To determine the expenditure on advertising that yields the maximum profit for the company, we need to analyze the given profit function:
[tex]\[ P(x) = 340 + 40x - 0.5x^2 \][/tex]
This is a quadratic function of the form:
[tex]\[ P(x) = ax^2 + bx + c \][/tex]
where [tex]\( a = -0.5 \)[/tex], [tex]\( b = 40 \)[/tex], and [tex]\( c = 340 \)[/tex].
To find the expenditure [tex]\( x \)[/tex] that yields the maximum profit, we need to locate the vertex of the parabola represented by this quadratic function. In a quadratic equation [tex]\( ax^2 + bx + c \)[/tex], the vertex formula for the x-coordinate is given by:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Here are the steps to find [tex]\( x \)[/tex]:
1. Identify the coefficients:
- [tex]\( a = -0.5 \)[/tex]
- [tex]\( b = 40 \)[/tex]
2. Substitute these values into the vertex formula:
[tex]\[ x = -\frac{40}{2 \times -0.5} \][/tex]
3. Simplify the expression:
[tex]\[ x = -\frac{40}{-1} \][/tex]
[tex]\[ x = 40 \][/tex]
Thus, the expenditure [tex]\( x \)[/tex] in hundreds of dollars that yields the maximum profit is [tex]\( 40 \)[/tex].
Now, to find the maximum profit, substitute [tex]\( x = 40 \)[/tex] back into the profit function:
[tex]\[ P(40) = 340 + 40(40) - 0.5(40)^2 \][/tex]
Calculate step by step:
[tex]\[ P(40) = 340 + 1600 - 0.5 \times 1600 \][/tex]
[tex]\[ P(40) = 340 + 1600 - 800 \][/tex]
[tex]\[ P(40) = 340 + 800 \][/tex]
[tex]\[ P(40) = 1140 \][/tex]
Therefore, the expenditure for advertising that yields the maximum profit is [tex]\( \$4000 \)[/tex] (since [tex]\( x = 40 \)[/tex] corresponds to [tex]\( 40 \times 100 \)[/tex] dollars) and the maximum profit is [tex]\( \$11,400 \)[/tex].
