sin46^\circ = \sin(46) \approx 0.7193398003\)

\(a=\frac{168\times\sin70^\circ}{\sin46^\circ}=\frac{168\times0.9396926208}{0.7193398003}\)

\(168\times0.9396926208 = 168\times0.9396926208 = 157.8683603\)

\(157.8683603\div0.7193398003\approx219.46\)

\(b=\frac{168\times\sin64^\circ}{\sin46^\circ}=\frac{168\times0.8987940463}{0.7193398003}\)

\(168\times0.8987940463 = 151.0003997\)

\(151.0003997\div0.7193398003\approx209.91\)

Sum: \(219.46 + 209.91 = 429.37\approx429\) m. Wait, but maybe the answer is different? Wait, let's check the Law of Sines again. The triangle: vertices D1 (dock1), D2 (dock2), I (island). So D1D2 = 168 m. Angle at D1: between D1D2 and D1I is \(64^\circ\), angle at D2: between D1D2 and D2I is \(70^\circ\). So angle at I: \(180 - 64 - 70 = 46^\circ\). So sides: D1I = \(a\), D2I = \(b\), D1D2 = \(c = 168\). So by Law of Sines: \(\frac{a}{\sin(\angle D2)}=\frac{b}{\sin(\angle D1)}=\frac{c}{\sin(\angle I)}\)

Because \(\angle D2 = 70^\circ\) is opposite side \(a\) (D1I), \(\angle D1 = 64^\circ\) is opposite side \(b\) (D2I), and \(\angle I = 46^\circ\) is opposite side \(c\) (D1D2). So that's correct. So \(a=\frac{c\sin\angle D2}{\sin\angle I}\), \(b=\frac{c\sin\angle D1}{\sin\angle I}\). Then total distance is \(a + b=\frac{c(\sin\angle D2+\sin\angle D1)}{\sin\angle I}\)

Let's compute that:

\(\sin70^\circ+\sin64^\circ\approx0.9396926 + 0.8987940\approx1.8384866\)

\(c = 168\), \(\sin46^\circ\approx0.7193398\)

So total distance \(=\frac{168\times1.8384866}{0.7193398}\)

\(168\times1.8384866 = 168\times1.8384866\approx308.8657\)

\(308.8657\div0.7193398\approx429.37\approx429\) m. So the total distance Dave swims is approximately 429 metres.

Answer:

\(429\)