To find the surface area (S.A.) of a triangular prism, we can use the formula:
\[ \text{S.A.} = \text{Base Area} + \text{Lateral Area} \]
Step 1: Calculate the area of the triangular base
The triangular base is a right triangle with base \( b \) and height \( h \). The area \( A \) of a right triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times b \times h \]
In this case, the base \( b = 27.9 \) cm and the height \( h = 5.1 \) cm.
Substituting the values:
\[ A = \frac{1}{2} \times 27.9 \times 5.1 = \frac{1}{2} \times 142.29 = 71.145 , \text{cm}^2 \]
Step 2: Calculate the lateral area
The lateral area of a triangular prism is given by the formula:
\[ \text{Lateral Area} = \text{Perimeter of the base} \times \text{Length} \]
Calculate the perimeter of the triangular base:
The perimeter \( P \) of a right triangle is the sum of all its sides:
\[ P = b + h + \text{hypotenuse} \]
Where
- \( b = 27.9 \) cm
- \( h = 5.1 \) cm
- The hypotenuse \( c = 28.4 \) cm
Calculating the perimeter:
\[ P = 27.9 + 5.1 + 28.4 = 61.4 , \text{cm} \]
Calculate the length of the prism:
The length \( L = 30.5 , \text{cm} \).
Now calculate the lateral area:
\[ \text{Lateral Area} = P \times L \] \[ \text{Lateral Area} = 61.4 \times 30.5 = 1873.7 , \text{cm}^2 \]
Step 3: Calculate the total surface area
Now we can add the base area and the lateral area:
\[ \text{S.A.} = \text{Base Area} + \text{Lateral Area} \] \[ \text{S.A.} = 71.145 + 1873.7 = 1944.845 , \text{cm}^2 \]
Step 4: Round to the nearest whole number
\[ \text{S.A.} \approx 1945 , \text{cm}^2 \]
So, the total amount of wrapping paper needed to wrap the toy is:
\[ \boxed{1945} , \text{cm}^2 \]