Cuboid and eq2

Calculate the volume of cuboid with square base and height 6 cm if the surface area is 48 cm².

Correct result:

V =  18.2904 cm³

Solution:

$$\begin{gathered} h=6\text{ cm } \\ S=48{\text{cm}}^2 \\ S=4ah+2a^2 \\ 48=4\ast 6\ast a+2\ast a^2 \\ 48=4\cdot 6\cdot a+2\cdot a^2 \\ -2a^2-24a+48=0 \\ 2a^2+24a-48=0 \\ p=2;q=24;r=-48 \\ D=q^2-4pr=24^2-4\cdot 2\cdot (-48)=960 \\ D>0 \\ {a}_{1,2}={\displaystyle \frac{-q\pm \sqrt{D}}{2p}}={\displaystyle \frac{-24\pm \sqrt{960}}{4}}={\displaystyle \frac{-24\pm 8\sqrt{15}}{4}} \\ {a}_{1,2}=-6\pm 7.7459666924148 \\ {a}_1=1.7459666924148 \\ {a}_2=-13.745966692415 \\ \text{ Factored form of the equation: } \\ 2(a-1.7459666924148)(a+13.745966692415)=0 \\ a=a_1=1.746\doteq 1.746 \\ V=a^2\cdot h=1.746^2\cdot 6\doteq 18.2904=18.2904{\text{cm}}^3 \\ \text{ Verifying Solution: } \\ {S}_2=4\cdot a\cdot h+2\cdot a^2=4\cdot 1.746\cdot 6+2\cdot 1.746^2=48{\text{cm}}^2 \\ {S}_2=S \end{gathered}$$

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