why would I have uarrived at square root 2 over 2 in this question Chang is participating in a charity bicycle road race. The route starts at Centreville and travels east for 13 km to Eastdale. He then makes a 135° turn and heads northwest for another 18 km, arriving at Northcote. The final leg of the race returns to Centreville. a) What is the total length of the race, to the nearest tenth of a kilometre? (2 marks) b) What are the angles in the triangle formed by the three towns, to the nearest degree? (2 marks)
Here's my work for the question:
a)
CN2=132+182-2-13-18-cos 45
CN2= 169+324-234√2
CN= 12.7308 km.
The total distance of the race would be 13+18+12.7=43.7 km.
b)
I already know the angle of Eastdale which is 45° due to the turn from due east to the northwest. So therefore so I need to find the measures of angle c and angle n. I can use the Law of Sines.
sin‹C/c=sin‹E/e
sin‹C/18=(√2/2)/12.7308
sin‹C=18(√2/2)/12.708=.9998
angle c= 88.8°
So therefore m‹N= 180°-45°-88.8°=46.2°
a. d = 13 + 18 +13 = 44 km
b. 45 + 45 + 90 = 180o
To understand why you arrived at square root 2 over 2 in this question, let's break down the steps involved.
In part a), you are trying to find the total length of the race. You start by using the Law of Cosines to find the length of the side CN. The formula you used is CN^2 = AB^2 + BC^2 - 2(AB)(BC)cos(angle B).
In this formula, AB represents the length of the side from Centreville to Eastdale, which is 13 km, and BC represents the length of the side from Eastdale to Northcote, which is 18 km. The angle B is the known angle of 45°.
By substituting these values into the formula, you get CN^2 = 13^2 + 18^2 - 2(13)(18)cos(45°).
Simplifying further, you have CN^2 = 169 + 324 - 234√2.
To find CN, you need to take the square root of both sides, giving you CN ≈ √(169 + 324 - 234√2).
Evaluating this expression, you get CN ≈ 12.7308 km.
To find the total distance of the race, you add the lengths of the three sides: 13 km + 18 km + 12.7308 km. This equals approximately 43.7 km.
In part b), you are trying to find the angles in the triangle formed by the three towns. You already know one angle, which is the angle at Eastdale and is given as 45°. To find the other two angles, C and N, you can use the Law of Sines.
The Law of Sines states that the ratio of the sine of an angle to the length of the opposite side is constant for all angles in a triangle.
Using this law, you can set up the following equation: sin(angle C) / 18 = sin(angle E) / 12.7308.
In this equation, angle E is 45°, and the length of side CN is 12.7308 km, as found in part a).
By solving this equation, you find that sin(angle C) ≈ (18 * √2 / 2) / 12.7308.
Evaluating this expression gives sin(angle C) ≈ 0.9998.
To find angle C, you can use the inverse sine function (sin^-1) or arcsin to find the angle whose sine is approximately 0.9998. The value you get is approximately 88.8°.
Finally, to find angle N, you subtract the known angles at Eastdale (45°) and angle C (88.8°) from 180°. This gives you angle N ≈ 180° - 45° - 88.8° = 46.2°.
So, you arrived at square root 2 over 2 while using the Law of Cosines to find the length of the side CN, and you used the Law of Sines to find the angles in the triangle formed by the three towns.