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