The base of a right prism, whose height is 2 cm, is a square. If the total surface area of the prism is `10 cm^(2)`, then its volume is :

Options

A. `3 cm^(3)`
B. `1 cm^(3)`
C. `2 cm^(3)`
D. `4 cm^(3)`

Answer

To solve the problem step by step, we will find the volume of the right prism given its height and total surface area.

### Step 1: Understand the given information
- The height (h) of the prism is 2 cm.
- The total surface area (TSA) of the prism is 10 cm².
- The base of the prism is a square.

### Step 2: Write 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:
\[ \text{TSA} = 2 \times \text{Area of Base} + \text{Perimeter of Base} \times \text{Height} \]

### Step 3: Define the side length of the square base
Let the side length of the square base be \( a \).

### Step 4: Calculate the area and perimeter of the square base
- The area of the square base is:
\[ \text{Area of Base} = a^2 \]
- The perimeter of the square base is:
\[ \text{Perimeter of Base} = 4a \]

### Step 5: Substitute into the TSA formula
Substituting the area and perimeter into the TSA formula gives:
\[ 10 = 2 \times a^2 + 4a \times 2 \]
This simplifies to:
\[ 10 = 2a^2 + 8a \]

### Step 6: Rearrange the equation
Rearranging the equation, we get:
\[ 2a^2 + 8a - 10 = 0 \]
Dividing the entire equation by 2:
\[ a^2 + 4a - 5 = 0 \]

### Step 7: Factor the quadratic equation
Now, we can factor the quadratic:
\[ (a + 5)(a - 1) = 0 \]
This gives us two possible solutions for \( a \):
\[ a + 5 = 0 \quad \Rightarrow \quad a = -5 \quad (\text{not valid since side length cannot be negative}) \]
\[ a - 1 = 0 \quad \Rightarrow \quad a = 1 \]

### Step 8: Calculate the volume of the prism
Now that we have the side length \( a = 1 \) cm, we can find the volume (V) of the prism using the formula:
\[ V = \text{Area of Base} \times \text{Height} = a^2 \times h \]
Substituting the values:
\[ V = 1^2 \times 2 = 1 \times 2 = 2 \text{ cm}^3 \]

### Final Answer
The volume of the prism is \( 2 \text{ cm}^3 \).

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