Question 1207302
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Let x be the side length of the equilateral triangle.  Then<br>
{{{log(4,a)=x}}} --> {{{4^x=a}}}<br>
{{{log(10,b)=x}}} --> {{{10^x=b}}}<br>
{{{log(25,a+b)=x}}} --> {{{25^x=a+b}}}<br>
Now look for a relationship between the three bases 4, 10, and 25: {{{4*25 = 10^2}}}.  So<br>
{{{(10^x)^2=b^2}}}
{{{100^x=b^2}}}<br>
{{{25^x=(100/4)^x=100^x/4^x}}}
{{{a+b=b^2/a}}}
{{{a^2+ab=b^2}}}<br>
Treat this as a quadratic equation with a as the variable and solve for a using the quadratic formula.<br>
{{{a^2+ab-b^2=0}}}<br>
{{{a=(-b+sqrt(b^2+4b^2))/2}}}  (ignore the other solution, since a has to be positive)<br>
{{{a=(-b+sqrt(5b^2))/2}}}<br>
{{{a=(-b+b*sqrt(5))/2}}}<br>
{{{a=b(-1+sqrt(5)/2)}}}<br>
Divide by b to find the value of a/b.<br>
{{{a/b=(sqrt(5)-1)/2}}}<br>
ANSWER: {{{a/b=(sqrt(5)-1)/2}}}<br>
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NOTE added after seeing the response from tutor @ikleyn...<br>
This provides a good example of how a given problem is open to solving using very different equally good paths.<br>
ALWAYS be open to the possibility of solving any given problem in different ways.  Finding a different (and sometimes better) way to do something is how human knowledge increases.<br>