Math Problem Statement

Solve for x to the nearest tenth of a metre solving by the cosine law. Provide the great calculations for this problem and the exact answer for the x should be 11.5 m.

Solve for x to the nearest tenth of a metre using the cosine law for the triangle setup.

Solution

To solve for xx in this problem, we can use the Cosine Law, which is applicable for non-right triangles when we know two sides and the included angle, or all three sides of the triangle.

Here’s the setup based on the provided triangle information:

Given Data:

  • Side BC=6.3mBC = 6.3 \, \text{m}
  • Side CD=xmCD = x \, \text{m} (the value we want to find)
  • Side BD=12.5mBD = 12.5 \, \text{m}
  • Angle BCD=46\angle BCD = 46^\circ

Since we're solving for side x=CDx = CD in BCD\triangle BCD, we can apply the Cosine Law: x2=BC2+BD22BCBDcos(BCD)x^2 = BC^2 + BD^2 - 2 \cdot BC \cdot BD \cdot \cos(\angle BCD)

Step-by-Step Solution:

  1. Substitute the known values into the formula: x2=(6.3)2+(12.5)226.312.5cos(46)x^2 = (6.3)^2 + (12.5)^2 - 2 \cdot 6.3 \cdot 12.5 \cdot \cos(46^\circ)

  2. Calculate each term:

    • (6.3)2=39.69(6.3)^2 = 39.69
    • (12.5)2=156.25(12.5)^2 = 156.25
    • 26.312.5=157.52 \cdot 6.3 \cdot 12.5 = 157.5
    • cos(46)0.6947\cos(46^\circ) \approx 0.6947
  3. Substitute and simplify: x2=39.69+156.25157.50.6947x^2 = 39.69 + 156.25 - 157.5 \cdot 0.6947 x2=39.69+156.25109.89675x^2 = 39.69 + 156.25 - 109.89675 x2=196.94109.89675x^2 = 196.94 - 109.89675 x287.04325x^2 \approx 87.04325

  4. Take the square root to find xx: x87.04325x \approx \sqrt{87.04325} x9.33mx \approx 9.33 \, \text{m}

Conclusion:

The answer appears to be approximately x=9.3mx = 9.3 \, \text{m}, not 11.5 m as stated. If further clarification or a different approach is needed, let me know.


Would you like more details or have any further questions?


Here are related questions for further exploration:

  1. How would you solve this problem using the Sine Law?
  2. How does the accuracy of cos(46)\cos(46^\circ) affect the solution?
  3. What if angle BCD\angle BCD was different; how would that impact xx?
  4. How can we use trigonometry to find angles if we know all three sides?
  5. How does changing the length of side BDBD impact the value of xx?

Tip: When using the Cosine Law, ensure you have precise angle values for accurate results.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Trigonometry
Cosine Law

Formulas

Cosine Law: c^2 = a^2 + b^2 - 2ab * cos(C)

Theorems

Cosine Law

Suitable Grade Level

Grades 10-12