To solve for $y$, let us break this problem into steps.

Key Information:

• The triangle has an area $A = 72 \, \text{m}^2$.
• The given angle is $51^\circ$.
• $x$ is the base, and $y$ is the height.
• The formula for the area of a triangle is: $A = \frac{1}{2} \cdot \text{base} \cdot \text{height}.$ Here, the base and height are related to the sides of the triangle and the angle.

Step 1: Express the area in terms of $x$ and $y$:

From the triangle area formula: $72 = \frac{1}{2} \cdot x \cdot y,$ which simplifies to: x \cdot y = 144. \tag{1}

Step 2: Relate $x$ and $y$ using trigonometry:

• From the triangle, we know: $x = y \cdot \tan(51^\circ).$

Step 3: Substitute $x$ in Equation (1):

Substituting $x = y \cdot \tan(51^\circ)$ into $x \cdot y = 144$: $(y \cdot \tan(51^\circ)) \cdot y = 144.$ Simplify: $y^2 \cdot \tan(51^\circ) = 144.$ Solve for $y^2$: $y^2 = \frac{144}{\tan(51^\circ)}.$ $y = \sqrt{\frac{144}{\tan(51^\circ)}}.$

Step 4: Calculate $y$:

Using $\tan(51^\circ) \approx 1.2349$: $y = \sqrt{\frac{144}{1.2349}}.$ $y = \sqrt{116.58}.$ $y \approx 10.8 \, \text{m}.$

Final Answer:

$y \approx 10.8 \, \text{m}.$

Would you like a detailed explanation of any step, or should I calculate $x$ for you?

Related Questions:

1. How can you find the value of $x$ after determining $y$?
2. What is the length of the hypotenuse of this triangle?
3. How do trigonometric functions like $\tan$ relate to angles and sides in a triangle?
4. What are the properties of a right triangle with a given angle and area?
5. Can you find the perimeter of this triangle?

Tip:

When solving problems involving areas of triangles, always identify the base, height, and angle relationships clearly before applying formulas!