0000:00:00:00:00Start4Number 4ExplanationA rectangular prism (cuboid) ABCD.EFGHABCD.EFGHABCD.EFGH has a length that is twice its width. Its height is equal to its width. The length of DFDFDF is 999 cm. Then the width of the prism is...362\frac{3\sqrt{6}}{2}236 cm332\frac{3\sqrt{3}}{2}233 cm123\frac{1}{2}\sqrt{3}213 cm133\frac{1}{3}\sqrt{3}313 cm322\frac{3}{2}\sqrt{2}232 cmExplanationProblem Visualization Rectangular Prism ABCD.EFGHABCD.EFGHABCD.EFGHPrism ABCD.EFGHABCD.EFGHABCD.EFGH with length p=2lp=2lp=2l, width lll, and height t=lt=lt=l. Space diagonal DF=9DF=9DF=9.Toggle GridToggle Play Dimension Analysis Given the prism dimensions: Length p=2lp = 2lp=2l Height t=lt = lt=l Width lll Space diagonal DF=9DF = 9DF=9 cm Calculation The formula for the space diagonal is d2=p2+l2+t2d^2 = p^2 + l^2 + t^2d2=p2+l2+t2. Substitute the known values: DF2=p2+l2+t2DF^2 = p^2 + l^2 + t^2DF2=p2+l2+t292=(2l)2+l2+l29^2 = (2l)^2 + l^2 + l^292=(2l)2+l2+l281=4l2+l2+l281 = 4l^2 + l^2 + l^281=4l2+l2+l281=6l281 = 6l^281=6l2 Solve for lll: l2=816=272l^2 = \frac{81}{6} = \frac{27}{2}l2=681=227l=272l = \sqrt{\frac{27}{2}}l=227l=272=332l = \frac{\sqrt{27}}{\sqrt{2}} = \frac{3\sqrt{3}}{\sqrt{2}}l=227=233 Rationalize the denominator: l=332×22l = \frac{3\sqrt{3}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}l=233×22l=362l = \frac{3\sqrt{6}}{2}l=236 Thus, the width of the prism is 362\frac{3\sqrt{6}}{2}236 cm.
4Number 4ExplanationA rectangular prism (cuboid) ABCD.EFGHABCD.EFGHABCD.EFGH has a length that is twice its width. Its height is equal to its width. The length of DFDFDF is 999 cm. Then the width of the prism is...362\frac{3\sqrt{6}}{2}236 cm332\frac{3\sqrt{3}}{2}233 cm123\frac{1}{2}\sqrt{3}213 cm133\frac{1}{3}\sqrt{3}313 cm322\frac{3}{2}\sqrt{2}232 cmExplanationProblem Visualization Rectangular Prism ABCD.EFGHABCD.EFGHABCD.EFGHPrism ABCD.EFGHABCD.EFGHABCD.EFGH with length p=2lp=2lp=2l, width lll, and height t=lt=lt=l. Space diagonal DF=9DF=9DF=9.Toggle GridToggle Play Dimension Analysis Given the prism dimensions: Length p=2lp = 2lp=2l Height t=lt = lt=l Width lll Space diagonal DF=9DF = 9DF=9 cm Calculation The formula for the space diagonal is d2=p2+l2+t2d^2 = p^2 + l^2 + t^2d2=p2+l2+t2. Substitute the known values: DF2=p2+l2+t2DF^2 = p^2 + l^2 + t^2DF2=p2+l2+t292=(2l)2+l2+l29^2 = (2l)^2 + l^2 + l^292=(2l)2+l2+l281=4l2+l2+l281 = 4l^2 + l^2 + l^281=4l2+l2+l281=6l281 = 6l^281=6l2 Solve for lll: l2=816=272l^2 = \frac{81}{6} = \frac{27}{2}l2=681=227l=272l = \sqrt{\frac{27}{2}}l=227l=272=332l = \frac{\sqrt{27}}{\sqrt{2}} = \frac{3\sqrt{3}}{\sqrt{2}}l=227=233 Rationalize the denominator: l=332×22l = \frac{3\sqrt{3}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}l=233×22l=362l = \frac{3\sqrt{6}}{2}l=236 Thus, the width of the prism is 362\frac{3\sqrt{6}}{2}236 cm.
