Question

tom has a rectangular prism - shaped suitcase that measures 9 inches by 9 inches by 24 inches. he needs a second suitcase that has the same volume but smaller surface than his current suitcase. which suitcase size would fit Toms needs

Answer 1

18 inches by 9 inches by 12 inches

Step-by-step explanation:

The volume of Tom's rectangular prism-shaped suitcase which measures 9 inches by 9 inches by 24 inches is;

`\begin{gathered} Volume=l*w*h \\ \\ =9*9*24 \\ \\ =1944\text{ }square\text{ }inches \end{gathered}`

So the volume of Tom's suitcase is 1944 cubic inches

The surface area will be;

`\begin{gathered} SA=2(lw+wh+hl) \\ \\ =2(9*9+9*24+24*9) \\ \\ =2(81+216+216) \\ \\ =2(513) \\ \\ =1026\text{ }square\text{ }inches \end{gathered}`

So the volume of the suitcase is 1026 square inches

*Let's go ahead and determine the volume and surface area of a suitcase that measures 18 inches by 18 inches by 6 inches;

`\begin{gathered} Volume=l*w*h \\ \\ =18*18*6 \\ \\ =1944\text{ cubic inches} \end{gathered}`
`\begin{gathered} Surface\text{ }Area=2(18*18+18*6+6*18) \\ \\ =2(324+108+108) \\ \\ =2(540) \\ \\ =1080\text{ square inches} \end{gathered}`

We can see that the suitcase that measures 18 inches by 18 inches by 6 inches has the same volume as the first one but a higher surface area which doesn't fit Tom's needs

*Let's go ahead and determine the volume of a suitcase that measures 12 inches by 10 inches by 9 inches;

`\begin{gathered} Volume=12*10*9 \\ \\ =1080\text{ cubic inches} \end{gathered}`

We can see that the suitcase that measures 12 inches by 10 inches by 9 inches has a different volume from the first one which doesn't fit Tom's needs.

Let's go ahead and determine the volume of a suitcase that measures 16 inches by 5 inches by 9 inches;

`\begin{gathered} Volume=16*5*9 \\ \\ =720\text{ cubic inches} \end{gathered}`

We can see that the suitcase that measures 16 inches by 5 inches by 9 inches has a different volume from the first one which doesn't fit Tom's needs.

*Let's go ahead and determine the volume and surface area of a suitcase that measures 18 inches by 9 inches by 12 inches;

`\begin{gathered} Volume=l*w*h \\ \\ =18*9*12 \\ \\ =1944\text{ cubic inches} \end{gathered}`
`\begin{gathered} Surface\text{ }Area=2(18*9+9*12+12*18) \\ \\ =2(162+108+216) \\ \\ =2(486) \\ \\ =972\text{ square inches} \end{gathered}`

We can see that the suitcase that measures 18 inches by 9 inches by 12 inches has the same volume as the first one and s smaller surface area which fits Tom's needs