Why would you use square roots in the following 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)
Here is my work for the question:
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.
Things would be a lot clearer if you wrote
CN^2 = 13^2 + 18^2 - 2*13*18 cos 45
= 169 + 324 - 468*√2/2
= 162.07
CN = 12.73
In the given question, square roots are used to calculate the distance between two points using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
To calculate the distance between Centreville and Northcote (CN), you need to find the length of the side opposite the 90-degree angle in the right triangle formed by the route. This can be done by applying the Pythagorean theorem.
First, you determine the lengths of the two other sides of the triangle. The first leg of the race is 13 km (Centreville to Eastdale) and the second leg is 18 km (Eastdale to Northcote).
Next, you square each of these lengths: 13^2 = 169 and 18^2 = 324.
To find CN^2 (the square of the distance between Centreville and Northcote), you sum these squared lengths: CN^2 = 169 + 324.
Now, you can simplify this equation: CN^2 = 493.
Finally, to find the distance CN, you take the square root of CN^2: CN = √493 ≈ 22.18 km (rounded to two decimal places).
To calculate the total distance of the race, you add the lengths of the legs together: 13 km + 18 km + 22.18 km ≈ 53.18 km (rounded to the nearest tenth).