Question

application/thinking

1. a satellite calculates the distance and angle shown. find the distance between the two cities.

round answers to the nearest tenth.(4 marks)

1. while hiking along the top edge of the scarborough bluffs, bob stops at a lookout point (l).

from this point he can see bluffers beach (b) and the marina (m).
he estimates he is 3.2 km from bluffers beach.
he estimates he is 2.7 km from the marina.
the angle between his lines of sight to these two locations is 68°.
a) complete the diagram by labelling the given information.
3 marks

how far apart are bluffers beach and the marina?

Explanation:

Part 4 (Satellite and Cities Distance)

We can use the Law of Cosines here. The Law of Cosines states that for a triangle with sides \( a \), \( b \), \( c \) and the angle \( C \) opposite side \( c \), \( c^{2}=a^{2}+b^{2}-2ab\cos(C) \). Let the distance from the satellite to the first city be \( a = 370 \) km, to the second city be \( b = 360 \) km, and the included angle \( C=9.4^{\circ} \). We need to find the distance between the two cities, let's call it \( c \).

Step 1: Identify values for Law of Cosines

\( a = 370 \), \( b = 360 \), \( C = 9.4^{\circ} \)

Step 2: Apply Law of Cosines

\( c^{2}=370^{2}+360^{2}-2\times370\times360\times\cos(9.4^{\circ}) \)

First, calculate \( 370^{2}=136900 \), \( 360^{2} = 129600 \)

Then, \( 2\times370\times360=266400 \)

\( \cos(9.4^{\circ})\approx\cos(9.4)\approx0.986 \) (using calculator)

So, \( c^{2}=136900 + 129600-266400\times0.986 \)

\( c^{2}=266500-262670.4 \)

\( c^{2}=3829.6 \)

Step 3: Take square root

\( c=\sqrt{3829.6}\approx61.9 \) km (rounded to nearest tenth)

We can use the Law of Cosines here. Let the distance from lookout point \( L \) to Bluffers Beach \( B \) be \( a = 3.2 \) km, from \( L \) to marina \( M \) be \( b = 2.7 \) km, and the included angle \( C = 68^{\circ} \). We need to find the distance between \( B \) and \( M \), let's call it \( c \).

Step 1: Identify values for Law of Cosines

\( a = 3.2 \), \( b = 2.7 \), \( C = 68^{\circ} \)

Step 2: Apply Law of Cosines

\( c^{2}=a^{2}+b^{2}-2ab\cos(C) \)

Calculate \( a^{2}=3.2^{2} = 10.24 \), \( b^{2}=2.7^{2}=7.29 \)

\( 2ab = 2\times3.2\times2.7 = 17.28 \)

\( \cos(68^{\circ})\approx0.3746 \) (using calculator)

So, \( c^{2}=10.24 + 7.29-17.28\times0.3746 \)

\( c^{2}=17.53 - 6.47 \)

\( c^{2}=11.06 \)

Step 3: Take square root

\( c=\sqrt{11.06}\approx3.33 \) km (rounded to nearest hundredth)

Answer:

The distance between the two cities is approximately \( \boldsymbol{61.9} \) km.

Part 5 (Bluffers Beach and Marina Distance)