Math Problem Statement

determine the values of y.

Determine the value of y in the given triangle with an area of 72 m² and an angle of 51°.

Solution

To find the value of yy, let's solve the problem step by step based on the given triangle:


Given Data:

  1. The area of the triangle: A=72m2A = 72 \, \text{m}^2.
  2. One angle: θ=51\theta = 51^\circ.
  3. The base of the triangle: xx.
  4. The height of the triangle (relative to the base): yy.

Formula for the Area of a Triangle:

The area of a triangle can be written as: A=12xyA = \frac{1}{2} \cdot x \cdot y where xx is the base and yy is the height.

We need to express yy in terms of xx and solve further.


Relation Between Base, Height, and Angle:

In a triangle, the height yy relative to base xx can also be calculated using trigonometry: y=xtan(51)y = x \cdot \tan(51^\circ)

Substitute this yy into the area formula: A=12x(xtan(51))A = \frac{1}{2} \cdot x \cdot \left( x \cdot \tan(51^\circ) \right)

This simplifies to: 72=12x2tan(51)72 = \frac{1}{2} \cdot x^2 \cdot \tan(51^\circ)


Solve for xx:

Rearranging for x2x^2: x2=722tan(51)x^2 = \frac{72 \cdot 2}{\tan(51^\circ)}

Using tan(51)1.2349\tan(51^\circ) \approx 1.2349: x2=1441.2349116.6x^2 = \frac{144}{1.2349} \approx 116.6

So: x=116.610.8mx = \sqrt{116.6} \approx 10.8 \, \text{m}


Find yy:

Now that x10.8mx \approx 10.8 \, \text{m}, substitute it back into: y=xtan(51)y = x \cdot \tan(51^\circ)

y=10.81.234913.34my = 10.8 \cdot 1.2349 \approx 13.34 \, \text{m}


Final Answer:

y13.34my \approx 13.34 \, \text{m}


Would you like a more detailed breakdown of these steps? Let me know!


Related Questions to Expand:

  1. How do you compute the tangent of an angle without a calculator?
  2. What other trigonometric functions are useful in solving triangles?
  3. Can you calculate the base xx using only the area and height yy?
  4. What is the significance of the 5151^\circ angle in this problem?
  5. How does the formula for the area of a triangle change in non-right triangles?

Tip:

Always verify the units (e.g., meters, degrees) and ensure trigonometric functions are in the correct mode (degrees or radians) when solving problems like this!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Trigonometry
Geometry
Area of a Triangle

Formulas

Area of a triangle: A = 1/2 * base * height
Height in terms of trigonometry: height = base * tan(angle)

Theorems

Trigonometric Ratios
Relationship between area and base-height

Suitable Grade Level

Grades 9-12