Staircase

On a staircase 3.6 meters high, the number of steps would increase by three if the height of one step decreased by 4 cm. How high are the stairs?

Final Answer:

x =  24 cm

Step-by-step explanation:

$$\begin{gathered} h=3.6\text{m}\to \text{cm}=3.6\cdot 100\text{cm}=360\text{cm} \\ h=n\cdot x \\ h=(n+3)(x-4) \\ h=(h/x+3)\cdot (x-4) \\ h\cdot x=(h+3\cdot x)\cdot (x-4) \\ 360\cdot x=(360+3\cdot x)\cdot (x-4) \\ -3x^2+12x+1440=0 \\ 3x^2-12x-1440=0 \\ 3...\text{ prime number} \\ 12=2^2\cdot 3 \\ 1440=2^5\cdot 3^2\cdot 5 \\ \text{GCD}(3,12,1440)=3 \\ {x}^2-4x-480=0 \\ a=1;b=-4;c=-480 \\ D=b^2-4ac=4^2-4\cdot 1\cdot (-480)=1936 \\ D>0 \\ {x}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{4\pm \sqrt{1936}}{2} \\ {x}_{1,2}=\frac{4\pm 44}{2} \\ {x}_{1,2}=2\pm 22 \\ {x}_1=24 \\ {x}_2=-20 \\ x=x_1=24\text{cm} \\ n=h/x=360/24=15 \\ \text{ Verifying Solution: } \\ {h}_1=n\cdot x=15\cdot 24=360\text{cm} \\ {h}_2=(n+3)\cdot (x-4)=(15+3)\cdot (24-4)=360\text{cm} \end{gathered}$$

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