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Pierce removes the plug from a trough to drain the water. The volume, in gallons, in the trough after it has been unplugged can be modeled by [tex]f(x)=10 x^2 - 17 x + 3[/tex], where [tex]x[/tex] is the time in minutes. Which of the following equations will reveal the time in minutes when the trough is empty?

A. [tex]f(x)=10(x-3)^2-1[/tex]

B. [tex]f(x)=10(x-1)^2-3[/tex]

C. [tex]0=(5 x-1)(2 x-3)[/tex]

D. [tex]f(0)=10(0)^2-17(0)+3[/tex]



Answer :

GinnyAnswer
To determine when the trough is empty, we need to find the time [tex]\( x \)[/tex] when the volume [tex]\( f(x) \)[/tex] is zero. Given the function [tex]\( f(x) = 10x^2 - 17x + 3 \)[/tex], we set [tex]\( f(x) = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:

[tex]\[ 10x^2 - 17x + 3 = 0 \][/tex]

This is a quadratic equation, which can be factored as:

[tex]\[ 0 = (5x - 1)(2x - 3) \][/tex]

The factored form of the equation reveals the [tex]\( x \)[/tex]-values (or times) when the volume of the water in the trough is zero (i.e., when it is empty). Thus, the equation that reveals the time in minutes when the trough is empty is:

[tex]\[ 0 = (5x - 1)(2x - 3) \][/tex]

Hence, the answer is:

[tex]\[ 0 = (5x - 1)(2x - 3) \][/tex]

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