Infinity

In a square with side 19 is inscribed circle, the circle is inscribed next square, again circle, and so on to infinity. Calculate the sum of the area of all these squares.

Correct answer:

S =  722

Step-by-step explanation:

$$\begin{gathered} a_1=19 \\ {S}_1=a_1^2=19^2=361 \\ {d}_1=a_1=19 \\ {d}_1=\sqrt{2}{a}_2 \\ {a}_2=d_1\mathrm{/}\sqrt{2}=19\mathrm{/}\sqrt{2}\doteq 13.435 \\ {S}_2=a_2^2=13.435^2={\displaystyle \frac{361}{2}}=180\frac{1}{2}=180.5 \\ q=S_2\mathrm{/}{S}_1=180.5\mathrm{/}361={\displaystyle \frac{1}{2}}=0.5 \\ S=S_1\cdot {\displaystyle \frac{1}{1-q}}=361\cdot {\displaystyle \frac{1}{1-0.5}}=722 \\ \text{ Verifying Solution: } \\ {d}_2=a_2=13.435\doteq 13.435 \\ {d}_2=\sqrt{2}{a}_3 \\ {a}_3=d_2\mathrm{/}\sqrt{2}=13.435\mathrm{/}\sqrt{2}={\displaystyle \frac{19}{2}}=9\frac{1}{2}=9.5 \\ {S}_3=a_3^2=9.5^2={\displaystyle \frac{361}{4}}=90\frac{1}{4}=90.25 \\ {q}_2=S_3\mathrm{/}{S}_2=90.25\mathrm{/}180.5={\displaystyle \frac{1}{2}}=0.5 \end{gathered}$$

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Showing 2 comments:

Ann

I don't believe this answer is correct. The ratio (q) to find the area of the next square should be 1/8, not 1/2.

1 year ago  Like

Math student

no it is 1/2

1 year ago  Like

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