A company that manufactures a specialty line of high performance racing boats has determined that its profit margin can be represented by the equation, PM(x)=-x^3+7x^2-10x, where x is in thousands of dollars.
a) graph without technology.
b) determine the production numbers where the profit margin is negative.
c) determine the production numbers that generate a positive profit margin.
d)at what point is the profit margin greatest. ( do it in grade 12 ways)
PM(x)=-x^3+7x^2-10x
= -x(x^2 - 7x + 10)
= -x(x-5)(x-2)
so the x-intercepts of this function are
0, 5, and 2 and it is a cubic with a negative cubic term
Make a sketch and you should be able to answer all your questions quite readily.
thank u..
a) To graph the profit margin equation without using technology, you can follow these steps:
1. Plot a set of points on a coordinate plane, with the x-axis representing the production numbers and the y-axis representing the profit margin.
2. Choose a range for the x-values, for example, from 0 to 10.
3. Calculate the profit margin (PM(x)) for several values of x within the chosen range. For each x-value, substitute it into the equation PM(x) = -x^3 + 7x^2 - 10x and calculate the corresponding y-value.
4. Plot the points obtained from the calculations on the graph. Connect the plotted points with a smooth curve.
b) To determine the production numbers where the profit margin is negative, we need to find the values of x for which the profit margin equation yields negative values.
1. Set the profit margin equation equal to zero: -x^3 + 7x^2 - 10x = 0.
2. Factor out common terms: x(-x^2 + 7x - 10) = 0.
3. Set each factor equal to zero and solve for x:
- x = 0 (which means the x-value is zero, but it probably doesn't make sense in this context)
- -x^2 + 7x - 10 = 0 (quadratic equation).
To solve the quadratic equation, you can factorize or use the quadratic formula. Factoring is likely easier in this case if you can find factors that multiply to -10 and add up to 7.
-x^2 -3x + 10x -10 = 0
x(-x-3) + 2(5-x) = 0
(x-2)(-x-5) = 0
From this factorization, we get two potential solutions:
- x-2 = 0 => x = 2
- -x-5 = 0 => -x = 5 => x = -5
Hence, the production numbers where the profit margin is negative are x = 2 (in thousands of dollars) and x = -5 (in thousands of dollars).
c) To determine the production numbers that generate a positive profit margin, we need to find the values of x for which the profit margin equation yields positive values.
Since the profit margin equation is a cubic equation, it may have multiple real roots, making it challenging to find the positive roots without the aid of technology. However, we can make an estimation by examining the graph from part (a).
From the graph, identify the values of x where the curve lies above the x-axis. These are the production numbers that generate a positive profit margin.
d) To find the point at which the profit margin is greatest, we need to find the maximum value of the profit margin equation.
For this, we can use calculus. Take the derivative of the profit margin function PM(x) with respect to x:
PM'(x) = -3x^2 + 14x - 10.
Set the derivative equal to zero to find the critical point(s):
-3x^2 + 14x - 10 = 0.
Solve this quadratic equation to find the x-value(s) that correspond to the critical point(s). Substituting these x-values into the profit margin equation will give us the y-values or profit margins at those points. The largest y-value corresponds to the point where the profit margin is the greatest.