Question

the cost to produce metal statues for local parks is given by
\\( c = 3x^2 - 270x + 7000 \\),
where \\( x \\) represents the number of statues produced and \\( c \\) is the cost of producing them. complete parts a through f below.

(type an ordered pair. do not use commas in the individual coordinates.)
c. how do your answers in parts a and b compare?
\\( \boldsymbol{\checkmark} \\) a. the answers are the same.
\\( \circ \\) b. the answers are different because part a was an approximation.
d. is the vertex a minimum or a maximum point?
the vertex is a minimum point.
e. what is the practical meaning of the vertex in this situation?
\\( \circ \\) a. the minimum number of statues that can be produced is 45.
\\( \circ \\) b. the minimum number of statues that can be produced is 925.
\\( \circ \\) c. the cost of production is minimized when 45 statues are produced.
\\( \circ \\) d. the cost of production is minimized when 925 statues are produced.

Explanation:

1. The cost function is a quadratic function \( C = 3x^{2}-270x + 7000 \) with \( a = 3>0 \), so the parabola opens upward, and the vertex is the minimum point.
2. For a quadratic function \( y=ax^{2}+bx + c \), the x - coordinate of the vertex is \( x=-\frac{b}{2a} \). Here, \( a = 3 \), \( b=- 270 \), so \( x=-\frac{-270}{2\times3}=\frac{270}{6} = 45 \).
3. The vertex represents the minimum of the cost function (since the parabola opens upward). So the x - coordinate of the vertex (45) is the number of statues produced when the cost is minimized, and the y - coordinate is the minimum cost. Option A and B are wrong because the vertex's x - coordinate is about the number of statues to minimize cost, not the minimum number of statues that can be produced. Option D is wrong because the x - coordinate of the vertex is 45, not 925.

Answer:

C. The cost of production is minimized when 45 statues are produced.