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.

the vertex is a minimum point.

e. what is the practical meaning of the vertex in this situation?

a. the minimum number of statues that can be produced is 45.

b. the minimum number of statues that can be produced is 925.

c. the cost of production is minimized when 45 statues are produced.

d. the cost of production is minimized when 925 statues are produced.

f. what is the vertical intercept? what is the practical meaning of this intercept? select the correct choice below and fill in the answer box to complete your choice.
(type an integer or a decimal)

a. the cost to produce a single statue is $\square$.

b. the cost is $\square$, even if no statues are produced.

c. if $\square$ statues are produced, the cost is $\\$0$.

Explanation:

Part e

The cost function \( C = 3x^{2}-270x + 7000\) is a quadratic function with \(a = 3>0\), so its graph is a parabola opening upwards. The vertex of a parabola \(y=ax^{2}+bx + c\) has its \(x\) - coordinate given by \(x=-\frac{b}{2a}\). For \(C = 3x^{2}-270x + 7000\), \(a = 3\) and \(b=- 270\). Then \(x=-\frac{-270}{2\times3}=\frac{270}{6} = 45\). Since the parabola opens upwards, the vertex represents the minimum point of the function. So the \(x\) - value of the vertex (45) represents the number of statues produced that minimizes the cost \(C\), and the \(y\) - value (or \(C\) - value) at \(x = 45\) represents the minimum cost. Option A is wrong because 45 is not a minimum number of statues that can be produced (you can produce 0 or 1 or other numbers less than 45, but the cost will be higher). Option B is wrong as 925 is not related to the number of statues for minimum production. Option D is wrong as the \(x\) - coordinate of the vertex is 45, not 925.

Part f

To find the vertical intercept, we set \(x = 0\) in the cost function \(C=3x^{2}-270x + 7000\).

Step 1: Find the vertical intercept

The vertical intercept of a function \(y = f(x)\) is found by setting \(x = 0\) and solving for \(y\) (in our case, \(C\)).
For the cost function \(C=3x^{2}-270x + 7000\), substitute \(x = 0\):
\(C=3(0)^{2}-270(0)+7000\)
\(C = 7000\)

Step 2: Interpret the vertical intercept

• Option A: The cost to produce a single statue is not given by the vertical intercept. To find the cost per statue, we would look at the average cost or the derivative (marginal cost), not the value when \(x = 0\).
• Option B: When \(x = 0\), it means no statues are produced. And we found that when \(x = 0\), \(C = 7000\). So this option correctly interprets the vertical intercept as the cost when no statues are produced.
• Option C: If we set \(C = 0\) and solve \(3x^{2}-270x + 7000=0\), the discriminant \(\Delta=b^{2}-4ac=(-270)^{2}-4\times3\times7000=72900 - 84000=- 11100<0\), so there is no real - valued solution for \(x\) when \(C = 0\). So we can never have a cost of \(\$0\) by producing a non - negative number of statues.

Answer:

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