Dimensions of the trapezoid

One of the trapezoid bases is one-fifth larger than its height, and the second base is 1 cm larger than its height. Find the dimensions of the trapezoid if its area is 115 cm²

Correct answer:

a =  12 cm
c =  11 cm
h =  10 cm

Step-by-step explanation:

$$\begin{gathered} a=h+h/5 \\ c=1+h=1+10=1 \\ S=115{\text{cm}}^2 \\ S=\frac{a+c}{2}\cdot h \\ 2\cdot S=((h+h/5)+(1+h))\cdot h \\ 2\cdot 115=((h+h/5)+(1+h))\cdot h \\ -2.2h^2-h+230=0 \\ 2.2h^2+h-230=0 \\ a=2.2;b=1;c=-230 \\ D=b^2-4ac=1^2-4\cdot 2.2\cdot (-230)=2025 \\ D>0 \\ {h}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{-1\pm \sqrt{2025}}{4.4} \\ {h}_{1,2}=-0.227273\pm 10.227273 \\ {h}_1=10 \\ {h}_2=-10.454545455 \\ h=h_1=10 \\ a=h+h/5=10+10/5=12\text{cm} \end{gathered}$$

Our quadratic equation calculator calculates it.

$c=1+h=1+10=11\text{cm}$

$$\begin{gathered} h=10\text{cm} \\ \text{ Verifying Solution: } \\ {S}_1=\frac{a+c}{2}\cdot h=\frac{12+11}{2}\cdot 10=115{\text{cm}}^2 \end{gathered}$$

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