Answer:
[tex] \displaystyle V_{ \text{pyramid}}= 64 \: {cm}^{3} [/tex]
Step-by-step explanation:
we are given surface area and the length of the square base
we want to figure out the Volume
to do so
we need to figure out slant length first
recall the formula of surface area
[tex]\displaystyle A_{\text{surface}}=B+\dfrac{1}{2}\times P \times s[/tex]
where B stands for Base area
and P for Base Parimeter
so
[tex] \sf\displaystyle \: 144=(8 \times 8)+\dfrac{1}{2}\times (8 \times 4) \times s[/tex]
now we need our algebraic skills to figure out s
simplify parentheses:
[tex] \sf\displaystyle \: 64+\dfrac{1}{2}\times32\times s = 144[/tex]
reduce fraction:
[tex] \sf\displaystyle \: 64+\dfrac{1}{ \cancel{ \: 2}}\times \cancel{32} \: ^{16} \times s = 144 \\ 64 + 16 \times s = 144[/tex]
simplify multiplication:
[tex] \displaystyle \: 16s + 64 = 144[/tex]
cancel 64 from both sides;
[tex] \displaystyle \: 16s = 80[/tex]
divide both sides by 16:
[tex] \displaystyle \: \therefore \: s = 5[/tex]
now we'll use Pythagoras theorem to figure out height
according to the theorem
[tex] \displaystyle \: {h}^{2} + (\frac{l}{2} {)}^{2} = {s}^{2} [/tex]
substitute the value of l and s:
[tex] \displaystyle \: {h}^{2} + (\frac{8}{2} {)}^{2} = {5}^{2} [/tex]
simplify parentheses:
[tex] \displaystyle \: {h}^{2} + (4 {)}^{2} = {5}^{2} [/tex]
simplify squares:
[tex] \displaystyle \: {h}^{2} + 16 = 25[/tex]
cancel 16 from both sides:
[tex] \displaystyle \: {h}^{2} = 9[/tex]
square root both sides:
[tex] \displaystyle \: \therefore \: {h}^{} = 3[/tex]
recall the formula of a square pyramid
[tex]\displaystyle V_{pyramid}=\dfrac{1}{3}\times A\times h[/tex]
where A stands for Base area (l²)
substitute the value of h and l:
[tex] \sf\displaystyle V_{ \text{pyramid}}=\dfrac{1}{3}\times \{8 \times 8 \}\times 3[/tex]
simplify multiplication:
[tex] \sf\displaystyle V_{ \text{pyramid}}=\dfrac{1}{3}\times 64\times 3[/tex]
reduce fraction:
[tex] \sf\displaystyle V_{ \text{pyramid}}=\dfrac{1}{ \cancel{ 3 \: }}\times 64\times \cancel{ \: 3}[/tex]
hence,
[tex] \sf\displaystyle V_{ \text{pyramid}}= 64 \: {cm}^{3} [/tex]
