The height of a right prism with a square base is 15 cm. If the area of the total surface of the prism is 608 sq. cm, its volume is

Options

A. a)`910 cm^(3)`
B. b)`920 cm^(3)`
C. c)`960 cm^(3)`
D. d)`980 cm^(3)`

Answer

To find the volume of the right prism with a square base, we will follow these steps:

### Step 1: Understand the formula for the total surface area of a prism
The total surface area (TSA) of a right prism can be calculated using the formula:
\[ \text{TSA} = \text{Lateral Surface Area (LSA)} + 2 \times \text{Area of Base} \]
For a prism with a square base, the LSA can be calculated as:
\[ \text{LSA} = \text{Perimeter of Base} \times \text{Height} \]
The area of the base (which is a square) is:
\[ \text{Area of Base} = \text{side}^2 \]

### Step 2: Define the variables
Let the side of the square base be \( a \) cm. The height of the prism is given as \( h = 15 \) cm.

### Step 3: Calculate the perimeter of the base
The perimeter of the square base is:
\[ \text{Perimeter} = 4a \]

### Step 4: Substitute into the TSA formula
Substituting the values into the TSA formula:
\[ \text{TSA} = \text{LSA} + 2 \times \text{Area of Base} \]
\[ 608 = (4a \times 15) + 2(a^2) \]
\[ 608 = 60a + 2a^2 \]

### Step 5: Rearrange the equation
Rearranging gives us:
\[ 2a^2 + 60a - 608 = 0 \]
Dividing the entire equation by 2 simplifies it:
\[ a^2 + 30a - 304 = 0 \]

### Step 6: Factor the quadratic equation
Now we need to factor or use the quadratic formula to solve for \( a \). The equation can be factored as:
\[ (a + 38)(a - 8) = 0 \]
This gives us two possible solutions:
\[ a + 38 = 0 \quad \text{or} \quad a - 8 = 0 \]
Thus, \( a = -38 \) (not valid since side length cannot be negative) or \( a = 8 \) cm.

### Step 7: Calculate the volume of the prism
The volume \( V \) of the prism is given by:
\[ V = \text{Area of Base} \times \text{Height} \]
Substituting the values:
\[ V = a^2 \times h = 8^2 \times 15 = 64 \times 15 = 960 \text{ cm}^3 \]

### Final Answer
The volume of the prism is:
\[ \boxed{960 \text{ cm}^3} \]

---