Optimization problems (parabolas and lines)?

Question

Could someone pls help me with this? Thx.

The revenue generated by a new product can be modeled as R(x)= -5x^2 + 21x. The cost function is C(x)= 4x+14. How many items are required to maximize the profit? What is the maximum profit?

Answer 1

I assume x is the number of units.

Let
R = revenue
C = cost
P = profit

R = -5x² + 21x
C = 4x + 14

P = R - C = (-5x² + 21x) - (4x + 14)
P = -5x² + 17x - 14

P has a maximum value at the vertex of the parabola.
Complete the square.

P = (-5x² + 17x) - 14
P + 14 = -5[x² - (17/5)x]
P + 14 - 5(17/10)² = -5[x² - (17/5)x + (17/10)²]
P - 45/100 = -5(x - 17/10)²

The number of items to produce for maximum profit is
17/10 = 1.7 units

The maximum profit is
45/100 = 0.45

If you require an integral number of units to be produced maximum profit is at x = 2. The maximum profit is then zero.

Answer 2

Profit = Revenue - Cost
Profit = -5x^2 +21x -4x -14 = -5x^2 +17x -14
Find the zero value for the first derivative of this function:
P' = -10x + 17 = 0 (P'' = -10, this is a maximum)
x = 1.7 (must be an integer number, so only 1 and 2 could be the answer)

plug 1, and 2 into the profit function and it turns out that the maximum profit is zero for 2 items produced.

Answer 3

You want to find the apex or maximum point of the parabola from your revenue function. your x should 2.1 so use 2 then put 2 back into your revenue function to get maxium revenue and then put 2 into your cost function to get max cost then subtract max cost from max revenue and you get max profit

Answer 4

a million) it is worry-unfastened; ==> First opt for the component to the priority; precise right here it is paper dimensions, so as that its section would be minimum; then next comes decrease than what constraints this could desire to be finished? ==> genuine, precise right here on the chosen paper, you're leaving some margins for the period of and arriving on the section for printing, it is given some fixed fee. 2) Now, we are sparkling what we want; the owing to proceed for progression mathematical equations, so as that it would genuine be solved for determining to purchase the precise answer: 3) precise right here we are given the printing section as 50 squarewherein is consistent; for this reason enable us initiate from this suggestion; ==> enable the dimensions of the printing section be x in (height clever) via using y in (width clever) ==> Printing section = xy = 50; == y = 50/x -------(a million) 4) there's a margin of four in each and each at precise and backside are provided; for this reason worry-unfastened height of the paper is "x + 8" in; greater a margin of two in is equipped on the two sides; ==> worry-unfastened width = y + 4 in 5) for this reason the component to the paper is = (x+8)(y+4) 6) Substituting for y from equation (a million), A (x+8)(50/x + 4) = 50 + 4 hundred/x + 4x + 32 7) for this reason the function to be minimized is: A = 80 2 + 4x + 4 hundred/x 8) Differentiating this, A' = 0 + 4 - 200/x^2 = 4 - 4 hundred/x^2 9) Equating A' = 0, x^2 = one hundred; ==> x = +/- 10 in 10) yet a length won't be able to be destructive, for this reason we evaluate only + 10; So x = 10 in and y = 5 in 11) besides the fact that we want to confirm,no count if it is minimum or optimum; for which we are able to prepare 2d derivative attempt; So decrease back differentiating, A'' = 800/x^3; at x = 10, A'' is > 0; for this reason it is minimum subsequently we end for the printing the section to be 50 squarein, decrease than the given constraints of margins, the outer length of the paper could desire to be 18 in via using 9 in; So outer section is = 162 squarein. desire you're defined; have a smart time.