the profits of Mr/ Lucky's company can be represented by the equation y=-3x^2+18x-4, where y is the amount of profit in hundreds of thousands of dollars and x is the number of years of operation. He realizes his company is on the downturn and wishes to sell it before he ends up in debt.
a) when will Unlicky's buisness show the maximum of profit?
b)what is the maximum profit?
c)at what time will it be too late to sell his business?(when will he start losing money?)
Given:
f(x)=-3x^2+18x-4
a)
Find f'(x) and equate to zero.
solve for x=xm.
Since the leading coefficient is -3, the extremum is a maximum.
b) maximum profit=f(xm)
c) He starts losing money when f(x)=0, AND f'(x)<0. There are two roots to f(x)=0, make sure the correct one is chosen.
To find the answers, we need to apply some concepts from calculus. The equation y = -3x^2 + 18x - 4 represents a quadratic function, where the variable x represents the number of years the business has been in operation, and y represents the profit in hundreds of thousands of dollars.
a) To determine when the business will show the maximum profit, we need to find the vertex of the quadratic function. The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a, b, and c are the coefficients of the quadratic equation in the form ax^2 + bx + c = 0.
In this case, the quadratic equation is y = -3x^2 + 18x - 4, so we can identify a = -3 and b = 18.
Using the formula, we can calculate: x = -18 / (2 * -3) = -18 / -6 = 3. Therefore, the maximum profit will be attained after 3 years of operation.
b) To find the maximum profit, we substitute the value of x back into the equation y = -3x^2 + 18x - 4.
y = -3(3)^2 + 18(3) - 4
= -3(9) + 54 - 4
= -27 + 54 - 4
= 23.
So, the maximum profit will be $230,000.
c) To determine when it will be too late to sell the business, we need to find the x-coordinate at which the profit (y) becomes negative, indicating a loss.
Setting y = 0, we have:
0 = -3x^2 + 18x - 4.
Let's solve this quadratic equation using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
In this case, the quadratic equation is -3x^2 + 18x - 4 = 0, so a = -3, b = 18, and c = -4.
Plugging these values into the quadratic formula, we get:
x = (-18 ± √(18^2 - 4*(-3)*(-4))) / (2*(-3))
= (-18 ± √(324 - 48)) / (-6)
= (-18 ± √(276)) / (-6).
The discriminant (b^2 - 4ac) is positive, which means there are two real roots for x. Let's find those roots:
x = (-18 + √276) / (-6) and x = (-18 - √276) / (-6).
Using a calculator, we find that the roots are approximately x ≈ 0.738 and x ≈ 5.929.
Therefore, it will be too late to sell the business, and Mr. Lucky will begin losing money after approximately 5.929 years of operation.