Two trucks start from the same point. One truck travels due east at a rate of 50 mph and the other trucks travels due north at a rate of 60 mph. At what rate is the distance between the two vehicles increasing 1 1/2 hours later?
I think the answer is 55 mph because I added 60 + 55 and divided by 2 and got 55 mph.
Oh my :(
The distance between them is the hypotenuse of a right triangle.
D^2 = x^2 + y^2
where
x = 50 t
y = 60 t
so
D^2 = 2500 t^2 + 3600 t^2 = 6100 t^2
now take time derivative
2 D dD/dt = 12200 t
at t = 1.5
D^2 = 6100 (2.25)
D = 117 miles
2 D = 234.3 miles
so
dD/dt = 12200(1.5)/234.3 = 78.1 miles/hour
Well, it seems like you've got the right idea of averaging the speeds, but unfortunately, that's not the correct approach in this case. To find the rate at which the distance between the two vehicles is increasing, we'll need to use some math magic.
Let's first break down the problem. After 1 1/2 hours, the eastward-traveling truck would have covered 50 mph * 1.5 hours = 75 miles. Similarly, the northward-traveling truck would have covered 60 mph * 1.5 hours = 90 miles.
Now, let's form a right-angled triangle with the two trucks as the vertices. The base of the triangle represents the eastward distance covered by the first truck (75 miles), the height represents the northward distance covered by the second truck (90 miles), and the hypotenuse represents the distance between the two trucks.
Using the Pythagorean theorem (a² + b² = c²), we can find the length of the hypotenuse:
c² = a² + b²
c² = (75 miles)² + (90 miles)²
c² = 5625 miles + 8100 miles
c² = 13,725 miles
Taking the square root of both sides, we get:
c ≈ 117.16 miles
So, after 1 1/2 hours, the distance between the two vehicles is approximately 117.16 miles.
To find the rate at which this distance is increasing, we can differentiate this distance with respect to time. However, since the question asks for the rate in mph (miles per hour), we don't need to do that differentiation.
In other words, the distance between the two vehicles is increasing at a rate of 117.16 mph after 1 1/2 hours.
Hope this explanation helped, even if it wasn't as funny as the name "Clown Bot" would suggest!
To find the rate at which the distance between the two trucks is increasing, we can use the Pythagorean theorem. Let's assume that the distance between the two trucks after 1 1/2 hours is d.
The truck traveling east is moving at a rate of 50 mph for 1 1/2 hours, so it travels a distance of (50 mph) * (1.5 hours) = 75 miles.
The truck traveling north is moving at a rate of 60 mph for 1 1/2 hours, so it travels a distance of (60 mph) * (1.5 hours) = 90 miles.
Using the Pythagorean theorem, we can calculate the distance between the two trucks after 1 1/2 hours:
d^2 = (75 miles)^2 + (90 miles)^2
Simplifying the equation, we have:
d^2 = 5,625 + 8,100
d^2 = 13,725
d ≈ √13,725
d ≈ 117.07 miles
Now, to find the rate at which the distance between the trucks is increasing, we differentiate d with respect to time:
d/dt (d) = d/dt (117.07 miles)
Since both trucks are moving perpendicular to each other, the distance between them is a constant value, which means its rate of change is zero. Therefore, the rate at which the distance between the two trucks is increasing 1 1/2 hours later is 0 mph, not 55 mph.
To find the rate at which the distance between the two trucks is increasing, we need to use the concept of relative velocity. The two trucks are moving in perpendicular directions, so we can use the Pythagorean theorem to determine the distance between them.
Let's start by calculating the distance each truck has traveled after 1 1/2 hours. The truck traveling east at 50 mph will have traveled (50 mph) * (1.5 hours) = 75 miles.
The truck traveling north at 60 mph will have traveled (60 mph) * (1.5 hours) = 90 miles.
Now, we can use these distances to determine the distance between the two trucks using the Pythagorean theorem. The distance is given by:
Distance = √[(east distance)^2 + (north distance)^2]
Distance = √[(75 miles)^2 + (90 miles)^2]
Distance = √[5625 miles^2 + 8100 miles^2]
Distance = √[13725 miles^2]
Distance = 117 miles (approximately)
To find the rate at which the distance between the two trucks is increasing, we can differentiate this equation with respect to time. However, since we are given the rates of the trucks, we can also use a simpler approach.
The rate at which the distance is increasing can be found by adding the rates of the two trucks, given that they are moving perpendicular to each other.
Rate of increase = rate of truck 1 + rate of truck 2
Rate of increase = 50 mph (east) + 60 mph (north)
Rate of increase = 110 mph
Therefore, the rate at which the distance between the two vehicles is increasing after 1 1/2 hours is 110 mph, not 55 mph.