Question 257085
We have two formulas:
(i) {{{P = 2L + 2W}}}
(ii) {{{A = LW}}}
From the given information, (i) becomes
(iii) {{{52 = 2L + 2W}}}
or
(iv) {{{26 = L + W}}}
Solving (iv) for L, we get
(v) {{{L = 26 - W}}}
substituting (v) into (ii) we get
(vi) {{{A = (26-W)(W)}}}
or
(vii) {{{A = -W^2 + 26W}}}
applying the quadratic, we get
{{{-(w-13)^2 + 13^2}}}
so, the dimensions are 13 x 13 and the area is 169 sq. m.
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If one side is a house, then
(i) {{{P = L + 2W}}}
(ii) {{{A = LW}}}
From the given information, (i) becomes
(iii) {{{52 = L + 2W}}}
Solving (iii) for L, we get
(iv) {{{L = 52 - 2W}}}
substituting (iv) into (ii) we get
(v) {{{A = (52-2W)(W)}}}
or
(vii) {{{A = -2W^2 + 52W}}}
applying the quadratic, we get
{{{-2(w-13)^2 + 52^2/8}}}
so, the dimensions are 26 x 13 and the area is 338 sq. m.