Originally Posted by
Colonel
Rahab believed on accord of the miracles that God had performed during the Exodus :
Joshua 2:8 Now before they lay down, she came up to them on the roof, 9 and said to the men: "I know that the Lord has given you the land, that the terror of you has fallen on us, and that all the inhabitants of the land are fainthearted because of you. 10 For we have heard how the Lord dried up the water of the Red Sea for you when you came out of Egypt, and what you did to the two kings of the Amorites who were on the other side of the Jordan, Sihon and Og, whom you utterly destroyed. 11 And as soon as we heard these things, our hearts melted; neither did there remain any more courage in anyone because of you, for the Lord your God, He is God in heaven above and on earth beneath. 12 Now therefore, I beg you, swear to me by the Lord, since I have shown you kindness, that you also will show kindness to my father's house, and give me a true token, 13 and spare my father, my mother, my brothers, my sisters, and all that they have, and deliver our lives from death."
Many others probably did too and believed like demons do (James 2) yet acted differently. Like many Israelites did when they worshipped the golden calf while Moses was on the mountain with God, even though God was present in the fire up there while they were doing that. Rahab did not only believe from a position of only having heard about those things but she also became a proselyte to the Jewish religion by effectively abandoning her identity as belonging to the people inhabiting the city of Jericho, taking her place among the Jews.
There is no mention of Hezekiah exercising faith in God's will to heal him. To the contrary, Isaiah had just told him that God said to him that he was going to die. His prayer was a request for God to hear him, not a confirmation of his faith in God already having changed his mind. In addition, Hezekiah requests a sign that God is going to heal him which he wouldn't have bothered with if he had faith for healing. Maybe Hezekiah believed God after seeing the sign that he had requested but this didn't begin with Hezekiah having faith for healing. That is a matter of reading that into the passage.
Rahab had a revelation of God's intention to have the Israelites take the land and had something to base her faith on. Hezekiah had nothing to base faith for healing on, he even had Isaiah's statement to the contrary. There were the promises concerning health and healing for those who kept the law but that was works based rather than faith based. God's statement that he was going to die effectively declared that he had not been found worthy of seeing the fulfillment of those promises. Everything beyond that was a matter of pleading with God, just like you might plead with God for his setting up a better president in your country with the next election but you cannot simply exercise faith for that president based on a promise in the Word that God will simply do that. The prayers of the saints might still move God to tip the scales so that that happens though.