1. The problem states that the height of the chest is given by the function $H(x) = x + 6$ and the area of the base is given by $A(x) = \frac{20\sqrt{x}}{3}$, where $x$ is the width of the chest. 2. To find the volume $V(x)$ of the chest, we use the formula for the volume of a rectangular prism: $$V = \text{base area} \times \text{height}$$ 3. This means we multiply the functions $A(x)$ and $H(x)$ to get the volume function: $$V(x) = A(x) \times H(x) = \frac{20\sqrt{x}}{3} \times (x + 6)$$ 4. Distribute the multiplication: $$V(x) = \frac{20\sqrt{x}}{3} \times x + \frac{20\sqrt{x}}{3} \times 6 = \frac{20x\sqrt{x}}{3} + \frac{120\sqrt{x}}{3}$$ 5. Simplify the second term: $$\frac{120\sqrt{x}}{3} = 40\sqrt{x}$$ 6. So the volume function is: $$V(x) = \frac{20x\sqrt{x}}{3} + 40\sqrt{x}$$ 7. The problem's second dropdown likely asks about the form of the volume function. The term $\frac{20x\sqrt{x}}{3}$ can be rewritten as $\frac{20x^{3/2}}{3}$ since $\sqrt{x} = x^{1/2}$. 8. Therefore, the volume function is: $$V(x) = \frac{20x^{3/2}}{3} + 40x^{1/2}$$ Summary of dropdown answers: - Ronald should multiply functions $H(x)$ and $A(x)$. - The function that represents the volume is $V(x) = \frac{20x\sqrt{x}}{3} + 40\sqrt{x}$, which includes the term $\sqrt{x}$ as shown. Final answers: - First dropdown: "multiply" - Second dropdown: "plus 40\sqrt{x}"