Answer :
Let’s solve the problem step-by-step to find the height of the pyramid.
### Step 1: Find the side length of the square base
Given:
- Diagonal length [tex]\( d = 12\sqrt{2} \)[/tex] cm
For a square with side length [tex]\( s \)[/tex], the diagonal [tex]\( d \)[/tex] can be expressed as:
[tex]\[ d = s\sqrt{2} \][/tex]
Therefore:
[tex]\[ s = \frac{d}{\sqrt{2}} \][/tex]
[tex]\[ s = \frac{12\sqrt{2}}{\sqrt{2}} \][/tex]
[tex]\[ s = 12 \, \text{cm} \][/tex]
### Step 2: Calculate the perimeter of the base
The perimeter [tex]\( P \)[/tex] of a square with side length [tex]\( s \)[/tex] is:
[tex]\[ P = 4s \][/tex]
[tex]\[ P = 4 \times 12 \][/tex]
[tex]\[ P = 48 \, \text{cm} \][/tex]
### Step 3: Rearrange the formula for the curved surface area to find the slant height
Given:
- Curved Surface Area [tex]\( CSA = 1920 \, \text{cm}^2 \)[/tex]
The curved surface area of a right square pyramid is given by:
[tex]\[ CSA = \frac{1}{2} \times P \times l \][/tex]
where [tex]\( l \)[/tex] is the slant height.
Rearranging to solve for [tex]\( l \)[/tex]:
[tex]\[ 1920 = \frac{1}{2} \times 48 \times l \][/tex]
[tex]\[ 1920 = 24l \][/tex]
[tex]\[ l = \frac{1920}{24} \][/tex]
[tex]\[ l = 80 \, \text{cm} \][/tex]
### Step 4: Use the Pythagorean theorem to find the height of the pyramid
To find the height [tex]\( h \)[/tex] of the pyramid, we use the fact that the slant height forms a right triangle with half the side length and the height.
[tex]\[ l^2 = \left(\frac{s}{2}\right)^2 + h^2 \][/tex]
[tex]\[ 80^2 = \left(\frac{12}{2}\right)^2 + h^2 \][/tex]
[tex]\[ 6400 = 6^2 + h^2 \][/tex]
[tex]\[ 6400 = 36 + h^2 \][/tex]
[tex]\[ 6400 - 36 = h^2 \][/tex]
[tex]\[ 6364 = h^2 \][/tex]
[tex]\[ h = \sqrt{6364} \][/tex]
[tex]\[ h \approx 79.76 \][/tex]
### Step 5: Round the height to the nearest integer and compare with choices
Rounding off 79.76 to the nearest integer gives:
[tex]\[ h \approx 80 \][/tex]
The answer is:
[tex]\[ \boxed{B. 80 cm} \][/tex]
### Step 1: Find the side length of the square base
Given:
- Diagonal length [tex]\( d = 12\sqrt{2} \)[/tex] cm
For a square with side length [tex]\( s \)[/tex], the diagonal [tex]\( d \)[/tex] can be expressed as:
[tex]\[ d = s\sqrt{2} \][/tex]
Therefore:
[tex]\[ s = \frac{d}{\sqrt{2}} \][/tex]
[tex]\[ s = \frac{12\sqrt{2}}{\sqrt{2}} \][/tex]
[tex]\[ s = 12 \, \text{cm} \][/tex]
### Step 2: Calculate the perimeter of the base
The perimeter [tex]\( P \)[/tex] of a square with side length [tex]\( s \)[/tex] is:
[tex]\[ P = 4s \][/tex]
[tex]\[ P = 4 \times 12 \][/tex]
[tex]\[ P = 48 \, \text{cm} \][/tex]
### Step 3: Rearrange the formula for the curved surface area to find the slant height
Given:
- Curved Surface Area [tex]\( CSA = 1920 \, \text{cm}^2 \)[/tex]
The curved surface area of a right square pyramid is given by:
[tex]\[ CSA = \frac{1}{2} \times P \times l \][/tex]
where [tex]\( l \)[/tex] is the slant height.
Rearranging to solve for [tex]\( l \)[/tex]:
[tex]\[ 1920 = \frac{1}{2} \times 48 \times l \][/tex]
[tex]\[ 1920 = 24l \][/tex]
[tex]\[ l = \frac{1920}{24} \][/tex]
[tex]\[ l = 80 \, \text{cm} \][/tex]
### Step 4: Use the Pythagorean theorem to find the height of the pyramid
To find the height [tex]\( h \)[/tex] of the pyramid, we use the fact that the slant height forms a right triangle with half the side length and the height.
[tex]\[ l^2 = \left(\frac{s}{2}\right)^2 + h^2 \][/tex]
[tex]\[ 80^2 = \left(\frac{12}{2}\right)^2 + h^2 \][/tex]
[tex]\[ 6400 = 6^2 + h^2 \][/tex]
[tex]\[ 6400 = 36 + h^2 \][/tex]
[tex]\[ 6400 - 36 = h^2 \][/tex]
[tex]\[ 6364 = h^2 \][/tex]
[tex]\[ h = \sqrt{6364} \][/tex]
[tex]\[ h \approx 79.76 \][/tex]
### Step 5: Round the height to the nearest integer and compare with choices
Rounding off 79.76 to the nearest integer gives:
[tex]\[ h \approx 80 \][/tex]
The answer is:
[tex]\[ \boxed{B. 80 cm} \][/tex]
