A large birthday cake was baked to celebrate the 6th birthday of Mimi. It was a round cake, made of two layers of vanilla cake; each layer was 18 inches in diameter and 3.5 inches tall. Between the two layers was a quarter inch of chocolate frosting. And covering the top and side of the cake was 3/8" of buttercream frosting.
If the vanilla cake has 140 calories per cubic inch, and the chocolate frosting has 320 calories per cubic inch, and the buttercream frosting has 470 calories per cubic inch, how many 800-calorie servings are in the cake? Please round down to the nearest serving.
To determine the number of 800-calorie servings in the cake, we first need to calculate the volume of the cake and the volume of each layer.
The volume of a cylindrical cake can be calculated using the formula:
Volume = π * (radius^2) * height
Given that each layer of the vanilla cake has a diameter of 18 inches and a height of 3.5 inches, we can calculate its volume as follows:
Radius = Diameter / 2 = 18 / 2 = 9 inches
Volume of a layer of vanilla cake = π * (9^2) * 3.5
Next, we need to account for the quarter inch of chocolate frosting between the two layers. This forms a layer with a smaller diameter and the same height as the vanilla cake layer.
We subtract the quarter inch from the diameter of the vanilla cake layer:
Diameter of chocolate frosting layer = 18 - 0.25 = 17.75 inches
Radius of chocolate frosting layer = Diameter / 2 = 17.75 / 2 = 8.875 inches
Volume of a layer of chocolate frosting = π * (8.875^2) * 3.5
Now, we add the volumes of the two layers to find the total volume of the cake.
Total volume of the cake = Volume of vanilla cake layer + Volume of chocolate frosting layer
Finally, we divide the total volume of the cake by the volume of an 800-calorie serving to find the number of servings (rounding down to the nearest serving, as per the question).
Number of servings = Total volume of the cake / Volume of an 800-calorie serving (rounded down)
Let's calculate the values to determine the number of servings.
Total calories=140*cake volume+320Frosting volume.
Frosting volume= volume middle+volume top+ volumesides.
= PI*9^2*1/4 + PI*9^2*3/8 + (3.5+1/4+3.5+3/8)(PI*18)
You find the volume of the cake, then proceed to find total calories, and divide that by 800.
volume of vanilla cake = pi(9^2)(7)
that answer times 140 gives vanilla calories.
volume of chocolate frosting = pi(9^2)(.25)
that times 320 gives you chocolate calories.
frosting around the outside = pi(9.375^2)(7.25) - pi(9^2)(7.25)
= icing-1
for the top frosting
= pi(9.375^2)(.375)
= icing-2
calories for frosting = 470(icing-1 + icing-2)
add up all the calories.
you do the math!