My cake recipe calls for baking at 350°F for 55 minutes.
I had to bake it for 15-20 minutes longer for inside to be done.
Can I adjust the temperature to 360°F to bake for 55 minutes without burning the cake?
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If you used a reputable recipe, followed it without variation, including using the same size pan as was specified, and adjusted for altitude, if appropriate, then:
The problem was likely with the thermostat calibration on your oven. The best thing to do is get an oven thermometer and use it to find out how much you need to adjust your temperature control. This answer to a question similar to yours may help with that.
If you can't do that right now - adjusting the temperature up in small increments (10-25 degrees F) and noting what happens, until you get the desired results, is a good plan. Remember that there will always be variables that can affect the cooking time and that most times given are approximate.
I believe the answer is no. Heat moves insde food (the technical name is "conducts") in a roughly similar rate. Doubling the temperature won't cut the baking time in half, and other intuitive calculations might not work as well.
In fact, too many factors influence the heat and heating rate, from the calibration of the oven's thermostat (perhaps it's not really 350F?) to the tickness of the cake. Recipes work best in the kitchens of their composers.
I suggest you try the following things:
calibrate your oven. Sugar begins to melt in 320F. Heat your oven to 320, put a little sugar inside. Wait. If it doesn't melt. If it never melts, you need to fix your thermometer.
Let the cake stand for 10 minutes after you pull it out of the oven. Residue heat will actually keep cooking its inside, and may raise the temperatures in a few degrees.
Use 10% less batter, with the same time and temperature. Thinner cake needs less time.
finally, if and when you need to bake it longer - LOWER the oven's temperature. That way you'll give it time to finish the inside without overwhelming and drying the outside.
Think about the food, not the appliance.