Answer:
15. 80 benches
16. 45 cost units
Explanation:
The two problems together are asking you to find the coordinates of the vertex of the cost function. To do this you can use the procedure for writing a quadratic in vertex form.
C(x) = 0.1x^2 -16x +685
= 0.1(x^2 -160x) +685 . . . . . . factor out the leading coefficient
= 0.1(x^2 -160x +6400) +685 -(0.1)(6400) . . . add and subtract 0.1(6400)
C(x) = 0.1(x -80)² +45 . . . cost function in vertex form
__
15.
The number of benches that minimizes the cost function is the x-coordinate of the vertex. That is the value that makes the square equal to zero.
x = 80 . . . benches
__
16.
The minimum value of the cost function is the value obtained when the squared term is zero:
C(80) = 0.1(80 -80)² +45
C(80) = 45
The minimum cost per bench is 45. (The problem statement gives no units for cost. It could be pesos, or millions of rubles. We can't tell.)
_____
Additional comment
The square of a binomial is ...
(x +a)² = x² +2ax +a²
You will notice that the constant (a²) is the square of half the x-coefficient. To "complete the square", we put the equation in a form where we have (x² +2ax) in parentheses. Then we divide the x-coefficient by 2, and add its square inside parentheses: (x² +2ax +a²) and rewrite to the form of a square: (x +a)².
In order to keep the expression the same, we must subtract the same quantity elsewhere in the equation. Outside parentheses, it must be multiplied by the coefficient in front of the parentheses. In the above, we added (0.1)(6400) and subtracted (0.1)(6400) so that we could rearrange the cost function to vertex form: a(x -h)² +k has vertex (h, k).
__
We find a graphing calculator to be an extremely handy tool for dealing with polynomial functions. The attached shows the graph of the cost function. The calculator marks the vertex for us. Since it is a plot of cost versus number of benches, the units of the vertex are (benches, cost units).