Question

Find the number of units x that must be produced and sold in order to yield the maximum profit. Assume the revenue, R(x), and cost, C(x), are in dollars. Round your answer to the nearest unit. MathJ

Answer 1

we have a quadratic function in the form P(x)=ax2+bx+c, we can change it to the vertex form P(x)=a(x−h)2+k, where h=−b2a,k=c−b24a to find its maximum or minimum value. If a is negative, P(x) achieves its maximum value at x = h. Another easier way to find the answer is to use the function's first derivative. When the parabola associated with the quadratic function opens downward, its maximum point is the solution to the equation dPdx=0.

Answer 2

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Given that,

Cost, C(x) = 20 x + 4000

Revenue, R(x) = 60x + 2000

We know that, profit = Revenue – Cost

Now, subsitute the given data in the above formula,

Profit = R(x) – C(x)

Profit = (60x + 2000)-(20 x + 4000)

Now, simplify it:

Profit = 60x + 2000 -20x -4000

Profit = 40x – 2000

To earn some profit, 40x – 2000 > 0

⇒40x > 2000

⇒ x>2000/40

⇒ x > 50

Thus, the manufacturer should sell more than 50 items to realise some profit

Hope it's Helpful....:)