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PLEASE HELP!!
An ice cube is freezing in such a way that the side length s, in inches, is s of t equals one half times t plus 4 comma where t is in hours. The surface area of the ice cube is the function A(s) = 6s2.

Part A: Write an equation that gives the volume at t hours after freezing begins. (2 points)

Part B: Find the surface area as a function of time, using composition, and determine its range. (4 points)

Part C: After how many hours will the surface area equal 294 square inches? Show all necessary calculations. (4 points)

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Respuesta :

MrRoyal

The freezing of the ice cube is an illustration of composite functions.

  • The expression of volume is: [tex]\mathbf{V(t) = (\frac 12 t + 4)^3}[/tex]
  • The expression of area is: [tex]\mathbf{A(t) = 6(\frac{1}{2}t + 4)^2}[/tex]
  • The surface area will equal 294 square inches after 6 hours

The side length is given as:

[tex]\mathbf{s(t) = \frac{1}{2}t + 4}[/tex]

The area is given as:

[tex]\mathbf{A(s) = 6s^2}[/tex]

(a) Expression for volume, in terms of time (t)

The volume of a cube is:

[tex]\mathbf{V(s) = s^3}[/tex]

Substitute [tex]\mathbf{s(t) = \frac{1}{2}t + 4}[/tex]

[tex]\mathbf{V(t) = (\frac 12 t + 4)^3}[/tex]

(b) Surface area, in terms of t

We have:

[tex]\mathbf{A(s) = 6s^2}[/tex]

Substitute [tex]\mathbf{s(t) = \frac{1}{2}t + 4}[/tex]

[tex]\mathbf{A(t) = 6(\frac{1}{2}t + 4)^2}[/tex]

(c) At what time is the surface area, 294

Substitute 294 for A in [tex]\mathbf{A(t) = 6(\frac{1}{2}t + 4)^2}[/tex]

[tex]\mathbf{294) = 6(\frac{1}{2}t + 4)^2}[/tex]

Divide both sides by 6

[tex]\mathbf{49 = (\frac{1}{2}t + 4)^2}[/tex]

Take square roots of both sides

[tex]\mathbf{7 = \frac{1}{2}t + 4}[/tex]

Subtract 4 from both sides

[tex]\mathbf{3 = \frac{1}{2}t}[/tex]

Multiply both sides by 2

[tex]\mathbf{t = 6}[/tex]

Hence, the surface area will equal 294 square inches after 6 hours

Read more about composite functions at:

https://brainly.com/question/10830110

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