An ice cube is freezing in such a way that the side length [tex]$s$[/tex], in inches, is [tex]$s(t)=\frac{1}{2} t+4$[/tex], where [tex]$t$[/tex] is in hours. The surface area of the ice cube is the function [tex]$A(s)=6 s^2$[/tex].
Part A: Write an equation that gives the volume at [tex]$t$[/tex] hours after freezing begins.
Part B: Find the surface area as a function of time, using composition, and determine its range.
Part C: After how many hours will the surface area equal 294 square inches? Show all necessary calculations, and check for extraneous solutions.
Answer (2)
The volume of the ice cube as a function of time is V ( t ) = ( 2 1 t + 4 ) 3 .
The surface area of the ice cube as a function of time is A ( t ) = 6 ( 2 1 t + 4 ) 2 , with a range of [ 96 , ∞ ) .
Setting the surface area equal to 294 square inches, we solve for t and find t = 6 hours.
Discarding the extraneous solution t = − 22 , the surface area equals 294 square inches after 6 hours.
Explanation
Problem Analysis We are given the side length of an ice cube as a function of time, s ( t ) = 2 1 t + 4 , and the surface area as a function of side length, A ( s ) = 6 s 2 . We need to find the volume as a function of time, the surface area as a function of time and its range, and the time when the surface area is 294 square inches.
Finding Volume as a Function of Time Part A: The volume of a cube is given by V ( s ) = s 3 . To find the volume as a function of time, we substitute s ( t ) into the volume equation:
Volume Equation V ( t ) = ( 2 1 t + 4 ) 3
Finding Surface Area as a Function of Time Part B: To find the surface area as a function of time, we compose the functions A ( s ) and s ( t ) :
Surface Area Equation A ( t ) = A ( s ( t )) = 6 ( 2 1 t + 4 ) 2
Range of Surface Area Function To find the range of A ( t ) , we consider the domain of t . Since the ice cube is freezing, we can assume t ≥ 0 . As t increases, A ( t ) also increases. When t = 0 , A ( 0 ) = 6 ( 2 1 ( 0 ) + 4 ) 2 = 6 ( 4 ) 2 = 6 ( 16 ) = 96 . Since t can increase without bound, the surface area can also increase without bound. Therefore, the range of A ( t ) is [ 96 , ∞ ) .
Finding Time When Surface Area is 294 Part C: To find the time when the surface area is 294 square inches, we set A ( t ) = 294 and solve for t :
Setting up the Equation 6 ( 2 1 t + 4 ) 2 = 294
Simplifying the Equation Divide both sides by 6:
Further Simplification ( 2 1 t + 4 ) 2 = 49
Taking the Square Root Take the square root of both sides:
Two Possible Solutions 2 1 t + 4 = ± 7
Solving for t Solve for t for both cases: 2 1 t + 4 = 7 and 2 1 t + 4 = − 7 .
Finding the Valid Solution Case 1: 2 1 t + 4 = 7 ⇒ 2 1 t = 3 ⇒ t = 6 . Case 2: 2 1 t + 4 = − 7 ⇒ 2 1 t = − 11 ⇒ t = − 22 . Since time cannot be negative, t = − 22 is an extraneous solution.
Final Answer Therefore, the surface area will equal 294 square inches after 6 hours.
Examples
Imagine you're designing ice sculptures for a winter festival. Knowing how the volume and surface area of the ice change over time, as demonstrated in this problem, helps you predict the melting rate and adjust your designs to ensure they last through the event. This blend of math and practical knowledge is crucial for artists and engineers alike.
The volume of the ice cube at time t is given by V ( t ) = ( 2 1 t + 4 ) 3 . The surface area as a function of time is A ( t ) = 6 ( 2 1 t + 4 ) 2 , with a range of [ 96 , ∞ ) . The surface area reaches 294 square inches after 6 hours.
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