A rectangular box of internal base dimensions 2r, 3r and internal height 9r is filled with spheres of radius r. Find the maximum number of spheres that can be fitted into the box.

<p>Let L be the vertical distance between the centers of two adjacent balls.</p><p><span contenteditable="false"> <math> <mrow> <mi>L</mi> <mo>=</mo> <mroot> <mrow> <mrow> <mo> [</mo> <mrow> <msup> <mrow> <mo stretchy="false"> (</mo> <mn>2</mn> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mi>r</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mrow> <mrow></mrow> </mroot> <mo>=</mo> <mi>r</mi> <mroot> <mrow> <mn>3</mn> </mrow> <mrow></mrow> </mroot> </mrow> </math> </span></p><p>Also, if the box can hold n sphere then (n – 1) L + 2r</p><p>≤ 9r [ the last ball height = 2r]</p><p><span contenteditable="false"> <math> <mrow> <mo stretchy="false"> (</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mroot> <mrow> <mn>3</mn> </mrow> <mrow></mrow> </mroot> <mtext> </mtext> <mi>r</mi> <mo>≤</mo> <mn>7</mn> <mi>r</mi> </mrow> </math> </span></p><p><span contenteditable="false"> <math> <mrow> <mi>n</mi> <mo>≤</mo> <mfrac> <mrow> <mn>7</mn> </mrow> <mrow> <mroot> <mrow> <mn>3</mn> </mrow> <mrow></mrow> </mroot> </mrow> </mfrac> <mo>+</mo> <mn>1</mn> </mrow> </math> </span></p><p>n ≤ 5.04</p><p>Since, n is a natural number.</p><p>n = 5</p><p>So, the box can hold maximum of 5 spheres.</p>

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