two cyclists A and B start cycling at the same time from a certain point C. cyclist A is cycling at 45km/h in an easterly direction. cyclist B is cycling north at 21km/h. find the shortest distance between the two hours. round answer off to two decimal places
To find the shortest distance between the two cyclists, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
In this case, the two sides of the right triangle are the distance travelled by cyclist A and cyclist B. Since the time is the same for both cyclists, the distance travelled by cyclist A after time t will be equal to the speed of cyclist A multiplied by t. Similarly, the distance travelled by cyclist B after time t will be equal to the speed of cyclist B multiplied by t.
Let's assume that the time elapsed is t hours.
Distance travelled by cyclist A = 45 * t (in km)
Distance travelled by cyclist B = 21 * t (in km)
Using the Pythagorean theorem, the square of the shortest distance between the two cyclists is equal to the sum of the squares of the distances travelled by each cyclist:
(45t)^2 + (21t)^2 = shortest distance^2
2025t^2 + 441t^2 = shortest distance^2
2466t^2 = shortest distance^2
Taking the square root of both sides:
shortest distance = sqrt(2466t^2)
Now, we need to find the value of t that gives us the shortest distance. Since we want to find the shortest distance, we can assume that the two cyclists have travelled for the same amount of time. Therefore, t is the same for both cyclists.
To find the shortest distance, we can substitute t = 1 into the equation:
shortest distance = sqrt(2466 * 1^2) = sqrt(2466)
Rounded off to two decimal places, the shortest distance between the two cyclists is approximately 49.66 km.
To find the shortest distance between the two cyclists, we can use the Pythagorean theorem.
Let's assume that after t hours, cyclist A has traveled a distance of dA and cyclist B has traveled a distance of dB.
Since cyclist A is traveling at a constant speed of 45 km/h in an easterly direction, the distance traveled by cyclist A can be calculated as dA = 45t.
Similarly, since cyclist B is traveling at a constant speed of 21 km/h in a northerly direction, the distance traveled by cyclist B can be calculated as dB = 21t.
Now, let's consider the right triangle formed by the shortest distance between the two cyclists. The distance traveled by cyclist A represents the length of the adjacent side of the triangle, and the distance traveled by cyclist B represents the length of the opposite side of the triangle.
Using the Pythagorean theorem, we can calculate the shortest distance (hypotenuse) between the two cyclists:
shortest distance = sqrt((dA)^2 + (dB)^2)
= sqrt((45t)^2 + (21t)^2)
To find the value of t, we need to equate the distances traveled by cyclist A and cyclist B. Since both cyclists start at the same time, dA = dB. Therefore:
45t = 21t
Subtracting 21t from both sides:
24t = 0
Dividing both sides by 24:
t = 0
This means that the two cyclists will meet at the starting point C. Therefore, the shortest distance between them is zero.
Rounding off to two decimal places, the shortest distance between the two cyclists is 0.00 km.
To find the shortest distance between the two cyclists' positions, we can use the Pythagorean theorem. According to the theorem, in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this scenario, we can consider the position of cyclist A as one side of the triangle, the position of cyclist B as the other side, and the shortest distance between them as the hypotenuse.
Let's label the point where the cyclists start as C. We assume that cyclist A is moving along the x-axis (from left to right) and cyclist B is moving along the y-axis (from bottom to top).
Cyclist A is moving eastward at 45 km/h, so their position after a certain time can be represented as (distance, 0). Similarly, cyclist B is moving northward at 21 km/h, so their position can be represented as (0, distance).
Now, let's assume that they have been traveling for t hours. The distance covered by cyclist A would be 45t km, and the distance covered by cyclist B would be 21t km.
Using these coordinates, we can draw a right triangle with one side along the x-axis, one side along the y-axis, and the hypotenuse connecting the positions of A and B.
The length of the hypotenuse (d) can be calculated using the Pythagorean theorem:
d^2 = (45t)^2 + (21t)^2
Simplifying the equation:
d^2 = 2025t^2 + 441t^2
d^2 = 2466t^2
To find the shortest distance, we need to minimize d. This occurs when t is at its minimum value. Since t represents time, it cannot be negative. Therefore, the minimum value for t is 0.
Substituting t = 0 into the equation:
d^2 = 2466(0)^2
d^2 = 0
Therefore, the shortest distance between the two cyclists is 0 km.
This means that the two cyclists will meet at the starting point C.