1. **Problem 1: Express $x$ in the form $a + b\sqrt{c}$** Given a right triangle with hypotenuse $x + 16$, one leg $2x$, and the angle opposite the leg $2x$ is $30^\circ$. 2. Use the sine definition: $\sin 30^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{2x}{x+16}$. 3. Since $\sin 30^\circ = \frac{1}{2}$, set up the equation: $$\frac{1}{2} = \frac{2x}{x+16}$$ 4. Cross-multiply: $$x + 16 = 4x$$ 5. Rearrange: $$16 = 3x \implies x = \frac{16}{3}$$ 6. Check if $x$ can be expressed as $a + b\sqrt{c}$ with integers $a,b,c$. Here, $x = \frac{16}{3}$ is a rational number, so $a=\frac{16}{3}$, $b=0$, $c=0$. --- 1. **Problem 2: Total books read by the youth group** - Number of girls = 11, mean books per girl = 8 - Number of boys = 7, mean books per boy = 4 2. Total books by girls: $$11 \times 8 = 88$$ 3. Total books by boys: $$7 \times 4 = 28$$ 4. Total books by youth group: $$88 + 28 = 116$$ --- 1. **Problem 3: Bearing from A to B** - Coordinates: $A(1,3)$, $B(8,7)$ 2. Calculate differences: $$\Delta x = 8 - 1 = 7$$ $$\Delta y = 7 - 3 = 4$$ 3. Calculate angle $\theta$ from north clockwise: - Bearing is measured clockwise from north. - Calculate angle from east axis: $$\alpha = \arctan\left(\frac{\Delta y}{\Delta x}\right) = \arctan\left(\frac{4}{7}\right) \approx 29.74^\circ$$ 4. Bearing from north: $$\text{bearing} = 90^\circ - \alpha = 90^\circ - 29.74^\circ = 60.26^\circ$$ 5. Rounded to nearest degree: $$60^\circ$$ --- **Final answers:** - Problem 1: $a = \frac{16}{3}$, $b = 0$, $c = 0$ - Problem 2: Total books = 116 - Problem 3: Bearing from A to B = $60^\circ$