Two walkers set off at the same time from a crossroad and walk along flat straight roads inclined to each other at 68 degrees.If they both walk at a speed of 6km/h,find their distance apart 10 minutes later.
make a sketch, you should get an isosceles triangle with equal legs of 6(10/60) or 1 km, and vertex angle of 68°
let the base be 2x, and each of the base angles will be 56°
cos 56° = x/1
x = .55919..
2x = 1.118..
they will be 1.118 km apart
or
we could use the cosine law:
distance^2 = 1^1 + 1^2 - 2(1)(1)cos68°
= 2 - 2cos68° = 1.25078..
distance = √1.25078.. = 1.118 , same as above
To find the distance between the two walkers 10 minutes later, we can use the concept of relative velocity.
Given:
- The walkers set off at the same time and walk at a speed of 6 km/h
- The angle between their paths is 68 degrees
First, let's find the vertical component of their velocities:
Vertical component = Speed * sin(angle)
Vertical component = 6 km/h * sin(68 degrees)
To maintain consistency, let's convert the speed from km/h to km/min:
Vertical component = 6 km/min * sin(68 degrees)
Next, let's find the horizontal component of their velocities:
Horizontal component = Speed * cos(angle)
Horizontal component = 6 km/min * cos(68 degrees)
Now, we can calculate the distance between the two walkers using the Pythagorean theorem:
Distance = sqrt((Vertical component)^2 + (Horizontal component)^2)
Distance = sqrt((6 km/min * sin(68 degrees))^2 + (6 km/min * cos(68 degrees))^2)
Finally, let's calculate the distance between the two walkers:
Distance = sqrt((6 km/min * sin(68 degrees))^2 + (6 km/min * cos(68 degrees))^2)
Distance = sqrt((6)^2 * (sin(68 degrees))^2 + (6)^2 * (cos(68 degrees))^2) km
Using a calculator, we can find the value of Distance.
To find the distance between the walkers after 10 minutes, we can break down the problem into components.
1. Find the distance each walker has traveled after 10 minutes.
2. Use the law of cosines to find the distance between the walkers.
Step 1: Calculate the distance each walker has traveled after 10 minutes.
Since the walkers are traveling at a speed of 6 km/h, we can calculate the distance using the formula: distance = speed × time.
Distance traveled by each walker = 6 km/h × (10 minutes / 60 minutes)
To convert 10 minutes into hours, we divide by 60 (since there are 60 minutes in an hour).
Distance traveled by each walker = 6 km/h × (1/6) hours = 1 km
Therefore, each walker has traveled 1 kilometer after 10 minutes.
Step 2: Use the law of cosines to find the distance between the walkers.
The law of cosines states that for any triangle with sides a, b, and c, and angle C opposite side c:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, we have a triangle formed by the two walkers, with distances a and b as the two sides and an angle of 68 degrees.
Let's assume a = b = 1 km (since both walkers have traveled the same distance).
Plugging in the values into the law of cosines:
c^2 = 1^2 + 1^2 - 2(1)(1) * cos(68 degrees)
Using a calculator, evaluate cos(68 degrees) ≈ 0.3256.
c^2 = 1 + 1 - 2(1)(1) * 0.3256
c^2 = 2 - 2(0.3256)
c^2 = 2 - 0.6512
c^2 ≈ 1.3488
Taking the square root of both sides of the equation, we find:
c ≈ sqrt(1.3488)
c ≈ 1.1616
Hence, after 10 minutes, the distance between the two walkers is approximately 1.1616 kilometers.