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A point (3*sqrt(5), d + 6) is 3d units away from the origin. What is the smallest possible value of d?

 Jan 21, 2022
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To graph it out, we would see a right triangle with legs of length \(3\sqrt{5}\) and \(d + 6\). If the hypotenuse is \(3d\), then we can use the pythagorean theorem to get this equation:

 

\((3\sqrt{5})^2 + (d + 6)^2 = (3d)^2\)

Simplified:

\(45 + d^2 + 36 + 12d = 9d^2\)

Combine like terms and subtraction:

\(-8d^2 + 12d + 81 = 0\)

Now we have a quadratic, we can apply the quadratic formula \(d\) = \(-b {+\over} \sqrt{b^2 - 4ac}\over2a\) where a is the coefficient of \(d^2\), b is the coefficient of \(d\), and c is the constant of the equation.

 

Plugging in the values, we get:

 

\(-3 {+\over} 3\sqrt{19}\over-4\) = d

 

Since we need the smallest value of d, and d can't be a negative distance away from something, then we will use the subtraction operation. This simplifies to:

 

\(d = {3 + 3\sqrt{19}\over4}\)

 

 

 

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 Jan 23, 2022

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