A rectangle has an area of #30# #cm^2#. It has a width of #5+2x# #cm# and a length of #2x+4# #cm#. What is the length and width?

Question

Caroline Aucoin · Accepted Answer

The width is #5# cm and the length is #6# cm

If you know the length #l# and width #w# of a rectangle, the area #A# is given by their product. So, you have #A = w*l \implies 30 = (5+2x)(2x+4)# (we just translated the known values/expressions for area, width and length). We can expand the width-length product, multiplying each term in the first parenthesis times each term in the second parenthesis: #(5+2x)(2x+4) = 5*2x+5*4+2x*2x+2x*4# # = 10x+20+4x^2+8x# # = 4x^2+18x+20# We want this quantity to equal #30#, so we have #4x^2+18x+20 = 30# subtracting #30# from both sides... #4x^2+18x-10 = 0# You can solve this equation with the usual quadratic formula to get #x_{1,2} = \frac{-18\pm\sqrt{324+160}}{8}=\frac{-18\pm\sqrt{484}}{8} = \frac{-18\pm 22}{8}# If we choose the plus sign, we have #\frac{-18 + 22}{8} = \frac{4}{8} = \frac{1}{2}# If we choose the minus sign, we have #\frac{-18 - 22}{8} = \frac{-40}{8} = -5# Given those values for #x#, we can try to deduce the width and length: considering the first solution #x = 1/2# we have #w = 5+2x \implies w = 6# cm #h = 4+2x \implies h = 5# cm Considering the second solution #x = -5# we have #w = 5+2x \implies w = -5# cm #h = 4+2x \implies h = -6# cm This solution makes algebraically sense, since the area would be #(-5cm)(-6cm) = 30cm^2#, but we can't have sides of a rectangle with negative measures, so we must reject these solutions.

Noah Atkinson · Answer

#color(blue)("Length"=6cm)#

#color(blue)("Width"=5cm)# The area of a rectangle is given by: #"Area"="length"xx"width"# We are given #"Area"=30cm^2# Therefore: #"Area"=(2x+4)*(5+2x)=30# # \ \ \ \ \ \ \ \ \ \=4x^2+18x+20=30# # \ \ \ \ \ \ \ \ \ \=4x^2+18x-10=0# Factor: #(2x+10)(2x-1)=0=>x=-5 and x=1/2# We now check these, since we can't have a negative side to the rectangle: #x=-5# #(2(-5)+4)=-6# We discard this solution. #x=1/2# #(2(1/2)+4)=5# #(5+2(1/2))=6# So: #"Length"=6cm# #"Width"=5cm#

Aurora Ayala · Answer

undefined Length = $2x + 4$ cm, Width = $5 + 2x$ cm