Question

You have two rectangular prisms (Prism A and Prism B). Prism A has the following dimensions: length x, width y, and height z. Prism B is made by multiplying each dimension of Prism A by a factor of m, where m >0.
(a) Write a paragraph proof to show that the surface area of Prism B is 2 times the surface area of Prism A.

(b) Write a paragraph proof to show that the volume of Prism B is 3 times the volume of Prism A.

Answer 1

Answer:

Part A) The surface area of prism B is equal to the surface area of prism A multiplied by the scale factor (m) squared  

Part B) The Volume of prism B is equal to the Volume of prism A multiplied by the scale factor (m) elevated to the cube

Step-by-step explanation:

Part A) we know that

The scale factor is equal to m

The surface area of the prism is equal to

$S=2B+Ph$

where

B is the area of the base

P is the perimeter of the base

h is the height of the prism

we have

Prism A

$B=xy\ units^{2}$

$P=2(x+y)\ units$

$h=z\ units$

substitute

$SA=[2(xy)+2(x+y)z]\ units^{2}$

Prism B

$B=(mx)(my)=(xy)m^{2}\ units^{2}$

$P=2(mx+my)=2m(x+y)\ units$

$h=mz\ units$

substitute

$SB=[2(xym^{2})+2m(x+y)mz]\ units^{2}$

$SB=[2(xym^{2})+2m^{2}(x+y)z]\ units^{2}$

$SB=m^{2}[2(xy)+2(x+y)z]\ units^{2}$

therefore

The surface area of prism B is equal to the surface area of prism A multiplied by the scale factor (m) squared      

Part B) we know that

The volume of the prism is equal to

$V=Bh$

where

B is the area of the base

h is the height of the prism

we have

Prism A

$B=xy\ units^{2}$

$h=z\ units$

substitute

$VA=[(xyz]\ units^{3}$

Prism B

$B=(mx)(my)=(xy)m^{2}\ units^{2}$

$h=mz\ units$

substitute

$VB=[(xym^{2})mz]\ units^{3}$

$VB=[(xyzm^{3})]\ units^{3}$

therefore

The Volume of prism B is equal to the Volume of prism A multiplied by the scale factor (m) elevated to the cube