University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 1 - Section 1.1 - Functions and Their Graphs - Exercises - Page 13: 63


$V(x)=4x^3-72x^2+308x$ ($0 \lt x\lt 7$).

Work Step by Step

Let's say that we are going to cut equal squares of side length $\:x\:\mathrm{in}\:$ from each corner. By doing so, and folding up the sides, we will be left with the height of the box $\:x\:\mathrm{in}.$ We know that the volume of a rectangular box is the product of height $\:h\:$, width $\:w\:$ and length $\:l\:.$ $\mathrm{Volume}\:=\:l\cdot w\cdot h$ When we have cut out the equal squares, length and width of the box will be: $l=22-2x$ $w=14-2x$ So, the volume of the box will be written as: $V=(22-2x)(14-2x)\cdot x$ $\Rightarrow\:V=4x^3-72x^2+308x$
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