An barrels manufacturer can produce up to 300 barrels per day. The profit made fromthe sale of these barrels can be modelled by the function P (x) = − 10x2 + 3500x −66000 where P(x) is the profit in rupees and x is the number of barrels made and sold.(i) When no barrels are produce what is a profit /loss?(ii) What is the break even point ? (Zero profit point is called break even)(iii) What is the profit/loss if 175 barrels are produced(iv) What is the profit/loss if 400 barrels are produced

(i) When no barrels are produced, the profit/loss can be calculated by substituting x = 0 into the profit function P(x).

P(0) = -10(0)^2 + 3500(0) - 66000 P(0) = 0 - 0 - 66000 P(0) = -66000

Therefore, when no barrels are produced, there is a loss of 66,000 rupees.

(ii) The break-even point is the point at which the profit is zero. To find the break-even point, we need to solve the equation P(x) = 0.

-10x^2 + 3500x - 66000 = 0

To solve this quadratic equation, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = -10, b = 3500, and c = -66000. Plugging these values into the quadratic formula, we get:

x = (-3500 ± √(3500^2 - 4(-10)(-66000))) / (2(-10))

Simplifying further, we get:

x = (-3500 ± √(12250000 - 2640000)) / (-20) x = (-3500 ± √(9610000)) / (-20) x = (-3500 ± 3100) / (-20)

This gives us two possible solutions:

x1 = (-3500 + 3100) / (-20) = -400 / (-20) = 20 x2 = (-3500 - 3100) / (-20) = -6600 / (-20) = 330

Therefore, the break-even point is when 20 barrels are produced and sold, or when 330 barrels are produced and sold.

(iii) To find the profit/loss if 175 barrels are produced, we can substitute x = 175 into the profit function P(x).

P(175) = -10(175)^2 + 3500(175) - 66000 P(175) = -10(30625) + 612500 - 66000 P(175) = -306250 + 612500 - 66000 P(175) = 245250

Therefore, if 175 barrels are produced and sold, the profit is 245,250 rupees.

(iv) To find the profit/loss if 400 barrels are produced, we can substitute x = 400 into the profit function P(x).

P(400) = -10(400)^2 + 3500(400) - 66000 P(400) = -10(160000) + 1400000 - 66000 P(400) = -1600000 + 1400000 - 66000 P(400) = -260000

Therefore, if 400 barrels are produced and sold, the loss is 260,000 rupees.