The profit made when t units are sold, t>, is given by P²-36t+320. Determine the number of units to be sold in order for P>0 (a profit is made).
To determine the number of units to be sold in order for a profit to be made (P>0), we need to find the value of t for which the expression P² - 36t + 320 is greater than zero.
To do this, we can set up an inequality:
P² - 36t + 320 > 0
Now, let's solve this inequality step by step:
Step 1: Factor the quadratic expression P² - 36t + 320 if possible.
The quadratic expression P² - 36t + 320 cannot be factored easily without using the quadratic formula. So, we'll proceed with the quadratic formula to find the values of P:
t = (-b ± √(b² - 4ac)) / (2a)
For our equation P² - 36t + 320 > 0:
a = 1, b = -36, and c = 320
Step 2: Calculate the discriminant (b² - 4ac) to determine the nature of the solutions.
Discriminant = (-36)² - 4(1)(320) = 1296 - 1280 = 16
Since the discriminant is positive (greater than zero), the quadratic equation has two distinct real solutions.
Step 3: Use the quadratic formula to find the solutions for the equation P² - 36t + 320 = 0.
t = (-(-36) ± √(16)) / (2*1)
t = (36 ± 4) / 2
t₁ = (36 + 4) / 2 = 40 / 2 = 20
t₂ = (36 - 4) / 2 = 32 / 2 = 16
Step 4: Analyze the inequality P² - 36t + 320 > 0 based on the solutions obtained:
We have two solutions, t₁ = 20 and t₂ = 16. Now we need to determine when the inequality is true.
To do this, draw a number line and plot the values of t on it:
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 16 20
Now, select a test point within each interval and substitute it into the inequality:
For t < 16, we can choose t = 10:
P² - 36(10) + 320 > 0
P² - 360 + 320 > 0
P² - 40 > 0
Since P is squared, it must always be greater than or equal to zero. So, P > 0.
For 16 < t < 20, we can choose t = 18:
P² - 36(18) + 320 > 0
P² - 648 + 320 > 0
P² - 328 > 0
For t > 20, we can choose t = 25:
P² - 36(25) + 320 > 0
P² - 900 + 320 > 0
P² - 580 > 0
By analyzing the inequality at each interval, we see that t > 20 satisfies the condition P > 0. Therefore, the number of units to be sold in order for a profit to be made is t > 20.