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.

Result

a =  39.846 cm
b =  29.885 cm
c =  20 cm

Solution:

S1=a2 b=3/4⋅ a c=20 S2=b⋅ c S1−S2=990   a2−3/4⋅ a⋅ 20=990 a2−15a−990=0  p=1;q=−15;r=−990 D=q2−4pr=152−4⋅1⋅(−990)=4185 D>0  a1,2=−q±D2p=15±41852=15±34652 a1,2=7.5±32.345787979272 a1=39.845787979272 a2=−24.845787979272   Factored form of the equation:  (a−39.845787979272)(a+24.845787979272)=0  a=a1=39.8458≐39.8458≐39.846 cmS_{1}=a^2 \ \\ b=3/4 \cdot \ a \ \\ c=20 \ \\ S_{2}=b \cdot \ c \ \\ S_{1}-S_{2}=990 \ \\ \ \\ \ \\ a^2 - 3/4 \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}=\dfrac{ -q \pm \sqrt{ D } }{ 2p }=\dfrac{ 15 \pm \sqrt{ 4185 } }{ 2 }=\dfrac{ 15 \pm 3 \sqrt{ 465 } }{ 2 } \ \\ a_{1,2}=7.5 \pm 32.345787979272 \ \\ a_{1}=39.845787979272 \ \\ a_{2}=-24.845787979272 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ (a -39.845787979272) (a +24.845787979272)=0 \ \\ \ \\ a=a_{1}=39.8458 \doteq 39.8458 \doteq 39.846 \ \text{cm}

Checkout calculation with our calculator of quadratic equations.

b=3/4⋅ a=3/4⋅ 39.8458=29.8845=29.885 cmb=3/4 \cdot \ a=3/4 \cdot \ 39.8458=29.8845=29.885 \ \text{cm}
c=20  S1=a2=39.84582≐1587.7037 cm2 c=S2=b⋅ c=29.8845⋅ 20=597710=597.7 cm2=20 cmc=20 \ \\ \ \\ S_{1}=a^2=39.8458^2 \doteq 1587.7037 \ \text{cm}^2 \ \\ c=S_{2}=b \cdot \ c=29.8845 \cdot \ 20=\dfrac{ 5977 }{ 10 }=597.7 \ \text{cm}^2=20 \ \text{cm}

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