# The fuel of a rocket is launched is given by -x^2 - 140x +2000. During what period of time is the mass of the fuel greater than 500t?

Jun 19, 2018

The time period is:

$0 \text{ s" <= x < 10" s}$

#### Explanation:

I am assuming that the function gives the weight of the fuel (in tons) and that the time variable $x$ has the domain $x \ge 0$.

$w \left(x\right) = - {x}^{2} - 140 x + 2000 , x \ge 0$

Please observe that at $x = 0$ the weight of the fuel is $2000 \text{ tons}$:

$w \left(0\right) = - {0}^{2} - 140 \left(0\right) + 2000$

$w \left(0\right) = 2000 \text{ tons}$

Let's find the time where the weight of the fuel is $500 \text{ tons}$:

$500 = - {x}^{2} - 140 x + 2000 , x \ge 0$

$0 = - {x}^{2} - 140 x + 1500 , x \ge 0$

$0 = {x}^{2} + 140 x - 1500 , x \ge 0$

Factor:

$0 = \left(x - 10\right) \left(x + 150\right) , x \ge 0$

Discard the negative root:

$x = 10 \text{ s}$

The time period is:

$0 \text{ s" <= x < 10" s}$