What is the value of \( \sqrt{760-4 x^{2}} \) to the nearest tenth when \( x=6 \) ?

To find the value of \( \sqrt{760 - 4x^{2}} \) when \( x = 6 \): 1. **Substitute \( x = 6 \) into the expression:** \[ \sqrt{760 - 4(6)^2} = \sqrt{760 - 4 \times 36} = \sqrt{760 - 144} = \sqrt{616} \] 2. **Calculate \( \sqrt{616} \):** - We know that \( 24^2 = 576 \) and \( 25^2 = 625 \). So, \( \sqrt{616} \) is between 24 and 25. - To find a more precise value, consider: \[ 24.8^2 = (25 - 0.2)^2 = 625 - 2 \times 25 \times 0.2 + 0.04 = 615.04 \] - Since \( 24.8^2 = 615.04 \) and we need \( \sqrt{616} \), we can use a linear approximation: \[ \sqrt{616} \approx 24.8 + \frac{616 - 615.04}{2 \times 24.8} \approx 24.8 + \frac{0.96}{49.6} \approx 24.8 + 0.019 \approx 24.819 \] 3. **Round to the nearest tenth:** \[ 24.819 \approx 24.8 \] **Final Answer:** The value is approximately 24.8.