1. **State the problem:** Given vectors $a$ and $b$ with magnitudes $|a|=15$, $|b|=20$, and $|a-b|=15.5$, find the magnitude $|a+b|$. 2. **Recall the formula for the magnitude of the sum and difference of two vectors:** $$|a+b|^2 = |a|^2 + |b|^2 + 2|a||b|\cos\theta$$ $$|a-b|^2 = |a|^2 + |b|^2 - 2|a||b|\cos\theta$$ where $\theta$ is the angle between vectors $a$ and $b$. 3. **Use the given values in the difference formula to find $\cos\theta$:** $$|a-b|^2 = 15.5^2 = 240.25$$ $$240.25 = 15^2 + 20^2 - 2 \times 15 \times 20 \times \cos\theta$$ $$240.25 = 225 + 400 - 600 \cos\theta$$ $$240.25 = 625 - 600 \cos\theta$$ 4. **Isolate $\cos\theta$:** $$600 \cos\theta = 625 - 240.25 = 384.75$$ $$\cos\theta = \frac{384.75}{600} = 0.64125$$ 5. **Use $\cos\theta$ to find $|a+b|^2$:** $$|a+b|^2 = 15^2 + 20^2 + 2 \times 15 \times 20 \times 0.64125$$ $$= 225 + 400 + 600 \times 0.64125$$ $$= 625 + 384.75 = 1009.75$$ 6. **Find $|a+b|$ by taking the square root:** $$|a+b| = \sqrt{1009.75} \approx 31.77$$ **Final answer:** $$|a+b| \approx 31.77$$