Answer:
height of pyramid = 3 cm
height of prism = 9 cm
Step-by-step explanation:
Square Pyramid
surface area of 4 triangular sides of the pyramid = [tex]2a\sqrt{\dfrac{a^2}{4}+h^2}[/tex]
(where [tex]a[/tex] is the base edge and [tex]h[/tex] is the height)
Given [tex]a[/tex] = 8 cm
[tex]\implies \textsf{SA}=2\cdot8\sqrt{\dfrac{8^2}{4}+h^2}[/tex]
[tex]=16\sqrt{16+h^2}[/tex]
Square Prism
surface area of square prism with one base = [tex]a^2+4ah[/tex]
(where [tex]a[/tex] is the base edge and [tex]h[/tex] is the height)
Given [tex]a[/tex] = 8 cm and [tex]h[/tex] = 3h
[tex]\implies \textsf{SA}=8^2+4\cdot 8\cdot 3h[/tex]
[tex]=64+96h[/tex]
Total surface area
total SA = SA of pyramid + SA of square prism
Given total SA = 432 cm²
[tex]\implies 432=16\sqrt{16+h^2}+64+96h[/tex]
[tex]\implies 368=16\sqrt{16+h^2}+96h[/tex]
[tex]\implies 368=16(\sqrt{16+h^2}+6h)[/tex]
[tex]\implies 23=\sqrt{16+h^2}+6h[/tex]
[tex]\implies 23-6h=\sqrt{16+h^2}[/tex]
[tex]\implies (23-6h)^2=16+h^2[/tex]
[tex]\implies 529-276h+36h^2=16+h^2[/tex]
[tex]\implies 35h^2-276h+513=0[/tex]
[tex]\implies (h-3)(35h-171)=0[/tex]
[tex]\implies h=3, h=\dfrac{171}{35}[/tex]
Inputting both values of [tex]h[/tex] into the equations for SA:
when [tex]h = 3[/tex]:
[tex]\textsf{pyramid}=16\sqrt{16+3^2}=80 \ \sf cm^2[/tex]
[tex]\textsf{square prism}=64+96(3)=352 \ \sf cm^2[/tex]
[tex]\textsf{total SA}=80+352=432 \ \sf cm^2[/tex]
when [tex]h=\dfrac{171}{35}[/tex]:
[tex]\textsf{pyramid}=16\sqrt{16+(\frac{171}{35})^2}=101.0285714... \sf cm^2[/tex]
[tex]\textsf{square prism}=64+96(\frac{171}{35})=533.0285714... \ \sf cm^2[/tex]
[tex]\textsf{total SA}=101.0285714...+533.0285714... =634.0571428... \ \sf cm^2[/tex]
Therefore, h = 3 only (since the total area is 432 cm²)
⇒ height of pyramid = 3 cm
⇒ height of prism = 3h = 3 × 3 = 9 cm
