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zoeheskett
11.08.2021 •
Mathematics
A bicyclist is at point A on a paved road and must ride to point C on another paved road. The two roads meet at
an angle of 38° at point B. The distance from A to B is 18 mi, and the distance from B to C is 12 mi (see
the figure). If the bicyclist can ride 22 mph on the paved roads and 6.8 mph off-road, would it be faster for the bicyclist to ride from A to C on the paved roads or to ride a direct line from A to C off-road? Explain.
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Ответ:
Step-by-step explanation:
The diagrammatic expression to understand this question very well is attached in the image below.
By applying the law of cosine rule; we have:
a² = b² + c² - 2bc Cos A --- (1)b² = a² + c² - 2ac Cos B --- (2)c² = a² + b² - 2ab Cos C --- (3)From the diagram attached below, we need to determine the side "b" by using equation (2) from above:
b² = a² + c² - 2ac Cos B
From the information given:
a = 12 miles; c = 18 miles; ∠B = 38°
∴
replacing the values into the above equation:
b² = 12² + 18² - 2(12)(18) Cos (38°)
b² = 144 + 324 - 432 × (0.7880)
b² = 468 - 340.416
b² = 127.584
b = 11.30 miles
However, we are also being told that the speed from A → C = 6.8 mph
Thus, the time required to go from A → C can be determined by using the relation:
making time the subject of the formula, we have:
time = 1.66 hours
By using the paved roads, the speed is given as = 22 mph
thus, the total distance covered = |AB| + |BC|
= (18+12) miles
= 30 miles
∴
time = 1.36 hours
Therefore, the time used off-road = 1.661 hours while the time used on the paved road is 1.36 hours.
Since we are considering the shortest time possible;
We can conclude that it would be faster for the bicyclist to ride from A to C on the paved roads since it takes a shorter time to reach its destination compared to the time used off-road.
Learn more about Law of cosine here:
Ответ:
It would be faster for the bicyclist to ride from A to C on the paved roads since the time to go from A to C on the paved roads is 1.4 h and the time to go from A to C off-road is 1.7 h.
To calculate which way would be faster we need to find the distance from point A to C with the law of cosines:
Where:
Now, let's find the time for the two following cases.
1. From point A to C on the paved roads (t₁)
The time can be calculated with the following equation:
Where:
d: is the distance
v: is the velocity
Then, the total time that it takes the bicyclist to go from point A to C on the paved roads is:
2. From point A to C off-road (t₂)
With equation (1) we can calculate the time to go from point A to C off-road:
Therefore, it would be faster for the bicyclist to ride from A to C on the paved roads.
To find more about the law of cosines, go here:
I hope it helps you!
Ответ: