# 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

 Oct 22, 2012 Dimensions of a Box by: Staff AnswerPart 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 in³Volume 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 + 105The 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 = 0Solve 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 ` --------------------------------------------

 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