1. **Problem statement:** We are given a quadrilateral with sides labeled as $20\sqrt{35}$, $15\sqrt{45}$, $72$, and $30\sqrt{5}$, and an internal segment labeled $2$. We need to show that the total length of the fence (السياج الشائك) can be expressed as a real number in the form $a\sqrt{5} + b$ where $a$ and $b$ are natural numbers. Then, calculate the total length and find the price of the fence rounded to the nearest unit, given that 10 meters cost 2500 and write the price in scientific notation. 2. **Simplify each radical term:** - $20\sqrt{35} = 20\sqrt{5 \times 7} = 20\sqrt{5}\sqrt{7}$ (keep as is for now) - $15\sqrt{45} = 15\sqrt{9 \times 5} = 15 \times 3 \sqrt{5} = 45\sqrt{5}$ - $30\sqrt{5}$ is already in the form $30\sqrt{5}$ 3. **Express $20\sqrt{35}$ in terms of $\sqrt{5}$:** Since $\sqrt{35} = \sqrt{5 \times 7} = \sqrt{5} \times \sqrt{7}$, we write: $$20\sqrt{35} = 20 \sqrt{5} \sqrt{7} = 20 \sqrt{7} \sqrt{5}$$ We can treat $\sqrt{7}$ as a constant multiplier. 4. **Sum all sides and the internal segment:** Total length $L = 20\sqrt{35} + 15\sqrt{45} + 72 + 30\sqrt{5} + 2$ Substitute simplified terms: $$L = 20 \sqrt{7} \sqrt{5} + 45 \sqrt{5} + 72 + 30 \sqrt{5} + 2$$ Group terms with $\sqrt{5}$ and constants: $$L = (20 \sqrt{7} + 45 + 30) \sqrt{5} + (72 + 2)$$ Calculate constants: $$L = (20 \sqrt{7} + 75) \sqrt{5} + 74$$ 5. **Approximate $\sqrt{7}$ to express $L$ as $a\sqrt{5} + b$ with $a,b$ natural numbers:** $\sqrt{7} \approx 2.6458$ Calculate $20 \sqrt{7} = 20 \times 2.6458 = 52.916$ Sum inside parentheses: $$52.916 + 75 = 127.916$$ So, $$L \approx 127.916 \sqrt{5} + 74$$ Since $a$ and $b$ must be natural numbers, approximate $a = 128$, $b = 74$. 6. **Calculate numerical value of $L$:** $\sqrt{5} \approx 2.236$ $$L \approx 128 \times 2.236 + 74 = 286.208 + 74 = 360.208$$ 7. **Calculate the price of the fence:** Price per 10 meters = 2500 Length $\approx 360.208$ meters Price $= \frac{360.208}{10} \times 2500 = 36.0208 \times 2500 = 90052$ (rounded to nearest unit) 8. **Write price in scientific notation:** $$90052 \approx 9.0052 \times 10^{4}$$ **Final answers:** - Length of fence $\approx 128 \sqrt{5} + 74$ - Price $= 90052$ (rounded) - Price in scientific notation $= 9.0052 \times 10^{4}$