A photograph

A photograph will stick to a white square letter with a x cm length. The photo is 3/4 x cm long and 20 cm wide than the width of the paper. The surface of the remaining paper surrounding the photograph is 990 cm². Find the size of paper and photo.

Correct result:

a =  39.8458 cm
b =  29.8843 cm
c =  20 cm

Solution:

$$\begin{gathered} S_1=a^2 \\ b=3\mathrm{/}4\cdot a \\ c=20 \\ {S}_2=b\cdot c \\ {S}_1-S_2=990{\text{cm}}^2 \\ {a}^2-3\mathrm{/}4\ast a\ast 20=990 \\ {a}^2-3\mathrm{/}4\cdot a\cdot 20=990 \\ {a}^2-15a-990=0 \\ p=1;q=-15;r=-990 \\ D=q^2-4pr=15^2-4\cdot 1\cdot (-990)=4185 \\ D>0 \\ {a}_{1,2}={\displaystyle \frac{-q\pm \sqrt{D}}{2p}}={\displaystyle \frac{15\pm \sqrt{4185}}{2}}={\displaystyle \frac{15\pm 3\sqrt{465}}{2}} \\ {a}_{1,2}=7.5\pm 32.3457879793 \\ {a}_1=39.8457879793 \\ {a}_2=-24.8457879793 \\ \text{ Factored form of the equation: } \\ (a-39.8457879793)(a+24.8457879793)=0 \\ a=a_1=39.8458=39.8458\text{ cm} \end{gathered}$$

Our quadratic equation calculator calculates it.

$b=3\mathrm{/}4\cdot a=3\mathrm{/}4\cdot 39.8458=29.8843\text{ cm}$

$$\begin{gathered} c=20=20\text{ cm } \\ {S}_1=a^2=39.8458^2\doteq 1587.6868{\text{cm}}^2 \\ {S}_2=b\cdot c=29.8843\cdot 20\doteq 597.6868{\text{cm}}^2 \end{gathered}$$

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