Question

You construct an open box from a square piece of cardboard, 24 inches on a side, by cutting out equal squares with sides of length from the corners and turning up the sides (see figure below). Write a function V, in terms of 2, that represents the volume of the box. Then use a calculator to graph V and use the graph to estimate the value of that produces a maximum volume. - - - - x - - x 24 - 2x -> | * OA) V = (24 – 2x)"; x = 12 inches produces a maximum volume B) V = < (242 – (2x)?): 7 inches produces a maximum volume e : 2 = Type here to search BI

Answer 1

a Som: Given an open box is constructed from square piece of cardboard, 24 inches on a Side by cutting out equal square with sides of Length x from the corner and turning up the sides, then we have to find volume w in the function of x and the value of a such that volume is makimum. 2 from given Sketch of box, we can say that length of box = (24-2x) Breadth of box = (24-2x) and height of box = x. then, Nolume of box Length & breadth x height (24-24J* (24-2x) *Z X (24-2x) ² را) - Now, for maximum volupne, we differentiate v Wirto x, we get dr d (ve (24-2XJ2) da da = (24-2x)2 + Xx 2(24-225X-2

(24-23)( 24-22 -4x) (24-28) ( 24 -6x) du dx co Now, let dr dx -> (24-2%) (24-6x)=0 24-2X=0 X=12 » 24-61=0 1 Wirto a we again, we differentiale dr de dev - (24-22} *-6 -2 (24-6x) dx2 get theon 'at x=4 (24-2x4)*-6-2 -2 (24*=24) die 2 =4 - 96-0 dx² 8074 -96 co-> at x=4 ,vaftains its makina. -(2) at X=12, cd2w da? ( 24-248-6-2(24-72) 22 - 0 +96 =) (der dx² = 96 >0 = atx=12, w attains 8=12 its minimum. them from eaun cig and 1 = x ( 24-2x)? , 3=4 inches Phadures a meckimum volume - option (E) correct.