An automobile traveling at the rate of 30 ft/sec is approaching an intersection. When the automobile is 120 ft. from the intersection, a truck, traveling at the rate of 40 ft/sec on a road at right angles with the automobile’s road crosses the intersection. How long after the truck leaves the intersection the two vehicles are closest?

Let the time passed since the truck went through the intersection be t seconds

Make a diagram
Let the distance between them be D ft
I see that

D^2 = (40t)^2 + (120-30t)^2
= 1600t^2 + 14400 - 720t + 900t^2
= 2500t^2 - 7200t + 14400

2D dD/dt = 5000t -7200
but at a minimum of D , dD/dt = 0 , so

5000t = 7200
t = 7200/5000
= 1.44 sec after the truck entered the intersection

check:
when t = 0 , D = 120
when t = 1 , D = 98.5
when t = 1.44 , D = 96
when t = 1.5 , D = 96.05 , getting bigger again

thx reiny

thanks mike

Well, this sounds like a classic case of "Who's in a Rush Wreck"! Let's break it down and calculate the clowny details.

The distance between the automobile and the intersection is decreasing at a rate of 30 ft/sec. So, we can say that the position of the automobile relative to the intersection is given by the equation d = 120 - 30t, where t is the time in seconds.

Now, let's focus on the truck. It's traveling at a rate of 40 ft/sec on a road at right angles to the automobile's road. So, the position of the truck relative to the intersection is given by the equation d = 40t.

To find when the two vehicles are closest, we need to find the time (t) when the difference between their positions is the smallest. So, we can subtract the two position equations:

(120 - 30t) - (40t) = 120 - 70t.

Now, let's set this difference equation equal to zero and solve:

120 - 70t = 0
70t = 120
t = 120/70
t ≈ 1.71 seconds.

Therefore, approximately 1.71 seconds after the truck leaves the intersection, the two vehicles will be closest. Just make sure they exchange friendly honks and not friendly bumps!

Remember, folks, keep it clowny and safe on the roads!

To find out how long after the truck leaves the intersection the two vehicles are closest, we need to analyze the relative motion of the two vehicles.

Let's start by finding the equation for the distance between the two vehicles. Let's call the time elapsed since the truck left the intersection t(seconds).

The distance the automobile has traveled when t seconds have passed is equal to the product of the automobile's speed (30 ft/sec) and t (the time elapsed):
Distance traveled by automobile = 30t

Now, we need to find the distance between the truck and the automobile at time t. The distance traveled by the truck when t seconds have passed is equal to the product of the truck's speed (40 ft/sec) and t (the time elapsed):
Distance traveled by truck = 40t

The distance between the two vehicles at time t is the difference between the distances traveled by the truck and the automobile:
Distance between the two vehicles = 40t - 30t = 10t

Now, we know that when the automobile is 120 ft. from the intersection, the distance between the two vehicles is 120 ft. So we can set up the following equation:
10t = 120

To solve for t, we divide both sides of the equation by 10:
t = 120/10
t = 12

Therefore, the two vehicles are closest 12 seconds after the truck leaves the intersection.