At time t = 0, a set of three points A, B and C forms an equilateral triangle of side 6 cm. If two of the sides
of the triangle are decreasing in length at the rate of 0.1 cm/sec and the third side is increasing at the rate of
0.1 cm/sec, how long will it take for the system to just cease to form a triangle?
After a time of t seconds, the lengths of the two decreasing sides are 6+.1t cm each and the side which is increasing is 6 + .1t
It will cease being a triangle if the sum of two sides equals the third side
6 + .1t = 2(6 - .1t)
6 + .1t = 12 - .2t
.3t = 6
t = 6/.3 = 20
it will take 20 seconds
Well, well, well, looks like we have a disappearing triangle mystery on our hands! Let's take a plunge into this enigma, shall we?
First things first, let's calculate the time it takes for each side of the triangle to vanish. Since two sides are decreasing at a rate of 0.1 cm/sec, it will take:
6 cm / 0.1 cm/sec = 60 seconds for them to disappear.
On the other hand, the remaining side is increasing at a rate of 0.1 cm/sec. To find out when it becomes larger than the sum of the other two sides, we need to determine the maximum possible size of the other two sides.
In an equilateral triangle, each side is equal to 6 cm. So, the sum of the decreasing sides would be 6 cm + 6 cm = 12 cm.
Now, let's see how long it takes for the increasing side to surpass the maximum size of the decreasing sides:
(12 cm) / (0.1 cm/sec) = 120 seconds.
Now, the crucial part is when the increasing side finally overtakes the sum of the decreasing sides, the triangle will cease to exist.
Therefore, the longest time for this triangular adventure to come to an end is the larger of the two times, which is 120 seconds.
Voila! After 120 seconds, our triangle will bid us farewell and fade away into non-existence. Let's hope it doesn't vanish without leaving a note!
To determine how long it will take for the system to just cease to form a triangle, we need to find the time at which the sum of two sides becomes equal to or smaller than the third side.
Let's denote the side lengths as follows:
AB = BC = 6 cm (constant)
AC = x cm (variable)
Given that two sides are decreasing at the rate of 0.1 cm/sec and one side is increasing at the same rate, we can write the equations for their rates of change as follows:
d(AB)/dt = d(BC)/dt = 0 (constant)
d(AC)/dt = 0.1 cm/sec (variable)
We want to find the time t at which the sum of two sides becomes equal to or smaller than the third side:
AB + AC <= BC or BC + AC <= AB
Substituting the values into the equations, we get:
6 + x <= 6 or 6 + x <= 6
Simplifying the equations, we find:
x <= 0 or x <= 0
Since the side length cannot be negative, the system will cease to form a triangle when x becomes zero.
To find the time it takes for x to reach zero, we can use the equation:
t = |x / (d(AC)/dt)|
Substituting the values, we get:
t = |0 / 0.1| = 0 seconds
Hence, it will take 0 seconds for the system to just cease to form a triangle.
To find out how long it will take for the system to just cease to form a triangle, we need to determine when the length of at least one side becomes equal to or less than zero.
Let's consider the side AB, which is initially 6 cm. It is decreasing at a rate of 0.1 cm/sec. We can find out the time taken for side AB to become zero by using the formula:
Time = (Initial Length - Final Length) / Rate
In this case, the initial length is 6 cm, the final length is 0 cm (since it needs to become zero), and the rate is -0.1 cm/sec (negative because it's decreasing).
Time taken for side AB to become zero:
Time_AB = (6 cm - 0 cm) / (-0.1 cm/sec) = 60 sec
Next, let's consider the side BC, which is also initially 6 cm. It is also decreasing at a rate of 0.1 cm/sec. We can find out the time taken for side BC to become zero using the same formula:
Time_BC = (6 cm - 0 cm) / (-0.1 cm/sec) = 60 sec
Finally, we need to consider the side AC, which is initially 6 cm and is increasing at a rate of 0.1 cm/sec. We can find out the time taken for side AC to become zero by using the formula:
Time_AC = (0 cm - 6 cm) / (0.1 cm/sec) = -60 sec
However, since we cannot have negative time, we can conclude that side AC never becomes zero.
Therefore, the system will cease to form a triangle when either side AB or side BC becomes zero. Since both sides AB and BC take 60 seconds to reach zero length, the system will cease to form a triangle in 60 seconds.