On the reality TV show Last Man Standing, a package is thrown from a plane to the ocean below. The contestants must swim to the package to receive the “free pass” located inside the package. The path that the package follows can be modelled by the quadratic function d t 4 9. t t 10 1200 2 ^ h =- + + , where d represents the distance above the water, in metres, and t is the time, in seconds.
How long has the package been in the air when it hits the water?
3 years ago
Answered By Emily D
This is a mean way to put physics into math! It's a lot easier to work with these problems by drawing out diagrams
$d\left(t\right)=-49t^2+10t+1200$d(t)=−49t2+10t+1200
To see the graph I added, you may want to adjust the window settings so your x-axis range is [0,10] and your y-axis range is [-50, 1250]
Since we're looking for when the package hits the water, we want the height above the water to be 0
$d\left(t\right)=0=-49t^2+10t+1200$d(t)=0=−49t2+10t+1200
$0=-49t^2+10t+1200$0=−49t2+10t+1200 now just need to solve for t
I don't know if 20-1 still uses the quadratic formula, so please let me know if this is something your teacher wants you to solve a different way:
$t=\frac{-10\pm\sqrt{10^2-4\left(-49\right)\left(1200\right)}}{2\left(-49\right)}$t=−10±√102−4(−49)(1200)2(−49)
$t\approx-4.8477$t≈−4.8477 or $t\approx5.0518$t≈5.0518
Now we need to decide which one of those answers (-4.8s or 5.1s) is correct. The negative answer would suggest our package had been in the water before we threw it, so that can't be right! That leaves us with only one possibility
$t\approx5.05$t≈5.05 seconds when the package hits the water
As an aside in case you take physics: the polynomial we just used is the same as
$h\left(t\right)=h_0+v_0\cdot t+\frac{1}{2}a\cdot t^2$h(t)=h0+v0·t+12 a·t2
Where h0 is your starting height, v0 is the initial velocity of the package, and a is the acceleration acting on the package (gravity, in your example)
Attached Graph: