Hi
Can anyone help me solve the quadratic equation for the following:-
The motion of a two stage rocket is stated by it height y in metres above the ground since t seconds since launch.
Flies vertically Ist stage burns for 5 seconds and cuts out. 2nd stage rocket is under gravity alone.
modelled by the equations
Stage 1 height y = 10t2
time 0< t <5
stage 2 height y = - 5t2 + 150t - 375
time t >5
I need this to program my Ti -83 but am lost ...
Thanks :-)
Hi there!
To solve this quadratic equation, you need to find the values of "t" when the height "y" equals zero. In other words, you need to find the values of "t" when the rocket hits the ground.
Let's start with the first stage equation:
Stage 1 height equation: y = 10t^2
To find the values of "t" when y equals zero, we can set the equation equal to zero:
0 = 10t^2
Next, divide both sides of the equation by 10:
0 = t^2
Since any number squared is always positive or zero, this equation tells us that "t" can only be zero to make the equation true. However, the constraint for the first stage is 0 < t < 5 (time between launch and when the first stage burns out). So, "t" cannot be zero in this case.
Moving on to the second stage equation:
Stage 2 height equation: y = -5t^2 + 150t - 375
Again, we need to set the equation equal to zero to find the values of "t" when y equals zero:
0 = -5t^2 + 150t - 375
To solve this equation, you can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = -5, b = 150, and c = -375.
Substituting the values into the quadratic formula:
t = (-(150) ± √((150)^2 - 4(-5)(-375))) / (2(-5))
Simplifying:
t = (-150 ± √(22500 - 30000)) / (-10)
t = (-150 ± √(-7500)) / (-10)
At this point, the term inside the square root is negative, which means there are no real solutions. However, we can still find the complex solutions by evaluating the square root of -7500.
Calculating the square root of -7500:
√(-7500) = √(-1 * 7500) = √(-1) * √(7500) = i√(7500)
Therefore, the solutions for "t" are:
t = (-150 ± i√(7500)) / (-10)
You can plug these complex solutions into your Ti-83 calculator to obtain the numerical values.
I hope this explanation helps you solve the quadratic equation for your two-stage rocket model. If you have any further questions, feel free to ask!