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.

Question

carealee carealee

19-02-2019
Mathematics

contestada

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.

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calculista calculista · Answer 1 · 2019-03-01T00:24:10+01:00

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

[tex]S=2B+Ph[/tex]

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

[tex]B=xy\ units^{2}[/tex]

[tex]P=2(x+y)\ units[/tex]

[tex]h=z\ units[/tex]

substitute

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

Prism B

[tex]B=(mx)(my)=(xy)m^{2}\ units^{2}[/tex]

[tex]P=2(mx+my)=2m(x+y)\ units[/tex]

[tex]h=mz\ units[/tex]

substitute

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

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

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

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

[tex]V=Bh[/tex]

where

B is the area of the base

h is the height of the prism

we have

Prism A

[tex]B=xy\ units^{2}[/tex]

[tex]h=z\ units[/tex]

substitute

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

Prism B

[tex]B=(mx)(my)=(xy)m^{2}\ units^{2}[/tex]

[tex]h=mz\ units[/tex]

substitute

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

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

therefore

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