A car starting at 12:00 noon travels west at a speed of 30 kph. Another car starting from the same point at 2:00 PM travels north at 45 kph. Find how fast the two are separating at 4:00 PM?
The two cars are separating at a rate of 75 kph (30 kph + 45 kph).
To find how fast the two cars are separating at 4:00 PM, we can use the concept of relative velocity.
Let's break down the problem into different steps:
Step 1: Determine the position of both cars at 4:00 PM.
The first car started traveling at 12:00 noon and travels west at a steady speed of 30 kph. Thus, in 4 hours, the first car will have traveled a distance of:
Distance traveled by the first car = Speed × Time = 30 kph × 4 hours = 120 km west.
The second car started traveling at 2:00 PM and travels north at a steady speed of 45 kph. Therefore, in 2 hours, the second car will have traveled a distance of:
Distance traveled by the second car = Speed × Time = 45 kph × 2 hours = 90 km north.
Thus, at 4:00 PM, the first car will be 120 km west and the second car will be 90 km north from the starting point.
Step 2: Determine the speed at which the two cars are separating.
To find the speed at which the two cars are separating, we can use the Pythagorean theorem.
The distance between the two cars is the hypotenuse of a right-angled triangle formed by the horizontal distance traveled by the first car (west) and the vertical distance traveled by the second car (north).
Using the Pythagorean theorem, we can find the distance separating the two cars:
Distance = √((120 km)² + (90 km)²) = √(14400 km² + 8100 km²) = √(22500 km²) = 150 km.
Thus, at 4:00 PM, the two cars are 150 km apart.
Step 3: Calculate the time derivative of the distance to find how fast the two cars are separating.
Differentiating the distance with respect to time will give us the rate at which the distance is changing (how fast the two cars are separating).
Derivative of Distance = d/dt (√((120 km)² + (90 km)²))
To calculate the derivative, we can use the chain rule. Let's find each term individually:
Derivative of (√((120 km)² + (90 km)²)) = d/dt ((120 km)² + (90 km)²)^0.5
Using the chain rule, the derivative can be rewritten as:
= 0.5((120 km)² + (90 km)²)^(-0.5) × d/dt ((120 km)² + (90 km)²)
Now, let's calculate the derivative of the square terms:
d/dt ((120 km)² + (90 km)²) = 2(120 km)(30 kph) + 2(90 km)(45 kph)
= 2(120 km)(30 kph) + 2(90 km)(45 kph) = 7200 km²ph + 8100 km²ph
Finally, substituting the calculated derivative into the previous expression:
Derivative of Distance = 0.5((120 km)² + (90 km)²)^(-0.5) × (7200 km²ph + 8100 km²ph)
= 0.5(14400 km² + 8100 km²)^(-0.5) × (7200 km²ph + 8100 km²ph)
Simplifying further, we get:
= 0.5(22500 km²)^(-0.5) × (7200 km²ph + 8100 km²ph)
= 0.5(150 km) × (7200 km²ph + 8100 km²ph)
= 75 km × (7200 km²ph + 8100 km²ph)
= 75(7200 km²ph + 8100 km²ph)
= 75 × 15300 km²ph
= 1,147,500 km²ph
Therefore, at 4:00 PM, the two cars are separating at a speed of 1,147,500 km²ph.
To find how fast the two cars are separating at 4:00 PM, we can use the concept of relative velocity.
Let's consider the position of the first car at 4:00 PM. Since it started at 12:00 noon and traveled west at a speed of 30 kph for 4 hours, the first car has traveled a distance of 30 kph * 4 hours = 120 km.
Now, consider the position of the second car at 4:00 PM. Since it started at 2:00 PM and traveled north at a speed of 45 kph, the second car has traveled a distance of 45 kph * 2 hours = 90 km.
To visualize the positions, we can draw a right-angled triangle, with the first car's position being the base (120 km) and the second car's position being the perpendicular (90 km).
Next, we can calculate the separation distance between the two cars at 4:00 PM using the Pythagorean theorem. The separation distance (d) is given by:
d^2 = (120 km)^2 + (90 km)^2
d^2 = 14400 km^2 + 8100 km^2
d^2 = 22500 km^2
d = sqrt(22500) km
d ≈ 150 km
Now, to find how fast the two cars are separating at 4:00 PM, we need to differentiate the separation distance equation with respect to time. However, since we are given the speeds of the cars, we can directly calculate the rate at which they are separating.
The first car is moving west at a speed of 30 kph, and the second car is moving north at a speed of 45 kph. Since the two cars are moving perpendicular to each other, their speeds do not affect each other directly.
Thus, the two cars are separating at a constant rate of 30 kph + 45 kph = 75 kph. Therefore, the two cars are separating at a speed of 75 kph at 4:00 PM.