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Pre Calculus – Dimensions of a box grow at 2 inches per second

by Ashley
(Mount Laurel, NJ, USA)











































Dimensions of a Growing Box

   • How long will it take the volume of the following box to grow to at least 6 times its original size?

           Initial dimensions of a box

                5 by 7 by 3 inches.

           Growth of box

                each dimension increases at the rate of 2 in/sec.

   • Show the algebraic representation of the process.

   • Graph the process.

Comments for Pre Calculus – Dimensions of a box grow at 2 inches per second

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Oct 22, 2012
Dimensions of a Box
by: Staff

Answer

Part I


         Volume of Box = length * width * height

Volume 1 (initial volume)

Volume 1 (initial volume)
= (7 inches) * (5 inches) * (3 inches)

Volume 1 (initial volume) = 105 in³


Volume 2

t = time in seconds


Length = (7 inches) + (2 inches per second) * t

Width = (5 inches) + (2 inches per second) * t

Height = (3 inches) + (2 inches per second) * t

Volume 2 = (7 + 2t) * (5 + 2t) * (3 + 2t)

Volume 2 = 8t³ + 60t² + 142t + 105


The ratio of Volume 2 : Volume 1 = 6

Volume 2
------------ = 6
Volume 1


8t³ + 60t² + 142t + 105
---------------------------- = 6
105

8t³ + 60t² + 142t + 105 = 6 * 105

8t³ + 60t² + 142t + 105 = 630

8t³ + 60t² + 142t - 525 = 0


Solve for “t”


         the most straightforward way to solve for “t” is to graph the function for Volume 2


The most straightforward way to solve for “t” is to graph
the function for Volume 2, and then read the approximate
values directly from the graph.

When the volume of the box = 630, it is six
times the initial volume of 105:

Volume 2 = 8t³ + 60t² + 142t + 105

Volume of expanding box vs time





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

Oct 22, 2012
Dimensions of a Box
by: Staff

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

Part II


         the value of “t”

We now know the approximate value of t in seconds:

when t ≈ 1.86, volume ≈ 630 in³ (a volume 6 times
the initial volume)

To refine the value of t, calculate the volume
using incremental changes to t in the equation
for volume (use a computer):

A refined value for t is:

t = 1.8640680412981 seconds




Thanks for writing.

Staff
www.solving-math-problems.com


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