In a large mixing bowl, combine flour, sugar, baking soda, cinnamon, nutmeg and salt.
In a separate small bowl, combine the applesauce, oil and eggs. Add to the flour mixture, stirring until the ingredients are well blended. Add the carrots and mix again.
Pour the batter into a greased 9" tube pan, or use a 9" x 13" rectangular cake pan, or two bread loaf pans, or a cupcake pan.
Bake the cake in a preheated 350° F oven for about 1 hour, 10 minutes, or until a toothpick inserted in the thickest part of the cake comes out clean. Set the cake on a wire rack for five minutes. Then, run a knife around the edges of the pan to loosen the cake, and turn the cake out onto the rack to cool.