If the area of an equilateral triangle is 60√3 m2, then what is the value (in metres) of its height?

Options

A. `4sqrt5`
B. `5sqrt5`
C. `sqrt30`
D. `6sqrt5`

Answer

To find the height of an equilateral triangle given its area, we can follow these steps:

### Step 1: Use the formula for the area of an equilateral triangle
The area \( A \) of an equilateral triangle can be expressed in terms of its height \( h \) as:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
For an equilateral triangle, the base can be expressed in terms of the height using the relationship:
\[ \text{base} = \frac{2h}{\sqrt{3}} \]
Thus, the area can also be expressed as:
\[ A = \frac{1}{2} \times \frac{2h}{\sqrt{3}} \times h = \frac{h^2}{\sqrt{3}} \]

### Step 2: Set the area equal to the given value
We know from the problem that the area \( A = 60\sqrt{3} \) m². Therefore, we can set up the equation:
\[ \frac{h^2}{\sqrt{3}} = 60\sqrt{3} \]

### Step 3: Solve for \( h^2 \)
To eliminate the fraction, we can multiply both sides by \( \sqrt{3} \):
\[ h^2 = 60\sqrt{3} \times \sqrt{3} \]
This simplifies to:
\[ h^2 = 60 \times 3 = 180 \]

### Step 4: Find \( h \)
Now, we take the square root of both sides to find \( h \):
\[ h = \sqrt{180} \]

### Step 5: Simplify \( \sqrt{180} \)
To simplify \( \sqrt{180} \), we can factor it:
\[ 180 = 36 \times 5 = 6^2 \times 5 \]
Thus,
\[ \sqrt{180} = \sqrt{36 \times 5} = \sqrt{36} \times \sqrt{5} = 6\sqrt{5} \]

### Final Answer
The height \( h \) of the equilateral triangle is:
\[ h = 6\sqrt{5} \text{ m} \]