Question

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?

Answer 1

The time period is:

`0" 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>= 0`.

`w(x) = -x^2 - 140x +2000, x >=0`

Please observe that at `x = 0` the weight of the fuel is `2000" tons"`:

`w(0) = -0^2 - 140(0) +2000`

`w(0) = 2000" tons"`

Let's find the time where the weight of the fuel is `500" tons"`:

`500 = -x^2 - 140x +2000, x >=0`

`0 = -x^2 - 140x +1500, x >=0`

`0 = x^2 + 140x -1500, x >=0`

Factor:

`0 = (x-10)(x+150), x>=0`

Discard the negative root:

`x = 10" s"`

The time period is:

`0" s" <= x < 10" s"`