If the area of a rectangle is 84 sqcm and the difference of its length and breadth is 5cm,find its length,breadth and perimeter?

The eighth qstn in the two sums I have doubt pls help

Question

The eighth qstn in the two sums I have doubt pls help

Avery Allen · Accepted Answer

Length, Breath #= 12, 7# (cm)
Perimeter #= 38# (cm)

If the longer side is called the length and the length is called #L#; call the breadth #B#. Then we are told: [1]#color(white)("XXX")B=L+5# (cm) and [2]#color(white)("XXX")B * L = 84# (sq.cm.) Substituting #(L+5)# from [1] for #B# in [2] [3]#color(white)("XXX")(L+5) * L =84# [4]#color(white)("XXX")L^2+5L =84# [5]#color(white)("XXX")L^2+5L-84=0# [6]#color(white)("XXX")(L+12)(L-7)=0# [7]#color(white)("XXX"){:(L=-12,color(white)("X")orcolor(white)("X"),L=7), ("impossible",,):}# Since #B=L+5# #color(white)("XXX")B=12# #"Perimeter " = 2xx ("Length" + "Breadth")=2xx(7+12)=38#

Sadie Allen · Answer

undefined Let the length of the rectangle be $ L $ cm and the breadth be $ B $ cm. Given that the area of the rectangle is 84 sqcm, we have the equation $ L \times B = 84 $ sqcm. Also, given that the difference of its length and breadth is 5 cm, we have the equation $ L - B = 5 $ cm.

Solving these two equations simultaneously, we find:

$ L \times B = 84 $

$ L - B = 5 $

From the second equation, we can rewrite $ L $ as $ B + 5 $, and substitute it into the first equation:

$ (B + 5) \times B = 84 $

$ B^2 + 5B - 84 = 0 $

Now, we can solve this quadratic equation. Factoring or using the quadratic formula, we find the values of $ B $:

$ (B - 7)(B + 12) = 0 $

This gives us two possible values for $ B $: $ B = 7 $ or $ B = -12 $. Since breadth cannot be negative, we discard $ B = -12 $.

So, $ B = 7 $ cm.

Now, we can find the length $ L $:

$ L = B + 5 = 7 + 5 = 12 $ cm.

The perimeter of the rectangle is given by $ P = 2(L + B) $:

$ P = 2(12 + 7) $

$ P = 2(19) $

$ P = 38 $ cm.

So, the length of the rectangle is 12 cm, the breadth is 7 cm, and the perimeter is 38 cm.