Janet bakes a cake in the shape of a cube with side length x (in inches). She makes a vertical cut parallel to one of the sides, resulting in a slice with width y where x is 5 times as large as y.

a) Define a function h that determines the volume of the cake before Janet makes the first cut in terms of the original width of the cake, x
b) Define a function g that determines the width of the slice of cake in terms of the original width of the cake, x.
c) Define a function f that determines the volume of the slice of cake in terms of the original width of the cake, x.
d. Using the functions you defined in part (a) and (c), represent the volume of the cake that remains after Janet eats this slice in terms of the original width of the cake, x.
A. h(x)-f(x)
B. f(h(x))
C. f(x)+h(x)
D. f(x)-h(x)
E. h(f(x))
F. None of the above
e) Define a function k that determines the remaining volume of cake in terms of the original width of the cake, x.
f) Use the function you defined in part (e) to determine the remaining volume of cake if the original width of the cake was 10 inches.

a) The volume of a cube is calculated by taking the length of one side and cubing it. Therefore, the volume of the cake before the first cut can be defined as:

h(x) = x^3

b) According to the problem, the width of the slice is y and the side length of the cube is 5 times as large as the width. Therefore, we can define the width of the slice in terms of the original width of the cake as:

g(x) = x / 5

c) The volume of a slice of cake can be calculated by taking the length, width, and height of the slice. The height of the slice is equal to the original width of the cake, x, while the width is y. The length can be determined by subtracting the width of the slice from the original width of the cake. Therefore, the volume of the slice can be defined as:

f(x) = (x - g(x)) * y * x

d) The volume of the cake that remains after Janet eats the slice can be found by subtracting the volume of the slice from the volume of the cake. Using the functions defined in parts (a) and (c), we can represent the remaining volume of the cake as:

h(x) - f(x)

Therefore, the correct answer is A: h(x) - f(x).

e) To determine the remaining volume of cake in terms of the original width of the cake, x, we need to determine the volume of the cake that remains after the first cut. This can be calculated by subtracting the volume of the slice, defined by function f(x), from the volume of the cake, defined by function h(x). Therefore, we can define the function k as:

k(x) = h(x) - f(x)

f) To determine the remaining volume of cake when the original width of the cake is 10 inches, we can substitute x = 10 into the function k(x):

k(10) = h(10) - f(10)

To find the values for h(10) and f(10), we can substitute x = 10 into the functions h(x) and f(x):

h(10) = 10^3
h(10) = 1000

f(10) = (10 - g(10)) * y * 10

To find g(10), we substitute x = 10 into the function g(x):

g(10) = 10 / 5
g(10) = 2

Substituting g(10) into the function f(10):

f(10) = (10 - 2) * y * 10

Now we have enough information to calculate the remaining volume of cake:

k(10) = h(10) - f(10)
k(10) = 1000 - (10 - 2) * y * 10

Simplifying further:

k(10) = 1000 - 8y

Therefore, the remaining volume of the cake when the original width is 10 inches is 1000 - 8y.