Tubes

Iron tubes in the warehouse are stored in layers so that each tube's top layer fits into the gaps of the lower layer.

How many layers are needed to deposit 100 tubes if the top layer has 9 tubes? How many tubes are in the bottom layer of tubes?

Correct answer:

n₁ =  8
x =  16

Step-by-step explanation:

$$\begin{gathered} 100=n_1(9+a_n)/2 \\ {a}_n=9+1(n_1-1)=8+n_1 \\ 200=n_1(9+8+n_1) \\ 200=n_1(17+n_1) \\ 200=n\cdot (17+n) \\ 200=n\cdot (17+n) \\ -n^2-17n+200=0 \\ {n}^2+17n-200=0 \\ a=1;b=17;c=-200 \\ D=b^2-4ac=17^2-4\cdot 1\cdot (-200)=1089 \\ D>0 \\ {n}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{-17\pm \sqrt{1089}}{2} \\ {n}_{1,2}=\frac{-17\pm 33}{2} \\ {n}_{1,2}=-8.5\pm 16.5 \\ {n}_1=8 \\ {n}_2=-25 \\ n>0 \end{gathered}$$

Our quadratic equation calculator calculates it.

$x=9+(n_1-1)=9+(8-1)=16$

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