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I'm planning to bake this cake.

http://sallysbakingaddiction.com/2015/03/23/my-favorite-carrot-cake-recipe/

BUT...Instead of making a two layer cake, I would like to make a one layer cake in a 9-inch round cake tin. I was thinking that I could make 3/4 of the recipe, which would give me one layer that is a little bit thicker than in the recipe.

Other than baking for a longer amount of time, do I have to make any adjustments to the oven temperature? Am I risking having a dense cake that doesn't bake through?

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Whenever you make a cake deeper you risk of a denser cake for a couple of reasons. The first is because the lift you get from the action of the leavening agents and expansion of air and steam will be partially offset by the increased weight that needs to be pushed up. Second, you may lose more lift than a thinner cake because the center of the cake will take longer to crystallize - it may start to sag back down before it hardens up.

Because it takes longer to cook the center of a thicker cake the outside of your cake might be more overcooked than it would with a thinner cake.

How much of a difference this actually makes is very hard to say, it depends on the recipe and other factors. Most cake recipes are pretty flexible, so there's no reason you can't make it thicker and get a good result. You'll want to reduce temperature and increase cooking time, how much is hard to say, I'd try decreasing your temperature by 15F, and as a guide add at least 10 minutes to the cooking time.

To take the guesswork out of it I would suggest using an instant read thermometer to check if it is done rather than a toothpick or skewer, the cake will be done between 205-210F.

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I would definitely reduce the heat as well as there is more dough to bake. At a higher temperature, you risk burning your cake from the outside while it doesn't cook through.

Maybe drop the temperature 10º and see what happens.

The cake is done when it separates from the mold and when you have a "toothpick inserted into the center come out clean".

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