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Ronald is buying a rectangular chest to store his carpentry tools. The height of the chest, in feet, is represented by the function [tex]$H(x) = x + 6$[/tex]. The area of the base of the chest, in square feet, is represented by the function [tex]$A(x) = \frac{20 \sqrt{3}}{3}$[/tex]. In both functions, [tex][tex]$x$[/tex][/tex] represents the width of the chest in feet.

To find a function that represents the volume of the chest, Ronald should \_\_\_\_\_ functions [tex]$H(x)$[/tex] and [tex]$A(x)$[/tex].

The function that represents the volume of the chest is [tex][tex]$V(x) = \frac{20 x \sqrt{3}}{3} + \_\_\_\_\_$[/tex][\tex]



Answer :

GinnyAnswer
To find a function that represents the volume of the chest, Ronald should multiply functions [tex]\( H(x) \)[/tex] and [tex]\( A(x) \)[/tex].

Let's break down the given functions:
[tex]\[ H(x) = x + 6 \][/tex]
[tex]\[ A(x) = \frac{20 \sqrt{x}}{3} \][/tex]

Now, to find the volume function [tex]\( V(x) \)[/tex], we need to multiply the height function [tex]\( H(x) \)[/tex] and the area function [tex]\( A(x) \)[/tex]:
[tex]\[ V(x) = H(x) \times A(x) = (x + 6) \times \frac{20 \sqrt{x}}{3} \][/tex]

Expand this product:
[tex]\[ V(x) = \left( x + 6 \right) \times \frac{20 \sqrt{x}}{3} \][/tex]
[tex]\[ V(x) = \frac{20 \sqrt{x}}{3} \times x + \frac{20 \sqrt{x}}{3} \times 6 \][/tex]
[tex]\[ V(x) = \frac{20 x \sqrt{x}}{3} + \frac{20 \cdot 6 \sqrt{x}}{3} \][/tex]
[tex]\[ V(x) = \frac{20 x \sqrt{x}}{3} + \frac{120 \sqrt{x}}{3} \][/tex]
[tex]\[ V(x) = \frac{20 x \sqrt{x}}{3} + 40 \sqrt{x} \][/tex]

Therefore, the function that represents the volume of the chest is:
[tex]\[ V(x) = \frac{20 x \sqrt{x}}{3} + 40 \sqrt{x} \][/tex]

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