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Message: <blockquote data-quote="Wall-eyes" data-source="post: 463148" data-attributes="member: 1902">What you're solving for   You are determining the fraction of a cord of wood represented by a single log with specific dimensions.   What's given in the problem  <ul> <li data-xf-list-type="ul">The log is round with a diameter of [ATTACH=full]78510[/ATTACH] 12 inches12  inches . </li> <li data-xf-list-type="ul">The log is [ATTACH=full]78515[/ATTACH] 9 feet9  feet tall. </li> <li data-xf-list-type="ul">A cord of wood is defined as [ATTACH=full]78519[/ATTACH] 128 cubic feet128  cubic feet . </li> </ul>Helpful information  <ul> <li data-xf-list-type="ul">The volume of a cylinder is calculated using the formula [ATTACH=full]78511[/ATTACH] V=πr2hcap V equals pi r squared h . </li> <li data-xf-list-type="ul">A cord of wood accounts for air space between logs, so the solid wood volume will be less than the total cord volume. </li> </ul>How to solve   Calculate the volume of the log and then determine its proportion relative to a cord of wood, accounting for air space.  <ol> <li data-xf-list-type="ol"> Step 1 . Convert the log's diameter to feet and calculate its radius. <ul> <li data-xf-list-type="ul">Diameter in feet: [ATTACH=full]78513[/ATTACH] D=12 inches12 inches/foot=1 footcap D equals the fraction with numerator 12  inches and denominator 12  inches/foot end-fraction equals 1  foot . </li> <li data-xf-list-type="ul">Radius in feet: [ATTACH=full]78516[/ATTACH] r=D2=1 foot2=0.5 feetr equals the fraction with numerator cap D and denominator 2 end-fraction equals the fraction with numerator 1  foot and denominator 2 end-fraction equals 0.5  feet . </li> </ul> Step 2 . Calculate the volume of the log in cubic feet. <ul> <li data-xf-list-type="ul">Volume of log: [ATTACH=full]78512[/ATTACH] Vlog=πr2hcap V sub log end-sub equals pi r squared h . </li> <li data-xf-list-type="ul">[ATTACH=full]78517[/ATTACH] Vlog=π×(0.5 feet)2×9 feetcap V sub log end-sub equals pi cross open paren 0.5  feet close paren squared cross 9  feet . </li> <li data-xf-list-type="ul">[ATTACH=full]78514[/ATTACH] Vlog=π×0.25 feet2×9 feet≈7.068 cubic feetcap V sub log end-sub equals pi cross 0.25  feet squared cross 9  feet is approximately equal to 7.068  cubic feet . </li> </ul> Step 3 . Estimate the solid wood volume within a cord. <ul> <li data-xf-list-type="ul">Assume solid wood occupies about [ATTACH=full]78518[/ATTACH] 80%80 % of a cord's volume due to air space. </li> <li data-xf-list-type="ul">Solid wood volume in a cord: [ATTACH=full]78520[/ATTACH] Vsolid cord=128 cubic feet×0.80=102.4 cubic feetcap V sub solid cord end-sub equals 128  cubic feet cross 0.80 equals 102.4  cubic feet . </li> </ul> Step 4 . Determine how close the log is to a cord of wood. <ul> <li data-xf-list-type="ul">Fraction of a cord: [ATTACH=full]78523[/ATTACH] Fraction=VlogVsolid cordFraction equals the fraction with numerator cap V sub log end-sub and denominator cap V sub solid cord end-sub end-fraction . </li> <li data-xf-list-type="ul">[ATTACH=full]78522[/ATTACH] Fraction=7.068 cubic feet102.4 cubic feet≈0.069Fraction equals the fraction with numerator 7.068  cubic feet and denominator 102.4  cubic feet end-fraction is approximately equal to 0.069 . </li> </ul></li> </ol> Solution   A single [ATTACH=full]78524[/ATTACH]12 inch12  inch diameter, [ATTACH=full]78521[/ATTACH]9 foot9  foot tall log is approximately [ATTACH=full]78525[/ATTACH]0.0690.069 of a cord of wood.</blockquote>

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