Tableau pyramid

Your class will invent an original tableau pyramid from photos. What minimum dimensions will it have to have if you want to place 50 9x13 photos there? You want a classic pyramid, i.e., Each next row is one photo-less, but in the last row, two photos (the class and the principal)

Correct answer:

a =  90 cm
b =  117 cm

Step-by-step explanation:

$$\begin{gathered} x=9\text{cm} \\ y=13\text{cm} \\ s=50 \\ {a}_1=2 \\ d=1 \\ s=a_{1}n+n(n-1)d/2 \\ 50=2\cdot n+n\cdot (n-1)/2 \\ 50=2\cdot n+n\cdot (n-1)/2 \\ -0.5n^2-1.5n+50=0 \\ 0.5n^2+1.5n-50=0 \\ a=0.5;b=1.5;c=-50 \\ D=b^2-4ac=1.5^2-4\cdot 0.5\cdot (-50)=102.25 \\ D>0 \\ {n}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{-1.5\pm \sqrt{102.25}}{1} \\ {n}_{1,2}=-1.5\pm 10.111874 \\ {n}_1=8.611874208 \\ {n}_2=-11.611874208 \\ n>0 \\ n=\lceil n_1\rceil =\lceil 8.6119\rceil =9 \\ c=a_1+(n-1)\cdot d=2+(9-1)\cdot 1=10 \\ a=c\cdot x=10\cdot 9=90\text{cm} \end{gathered}$$

Our quadratic equation calculator calculates it.

$b=n\cdot y=9\cdot 13=117\text{cm}$

Try another example