Mai wants to make an open top box by cutting out corners of a square piece of cardboard and folding up the sides. The cardboard is 10 cm by 10 cm. The volume V(x) in cubic cm of the open top box is a function of the side length x in cm of the square cutouts

Answer:[tex]V(x)=(4x^{3}-40x^{2}+100x)\ cm^3[/tex]The domain for x is all real numbers greater than zero and less than 5 comStep-by-step explanation:The question isWhat is the volume of the open top box as a function of the side length x in cm of the square cutouts?see the attached figure to better understand the problemLetx -----> the side length in cm of the square cutoutswe know thatThe volume of the open top box is[tex]V=LWH[/tex]we have[tex]L=(10-2x)\ cm[/tex][tex]W=(10-2x)\ cm[/tex][tex]H=x)\ cm[/tex]substitute[tex]V(x)=(10-2x)(10-2x)x\\\\V(x)=(100-40x+4x^{2})x\\\\V(x)=(4x^{3}-40x^{2}+100x)\ cm^3[/tex]Find the domain for xwe know that[tex](10-2x) > 0\\10> 2x\\ 5 > x\\x < 5\ cm[/tex]soThe domain is the interval (0,5)The domain is all real numbers greater than zero and less than 5 cmthereforeThe volume of the open top box as a function of the side length x in cm of the square cutouts is[tex]V(x)=(4x^{3}-40x^{2}+100x)\ cm^3[/tex]