Question

livestock watering troughs are often custom - made from long metal sheets by bending two sides upward and adding left and right ends. the trough shown is made from a foot - wide strip of metal with the cross - sectional area ( a(x)=x(12 - 2x) ) square inches. for what value of ( x ) will the trough have maximum cross - sectional area? options: ( \frac{1}{12} ), 3, 12, 1

Explanation:

Step1: Recognize the function type

The cross - sectional area function is \(A(x)=x(12 - 2x)=- 2x^{2}+12x\). This is a quadratic function in the form of \(y = ax^{2}+bx + c\), where \(a=-2\), \(b = 12\), and \(c = 0\).

Step2: Find the vertex of the quadratic function

For a quadratic function \(y=ax^{2}+bx + c\), the x - coordinate of the vertex (which gives the maximum or minimum value of the function, and since \(a=-2<0\), the vertex gives the maximum value) is given by the formula \(x=-\frac{b}{2a}\).

Substitute \(a=-2\) and \(b = 12\) into the formula:
\(x=-\frac{12}{2\times(-2)}=-\frac{12}{-4} = 3\)

So the value of \(x\) that maximizes the cross - sectional area is 3.

Answer:

3