1. **Problem 1:** Write the equation for the joint variation "x varies jointly as y and z". Since x varies jointly as y and z, the formula is: $$x = kyz$$ where $k$ is the constant of proportionality. 2. **Problem 2:** Write the equation for the joint variation "z varies jointly as x and the square root of y". Since z varies jointly as x and $\sqrt{y}$, the formula is: $$z = kx\sqrt{y}$$ where $k$ is the constant of proportionality. 3. **Problem 3:** Write the equation for the joint variation "w varies jointly as x and y and inversely as z". Since w varies jointly as x and y and inversely as z, the formula is: $$w = \frac{kxy}{z}$$ where $k$ is the constant of proportionality. 4. **Problem 4:** Suppose y varies jointly with x and z. Given $y = 36$ when $x = 4$ and $z = 3$, find $y$ when $x = 12$ and $z = 36$. 1. Write the joint variation formula: $$y = kxz$$ 2. Substitute known values to find $k$: $$36 = k \times 4 \times 3$$ $$36 = 12k$$ $$k = \frac{36}{12} = 3$$ 3. Use $k = 3$ to find new $y$ when $x = 12$ and $z = 36$: $$y = 3 \times 12 \times 36$$ $$y = 3 \times 432 = 1296$$ **Final answer:** - Problem 1: $x = kyz$ - Problem 2: $z = kx\sqrt{y}$ - Problem 3: $w = \frac{kxy}{z}$ - Problem 4: $y = 1296$ when $x = 12$ and $z = 36$.