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Answer: There are two problems with extracting hot water from the faucet when the water is heated by an electric boiler. One is that it will cause the thermostat to heat water sooner than it would have had you not used it. This is a problem in terms of connecting an electric circuit, and, more importantly, that a glowing filament or a gas flame will go on, as well as cold water being heated in the process. All of the latter involve Torah-level Shabbat melachot. It is a complicated question whether there is a Torah-level violation for the person who takes out the water, as the normally delayed reaction makes it likely that it is a form of gerama (indirect causation). We will not go further into this interesting question because you describe your system as off when you want to use it, until the end of Shabbat.
The other problem that occurs when you extract hot water from the tank is that cold water rushes into the tank to take its place and mixes in with the remaining hot water. The systems are designed in such a manner that the cold water enters on the bottom, whereas the hot water (due to the physical properties of heat rising and hot water being less dense than cold water) for the most part stays on top and does not fully heat the cold water below. However, it can be assumed that if the hot water is hot enough, at least a small amount of the cold water will reach the forbidden level of yad soledet bo (113°F, 45°C). Even in this case, there is some amount of leniency in that you do not place the cold water in the hot water tank but it goes in based on properties of physics after you open up your tap (see Yabia Omer IV, Orach Chayim 35). However, at least under normal circumstances, it is forbidden to remove water from a hot-water tank (Shemirat Shabbat K’hilchata 1:39). If you were referring to a tank that had only solar-heated hot water in it, then this would very likely be permitted (see the famous leniency and its partial withdrawal in Shemirat Shabbat K’hilchata (ed. I, 1:31, and ed. II 1:45), as water heated by the sun is not fundamentally deemed to cook other things (Shabbat 39a). But, here too, we will be brief because this is not your case.
There are several practical ways to obtain lukewarm water. One is to not have the heating system go on on Shabbat and prepare things before Shabbat so that the water left in the tank does not heat the incoming cold water to yad soledet bo. Based on the physics discussed above, if you remove a nice amount of hot water before Shabbat after shutting off the heating system, the incoming water will not come in contact with hot water (Shemirat Shabbat K’hilchata ed. I, ibid.).
Another way to have warm water is to use a different source of hot water. If you remove hot water from a Shabbat urn, you can use it to create warm water in a container in two different ways. Many are used to using an extra cup in order to have a kli shlishi in which they put their tea, as many poskim (see Igrot Moshe, OC IV:74.15; Shemirat Shabbat K’hilchata 1:57 is slightly more stringent) posit that kli shilishi does not cook even kalei habishul (easily cooked food). However, regarding water, all agree that it is permitted to put it into a kli sheini (Shabbat 40b; Shulchan Aruch, Orach Chayim 318:13), so that the extra cup is unnecessary. (Just as when making tea, one should make sure that there is not a small amount of (never heated) water in the cup before pouring into the kli sheini (Igrot Moshe ibid. 19).
If there is a nice amount of cold water in a cup one can pour in a small amount of hot water if the resulting mixture will clearly be less hot than yad soledet bo (Rama, OC 318:12; Mishna Berura ad loc. 84). This is a good idea only if one is confident he will remember to be careful about the amount.
Bemare Habazak - Rabbis Questions (654)
Rabbi Daniel Mann
499 - Beracha on Hearing Aids
500 - Using Hot Water on Shabbat
501 - Different Drinks for Kiddush
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Kuzari -Rabbi Ari Shvat Kuzari class 8- "Answering Questions on the Kuzari's Proof from Mass Revelation
Rabbis in Midrash
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Kuzari -Rabbi Ari Shvat Kuzari class 9 - "Seeing is Believing" (parag. 21-30)
These paragraphs elaborate on the theme that seeing and knowing is better than any attempt to prove logically, and begins explaining the difference between Israel and gentiles.
Ein Aya Various Universal Stages of the Geula Process
Rav Kook examines the various stages of redemption, explaining how (in addition to the obvious oft-mentioned stages of ingathering the exiles, reviving the Hebrew language, army, state etc.) the messianic dream of world prosperity, the State of Israel and world unity can and are realistically and logically gradually coming true.
Kuzari -Rabbi Ari Shvat Kuzari class 8- "Answering Questions on the Kuzari's Proof from Mass Revelation
How do we know that the "claim" of mass revelation to 2,000,000 witnesses at Mt. Sinai is really true? This important class answers all of the questions skeptics ask about this claim of the Kuzari.
Ein Aya Armies Still Necessary for Balance & the War Against Wars
Rav Kook explains why the world was originally divided into the various seemingly contradicting ideologies and cultures, in order to develop each one respectively. Swords or armies symbolize how each respective ideology defends themselves, as well as deters their opposing ideologies and cultures. On the other hand, the messianic era will be one of peace, and Rav Kook explains the transition to that stage, which mankind is already undergoing.
The Land of Israel LGBT'S IN ISRAEL
The question was asked, how can one make Aliyah with the LGBT parades?
Kuzari -Rabbi Ari Shvat Kuzari class 7 - Five Accumulative Proofs of G-d
As a preparation for the Kuzari's classic proof of G-d from the mass-revelation at Sinai, we start here with 5 other directions to strengthen our belief which also contribute to what the Kuzari will present as well.
Ein Aya Muscle & Meaning: The Dual Nature of Gevurah (Physical Strength)
Is physical strength and fitness a necessity or an ideal? Although it if often totally overlooked among topics of Judaism, Rav Kook writes that it clearly is also a necessity to deter the many enemies of Israel, but even in Y'mot HaMashiach, in the Messianic era, to a certain extent, it's ideal continues even after our enemies will have been finished off.
