Question

A rectangular prism has sides that are in the ratio of 3:5:9. If the volume of the prism is 8640 cm3, then the surface area of the prism, in square millimetres, is_______?

Answer 1

The surface area of the rectangular prism is 278400 square millimeters, if a rectangular prism has sides that are in the ratio of 3:5:9 and the volume of the prism is 8640 cubic centimeters.

Step-by-step explanation:

The given is,

                Volume of Rectangular prism - 8640 Cubic centimeters

                Ratio of rectangular prism sides are 3:5:9

Step:1

               Formula for volume of rectangular prism,

                                           $Volume, V = whl$...................................(1)

               Where, w - width of the prism

                            h - Height of the prism

                             l - Length of the prism

               Let, x - side of the rectangular prism

               From given values,

                       Volume, V = 8640 cubic centimeters

                          Width, w = $\frac{3}{17}x$

                         Height, h = $\frac{5}{17} x$

                          Length, l = $\frac{9}{17}x$

              Equation (1) becomes,

                                $8640=(\frac{3}{17}x)(\frac{5}{17} x)(\frac{9}{17}x)$

                                        $= 0.027478 x^{3}$

                                    $x^{3}= \frac{8640}{0.027478}$

                                      $x = \sqrt[3]{314432}$

                                         $= 68$

Step:2

              Width, w = $\frac{3}{17}x$  $= (\frac{3}{17}) (68)$  $= 12 cm$

             Height, h $= (\frac{5}{17} )(68) = 20 cm$

              Length, l $= (\frac{9}{17})(68)=36cm$

Step:3

            Formula for surface area of rectangular prim,

                                            $A= 2(wl+lh+hlw)$

            From the values of w, l and h,

                                               $= 2 [(12)(20)+(36)(20)+(36)(12)]$      

                                               $= 2 (240+720+432)$

                                               $= 2(1392)$

                                               = 2784 Square centimeters

                                               = 2784 × 100 square millimeters

                         ( 1 square centimeters = 100 square millimeters )

                                            A = 278400 square millimeters

Result:    

             The surface area of the rectangular prism is 278400 square millimeters, if a rectangular prism has sides that are in the ratio of 3:5:9 and the volume of the prism is 8640 cubic centimeters.