Question

You and your mother bakes and decorates sheet cakes for special occasions. one of your costumers wanted you to reduced the volume of a small cake to be 351 cubic inches. the cake is in the shape of a rectangular solid. She wanted the length of the cake to be four inches longer than the width of the cake and the height of the cake to be one-third of the width. what should the dimensions of the cake pan be?​​

Answer 1

Answers:

• Length = 13 inches
• Width = 9 inches
• Height = 3 inches

=========================================================

Step-by-step explanation:

• x = width, any positive real number
• x+4 = length, because it's 4 more than the width
• x/3 = one third the width = height

Dividing by 3 is the same as multiplying by 1/3.

Multiply those expressions to get the volume.

Length*width*height = volume

(x+4)*(x)*(x/3) = 351

(1/3)*x^2*(x+4) = 351

x^2(x+4) = 3*351

x^3+4x^2 = 1053

x^3+4x^2-1053 = 0

From here, we need to use a graphing calculator to find the x intercept(s).

We could use the rational root theorem to check each possible rational root (assuming there are any to begin with), but such a task is a bit of tedious busywork in my opinion. I prefer to stick to technology whenever possible.

On your graphing calculator, use the "root" or "x intercept" feature. If you typed y = x^3+4x^2-1053 into Desmos (a free online graphing tool), then you can simply click on the x intercept to have its coordinates show up. You may have to adjust the window.

In this case, there's only one real number root and it's when x = 9. The other two remaining roots are nonreal complex numbers, which we'll ignore.

So again, the only practical value is x = 9 which means the width is 9 inches.

Use that value of x to find the other dimensions

length = x+4 = 9+4 = 13 inches

height = x/3 = 9/3 = 3 inches

-----------------------

Check:

length*width*height = 13*9*3 = 351

This confirms the correct dimensions.

Answer 2

Dimensions of the cake: 13 in (L) × 9 in (W) × 3 in (H)

Step-by-step explanation:

Given the volume of the small cake, 351 in³:

Width (W)

Length (L) = 4 + w

Height (H) =
`(1)/(3)w`

The formula for the volume of a rectangular prism is:

L × W × H = V

Substitute the given values into the formula:

`(4 + w) (w) ((1)/(3)w) = 351`

`(4)/(3)w^(2) + (1)/(3)w^(3) = 351`

Multiply both sides by 3:

`(3)(4)/(3)w^(2) + (3)(1)/(3)w^(3) = 351(3)`

4w² + w³ = 1051

Subtract 1051 from both sides:

w³ + 4w² - 1051 = 1051 - 1051

w³ + 4w² - 1051 = 0

In reference to the Rational Zero Theorem, w = 9 provides a remainder of 0 through synthetic division.

w = 9 represents the width. Substitute this value into the given values:

V = L × W × H

Width (W) = 9 inches

Length (L) = 4 + w = 4 + 9 = 13 inches

Height (H) =
`(1)/(3)w` = 3 inches

Therefore, the dimensions of the cake are: 13 in (L) × 9 in (W) × 3 in (H)