Harry

Harry Thomson bought a large land in the shape of a rectangle with a circumference of 90 meters. He divided it into three rectangular plots. The shorter side has all three plots of equal length, their longer sides are three consecutive natural numbers. Find out the dimensions of each plot and the area of the whole plot.

Correct answer:

a₁ =  11 m
b₁ =  12 m
c₁ =  13 m
y₁ =  9 m
S₁ =  324 m²
a₂ =  12 m
b₂ =  13 m
c₂ =  14 m
y₂ =  6 m
S₂ =  234 m²
a₃ =  13 m
b₃ =  14 m
c₃ =  15 m
y₃ =  3 m
S₃ =  126 m²

Step-by-step explanation:

$$\begin{gathered} o=90\text{ m } \\ o=2y+2\cdot (a+b+c) \\ o=2y+2\cdot (a+a+1+a+2) \\ o=2y+2\cdot (3a+3) \\ 90=2\cdot y+6\cdot (a+1) \\ a>0,y>0,a>y \\ {a}_1=11;y_1=9=11\text{ m } \\ {a}_2=12;y_2=6 \\ {a}_3=13;y_3=3 \end{gathered}$$

$b_1=a_1+1=11+1=12\text{ m}$

$c_1=a_1+2=11+2=13\text{ m}$

$S_1=y_1\cdot (a_1+b_1+c_1)=9\cdot (11+12+13)=324{\text{m}}^2$

$b_2=a_2+1=12+1=13\text{ m}$

$c_2=a_2+2=12+2=14\text{ m}$

$S_2=y_2\cdot (a_2+b_2+c_2)=6\cdot (12+13+14)=234{\text{m}}^2$

$b_3=a_3+1=13+1=14\text{ m}$

$c_3=a_3+2=13+2=15\text{ m}$

$$\begin{gathered} S_3=y_3\cdot (a_3+b_3+c_3)=3\cdot (13+14+15)=126{\text{m}}^2 \\ \text{ Verifying Solution: } \\ {o}_1=2\cdot (y_1+a_1+b_1+c_1)=2\cdot (9+11+12+13)=90\text{ m } \\ {o}_2=2\cdot (y_2+a_2+b_2+c_2)=2\cdot (6+12+13+14)=90\text{ m } \\ {o}_3=2\cdot (y_3+a_3+b_3+c_3)=2\cdot (3+13+14+15)=90\text{ m} \end{gathered}$$

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