The Cake Is Not a Lie Behold this beautiful 5-layer cake, suitable for weddings,
ID: 2864162 • Letter: T
Question
The Cake Is Not a Lie Behold this beautiful 5-layer cake, suitable for weddings, bat/bar-mitzvahs, or Calculus class. The cake is 15 inches high, so each layer is exactly 3 inches. The width of each layer varies linearly from 7.5 inches at the bottom to 1.5 inches at the top. Write a formula for the width of the cake as a function of h. The length of each layer varies linearly from 10 inches at the bottom to 2 inches at the top. Write a formula for the length of the cake as a function of h. Pretend the height of each layer is delta h. Write an expression for the volume of any layer of the cake. Express your formula using only the variable h, using your equations for the previous two parts. Using sigma notation (with i = 1 to 5), write an expression for the volume of the entire cake.Explanation / Answer
We have,
W = mh + b
The points are : (0, 7.5) and (12, 1.5)
7.5 = m(0) + b
b = 7.5
Again,
W = mh + b
1.5 = m(12) + b
1.5 = 12m + 7.5
12m = 1.5 - 7.5
12m = -6
m = -0.5
Now we see that,
L = mh + b
The points are : (0, 10) and (12, 2)
10 = m(0) + b
10 = 0 + b
b = 10
Again,
2 = m(12) + b
2 = 12m + 10
12m = 2 - 10
12m = -8
m = -(8/12)
m = -(2/3)
The total volume will be:
V = (10)(7.5)(3) + (8)(6)(3) + (6)(4.5)(3) + (4)(3)(3) + (2)(1.5)(3)
V = 225 + 144 + 81 + 36 + 9
V = 495
Hence the entire volume of the cakke is 495 cubic inches.