Let
x = {5, 10, 15, 20, 25, 30, 35, 40}.
define a relation

on x as follows. for
a, b is in x,

a

b
if b can be obtained from a by adding a (possibly empty) collection of dimes (10 cents) and quarters (25 cents). so, for example,
25

35, but 25

30.

list all minimal elements of
(x,

).
(enter your answers as a comma-separated list.)

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?