Question

select the correct answer from each drop-down menu.
ronald is buying a rectangular chest to store his carpentry tools. the height of the chest, in feet, is represented by the function $h(x) = x + 6$. the area of the base of the chest, in square feet, is represented by the function $a(x) = \frac{20\sqrt{x}}{3}$. in both functions, $x$ represents the width of the chest in feet.
to find a function that represents the volume of the chest, ronald should functions $h(x)$ and $a(x)$.
the function that represents the volume of the chest is $v(x) = \frac{20x\sqrt{x}}{3} + \sqrt{x}$.

Explanation:

Step1: Recall Volume of Prism

The volume \( V \) of a rectangular prism (chest) is given by \( V=\text{Base Area} \times \text{Height} \). Here, base area is \( A(x) \) and height is \( H(x) \), so we need to multiply \( A(x) \) and \( H(x) \).

Step2: Multiply the Functions

Given \( H(x)=x + 6 \) and \( A(x)=\frac{20\sqrt{x}}{3} \), then \( V(x)=A(x)\times H(x)=\frac{20\sqrt{x}}{3}\times(x + 6)=\frac{20x\sqrt{x}}{3}+\frac{120\sqrt{x}}{3}=\frac{20x\sqrt{x}}{3}+40\sqrt{x} \). Wait, but let's check the first drop - down: the operation is multiplication (since volume is base area times height). For the second drop - down, when we multiply \( \frac{20\sqrt{x}}{3} \) by \( (x + 6) \), we get \( \frac{20x\sqrt{x}}{3}+\frac{120\sqrt{x}}{3}=\frac{20x\sqrt{x}}{3}+40\sqrt{x} \). But maybe there was a typo in the problem's second function, but following the volume formula (multiplication of base area and height), the first operation is multiply.

Answer:

First drop - down: multiply
Second drop - down: 40 (assuming the intended second term after multiplication is \( 40\sqrt{x} \), so the coefficient of \( \sqrt{x} \) is 40)