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Tapping an artifact won't cause its abilities to stop applying unless those abilities say so.
If an artifact you control has a mana ability with in the cost, activating that ability while casting a spell with improvise will result in the artifact being tapped when you pay the spell's costs. You won't be able to tap it again for improvise. Similarly, if you sacrifice an artifact to activate a mana ability while casting a spell with improvise, that artifact won't be on the battlefield when you pay the spell's costs, so you won't be able to tap it for improvise.
Improvise doesn't change a spell's mana cost or mana value.
Equipment attached to a creature doesn't become tapped when that creature becomes tapped, and tapping that Equipment doesn't cause the creature to become tapped.
When calculating a spell's total cost, include any alternative costs, additional costs, or anything else that increases or reduces the cost to cast the spell. Improvise applies after the total cost is calculated.
Improvise can't pay for in Metallic Rebuke's total cost.
