Answer
The energy required to boost the shuttle to the new orbit is $1.16\times 10^{11}~J$
Work Step by Step
We can find the change in mechanical energy when the shuttle moves to a higher orbit. Let $M_e$ be the earth's mass and let $M_s$ be the shuttle's mass. Let $R$ be the earth's radius.
$\Delta E_{mech} = \frac{1}{2}\Delta U$
$\Delta E_{mech} = \frac{1}{2}(U_f-U_0)$
$\Delta E_{mech} = \frac{1}{2}[-\frac{G~M_e~M_s}{R+610~km}-(-\frac{G~M_e~M_s}{R+250~km})]$
$\Delta E_{mech} = \frac{1}{2}(\frac{G~M_e~M_s}{R+250~km}-\frac{G~M_e~M_s}{R+610~km})$
$\Delta E_{mech} = \frac{1}{2}(G~M_e~M_s)(\frac{1}{R+250~km}-\frac{1}{R+610~km})$
$\Delta E_{mech} = \frac{1}{2}(6.67\times 10^{-11}~m^3/kg~s^2)(5.98\times 10^{24}~kg)(75,000~kg)(\frac{1}{6.63\times 10^6~m}-\frac{1}{6.99\times 10^6~m})$
$\Delta E_{mech} = 1.16\times 10^{11}~J$
The energy required to boost the shuttle to the new orbit is $1.16\times 10^{11}~J$