Math Problem Statement

Solve the problem

Solve the mathematical expression given in the image.

Solution

To solve this problem, let's break down the expression step-by-step. The equation reads:

0.5×20415[14.6×9.8×10620×1000×1102]×1000×110\frac{0.5 \times 20}{415} \left[ 1 - \sqrt{\frac{4.6 \times 9.8 \times 10^6}{20 \times 1000 \times 110^2}} \right] \times 1000 \times 110

Let's solve it in parts:

  1. Calculate the numerator of the first fraction:
    0.5×20=100.5 \times 20 = 10

  2. Evaluate the first fraction:
    104150.0241\frac{10}{415} \approx 0.0241

  3. Calculate the expression inside the square root:

    • Numerator: 4.6×9.8×106=4.508×1074.6 \times 9.8 \times 10^6 = 4.508 \times 10^7
    • Denominator: 20×1000×1102=20×1000×12100=24200000020 \times 1000 \times 110^2 = 20 \times 1000 \times 12100 = 242000000

    So, we have: 4.508×1072.42×1080.1862\frac{4.508 \times 10^7}{2.42 \times 10^8} \approx 0.1862

  4. Take the square root:
    0.18620.4316\sqrt{0.1862} \approx 0.4316

  5. Evaluate 10.43161 - 0.4316:
    10.4316=0.56841 - 0.4316 = 0.5684

  6. Multiply by 0.02410.0241, 10001000, and 110110:

    • 0.0241×0.56840.01370.0241 \times 0.5684 \approx 0.0137
    • Then, 0.0137×1000×110=1.507×1030.0137 \times 1000 \times 110 = 1.507 \times 10^3 or approximately 1507mm21507 \, \text{mm}^2.

This differs slightly from the given result of 264.34mm2264.34 \, \text{mm}^2. Please verify the problem structure or any intermediate values if possible. Let me know if you'd like further clarification on any step!


Would you like further details on any of these steps, or need help with similar questions? Here are some related questions to consider:

  1. How does changing one of the values (e.g., the 4.6 or 9.8) affect the final result?
  2. How do you interpret the units in this expression?
  3. What are common sources of error when calculating expressions with square roots and fractions?
  4. How could we simplify this equation further without calculating each step?
  5. What are some applications of such expressions in physics or engineering?

Tip: Breaking down complex expressions into parts makes it easier to solve without losing track of the calculations.

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Math Problem Analysis

Mathematical Concepts

Algebra
Square Roots
Fractions
Multiplication

Formulas

Fraction: a/b
Square Root: √x
Multiplication: a × b

Theorems

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Suitable Grade Level

Grades 9-12