The surface area of a cube is 150 `m^(2)`. The length of its diagonal is

Options

A. `5 sqrt(3)` m
B. 5m
C. `(10)/(sqrt(3)) m,`
D. 15 m

Answer

To find the length of the diagonal of a cube given its surface area, we can follow these steps:

### Step 1: Understand the formula for the surface area of a cube.
The surface area \( S \) of a cube is given by the formula:
\[ S = 6a^2 \]
where \( a \) is the length of one side of the cube.

### Step 2: Set up the equation using the given surface area.
We know the surface area of the cube is 150 m². Therefore, we can set up the equation:
\[ 6a^2 = 150 \]

### Step 3: Solve for \( a^2 \).
To find \( a^2 \), divide both sides of the equation by 6:
\[ a^2 = \frac{150}{6} \]
\[ a^2 = 25 \]

### Step 4: Solve for \( a \).
Now, take the square root of both sides to find \( a \):
\[ a = \sqrt{25} \]
\[ a = 5 \text{ m} \]

### Step 5: Use the formula for the diagonal of the cube.
The length of the diagonal \( d \) of a cube can be calculated using the formula:
\[ d = a\sqrt{3} \]

### Step 6: Substitute the value of \( a \) into the diagonal formula.
Now substitute \( a = 5 \) m into the diagonal formula:
\[ d = 5\sqrt{3} \]

### Step 7: State the final answer.
Thus, the length of the diagonal of the cube is:
\[ d = 5\sqrt{3} \text{ m} \]

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