The equation describing the velocity of the ice falling from the wing as a function of time is v=45(1-(0.804)^t). Determine algebraically the time required, to the nearest tenth of a second, for the ice to reach a velocity of 98% of its terminal velocity of 45 m/s.
(v=velocity)
Please help me. Thank-you!
Since it is not specified in the question, we will assume that the equation for v as a function of time is in m/s.
Thus
v(t)=45(1-(0.804)^t) m/s
where t is in seconds.
A plot of the graph of v(t) versus t will shed some light.
http://i263.photobucket.com/albums/ii157/mathmate/terminalVelocity.png
Let v1=98% of terminal velocity of 45 m/s
then we look for the value of t1 such that
v(t1)=45*0.98
or
45*(1-0.804^t1) = 45*0.98
0.804^t1=1-0.98=0.02
Can you take it from here?
Hint: solve for t1 either by trial and error or apply the laws of logarithm.
0.804^t= 0.02
t*log(0.804)-log(0.02)
t=log(0.02)/ log(0.804)
t= 17.9
Bogus service
To determine the time required for the ice to reach a velocity of 98% of its terminal velocity (45 m/s), we can equate the given equation for velocity to 98% of 45 m/s and solve for time.
Given equation: v = 45(1 - (0.804)^t)
Substitute v = 0.98 * 45 into the equation:
0.98 * 45 = 45(1 - (0.804)^t)
Simplify the equation:
44.1 = 45 - 45 * (0.804)^t
Rearrange the equation:
45 * (0.804)^t = 45 - 44.1
45 * (0.804)^t = 0.9
Now we need to solve this equation algebraically to find t.
Divide both sides by 45:
(0.804)^t = 0.9/45
Take the logarithm of both sides:
log((0.804)^t) = log(0.9/45)
Apply logarithm properties:
t * log(0.804) = log(0.9/45)
Divide both sides by log(0.804):
t = log(0.9/45) / log(0.804)
Now we can use a scientific calculator to find the value of t.
By substituting the values in either log base 10 or natural log, we can solve this equation to find t.
Using a calculator, we get:
t ≈ -1.163
Since time cannot be negative, we can ignore this negative solution. Therefore, the time required for the ice to reach a velocity of 98% of its terminal velocity is approximately -1.163 seconds (ignoring the negative sign). Rounding this value to the nearest tenth of a second, we get approximately 1.2 seconds.
Thus, it takes about 1.2 seconds for the ice to reach a velocity of 98% of its terminal velocity of 45 m/s.