Total cost to manufacture q microwaves a day is C(q)=3q^2+q+48.What level of product gives the least average cost and Can a microwaves be produces for less than $30?

To find the level of production that gives the least average cost, we need to find the minimum of the average cost function. The average cost is calculated by dividing the total cost by the quantity produced.

The average cost function can be determined by taking the derivative of the total cost function and then dividing it by the quantity produced. Let's find the derivative of the cost function:

C(q) = 3q^2 + q + 48

Now, we’ll take the derivative with respect to q:

C'(q) = (d/dq) (3q^2 + q + 48)
= 6q + 1

Next, we'll divide the derivative by the quantity produced (q) to obtain the average cost function:

AC(q) = C'(q)/q
= (6q + 1)/q
= 6 + 1/q

To find the level of production that gives the least average cost, we need to find the minimum of the average cost function. Since the value 1/q can never be negative or zero, we can conclude that the minimum average cost occurs when q approaches infinity. In practical terms, this means that there is no specific quantity of microwaves that gives the minimum average cost. The average cost decreases as the quantity produced increases.

Now, let's determine whether a microwave can be produced for less than $30. To do this, we need to find the value of q for which the total cost, C(q), is less than $30:

C(q) = 3q^2 + q + 48

Setting C(q) less than $30:
3q^2 + q + 48 < 30

Simplifying the inequality:
3q^2 + q + 48 - 30 < 0
3q^2 + q + 18 < 0

Unfortunately, we cannot directly solve this inequality to find the exact range of values for q. However, we can determine whether there are any values of q for which the inequality is true. By analyzing the equation, we can see that since the coefficient of the quadratic term (3) is positive, the parabola opens upward, which means that there may not be any values of q that satisfy the inequality.

Therefore, based on the provided information, it is not possible to produce a microwave for less than $30.