Consider f and c below. f(x, y, z) = yzexzi + exzj + xyexzk, c: r(t) = (t2 + 5)i + (t2 − 1)j + (t2 − 5t)k, 0 ≤ t ≤ 5(a) find a function f such that f = ∇f. f(x, y, z) = (b) use part (a) to evaluate cf · dr along the given curve c.

Aspacecraft can attain a stable orbit 300 kilometers above earth if it reaches a velocity of 7.7 kilometers per second. the formula for a rocket's maximum velocity v in kilometers per second is vequalsminus0.0098tplusc ln upper r, where t is the firing time in seconds, c is the velocity of the exhaust in kilometers per second, and r is the ratio of the mass of the rocket filled with fuel to the mass of the rocket without fuel. find the velocity of a spacecraft whose booster rocket has a mass ratio of 20, an exhaust velocity of 2.1 km/s, and a firing time of 15 s. can the spacecraft achieve a stable orbit 300 km above earth?