A rectangle contains 324 sq cm. If the length is nine less than three times the width, what are the dimensions of the rectangle?

2 Answers
Martin C.
Mar 13, 2018

#l=27cm#
#w=12cm#

Explanation:

#l="length"#
#w="width"#

#l=3*w-9#
#A=color(red)(l)*w#
#A=color(red)((3*w-9))*w#
#324=3w^2-9w|:3#
#108=w^2-3w|-108#
#0=w^2-3w-108#
#0=(w-3/2)^2-108-9/4|+441/4#
#441/4=(w-3/2)^2|sqrt()#
#+-21/2=w-3/2|+3/2#
#3/2+-21/2=w#
#w_1=12cm or cancel(w_2=-9cm)#

#l=3*12-9=27cm#

Jim G.
Mar 13, 2018

#"length "=27" cm and width "=12" cm"#

Explanation:

#"let the width "=w#

#"then length "l=3w-9to" 9 less than 3 times width"#

#"area of rectangle "="length"xx"width"#

#rArr"area "=w(3w-9)#

#"now area "=324" cm"^2#

#rArrw(3w-9)=324#

#"distribute and equate to zero"#

#rArr3w^2-9w-324=0larrcolor(blue)"in standard form"#

#rArr3(w^2-3w-108)=0#

#"factor "w^2-3w-108" using the a-c method"#

#"The factors of - 108 which sum to - 3 are - 12 and + 9"#

#rArr3(w-12)(w+9)=0#

#"equate each factor to zero and solve for w"#

#w-12=0rArrw=12#

#w+9=0rArrw=-9#

#"but "w>0rArrw=12#

#"Hence width "=w=12" cm and"#

#"length "=3w-9=(3xx12)-9=27" cm"#

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