Question

Find the volume and surface area of each prism. Prism A: 3 cm by 3 cm by 3 cm A cube whose length, width, and height are each 3 centimeters. ​ Prism B: 5 cm by 5 cm by 1 cm Compare the volumes of the prisms and then their surface areas. Does the prism with the greater volume also have the greater surface area?

Answer 1

Answer:

Volume of prism A and B are $27cm^3$ and $25cm^3$ respectively and Surface area of prism A and B are $54cm^2$ and $70cm^2$ respectively.

Step-by-step explanation:

The prism A is a cube and prism B is a rectangular prism.

The volume of the prism is given by

$Volume=length\times width\times height$

So, Volume of prism A is with each side 3cm is,

$V(A)=3\times 3\times 3=27cm^3$

Volume of prism B with measurements 5 ,5 and 1 is

$V(B)=5\times 5\times 1=25cm^3$

Surface are of cube is $6a^2$, where $a$ is one side of the cube. So

$Surface \ Area(A)=6\times 3^2=6\times 9=54cm^2$

Surface area of rectangular prism is $2(length\times width + width\times height + length\times height)$

So, surface area of given prism B is

$Surface \ area(B)=2(5\times 5+5\times 1+5\times 1)\\=2(25+5+5)=2\times 35=70cm^2$

From the values, it is clear that prism with greater volume need not have greater surface area.

Answer 2

Given: Prism A: 3 cm by 3 cm by 3 cm A cube whose length, width, and height are each 3 centimeters. ​ Prism B: 5 cm by 5 cm by 1 cm.

To find: The volume and surface area of each prism. Compare the volumes of the prisms and then their surface areas. Does the prism with the greater volume also have the greater surface area?

Solution:

The volume of a prism is given by the following formula.

$V = l * b * h$

Here, V is the volume, l is the length, b is the base and h is the height of the prism. Let V₁ be the volume of prism A and let V₂ be the volume of prism B.

$V_{1} = 3 * 3* 3$

    $= 27 cm^{3}$

$V_{2} = 5 * 5 * 1$

    $= 25 cm^{3}$

The surface area of the prism can be calculated by finding out the semi perimeter of the prism. Let s₁ be the semi-perimeter of prism A and let s₂ be the semi-perimeter of prism B.

$s_{1} = \frac{3+3+3}{2}$

    $= 4.5$

$s_{2} = \frac{5+5+1}{2}$

    $= 5.5$

Now, the areas of the present can be calculated as

$A_{1} = \sqrt{s(s-a)(s-b)(s-c)}$

     $= \sqrt{4.5(4.5-3)(4.5-3)(4.5-3)}$

     $= 20.25 cm_{2}$

$A_{2} = \sqrt{s(s-a)(s-b)(s-c)}$

     $= \sqrt{5.5(5.5-5)(5.5-5)(5.5-1)}$

     $= 24.75 cm{2}$

Therefore, the volume and surface area of prism A is 27 cm³ and 20.25 cm², respectively. The volume and surface area of prism B is 25 cm³ and 24.75 cm². The volume of prism A is greater than prism B. The surface area of prism B is greater than prism A. The prism with the greater volume does not have the greater surface area.