the profit made when t units are sold, t > 0, is given by P=t^2-36t+320. Determine the number of units in order for P > 0 (a profit is made).
P=t^2-36t+320>0
for zeroes, t=(36+-4)/2=20;16
well, if those are the zeroes, then between them are the P<0
example, t=18
P=18^2-36*18+320=320-18^2=-4
example, t=22, P=22^2-36*22+320
P=12
(t-20)(t+16) = 0
t = 20 is positive root
parabola opens up, anything above right hand root is positive p
so
t>20
(t-20)(t-16) = 0
t = 20 is right hand root
Ah, the thrilling world of profit calculations! Let's find out how many units we need to sell to make a profit.
We want to find the number of units for which P, the profit, is greater than 0. In mathematical terms, we're looking for the values of t where P > 0.
The profit equation is given by P = t^2 - 36t + 320. So we want to solve the inequality:
t^2 - 36t + 320 > 0
Now, let's put on our mathematical clown shoes, juggle some numbers, and solve this inequality!
First, we can factorize the quadratic expression: (t - 16)(t - 20) > 0.
Now we have two possibilities: either both factors are positive or both are negative. If t - 16 > 0 and t - 20 > 0, then t > 16 and t > 20.
However, we can't have t being greater than 20 and at the same time greater than 16 since those conditions are contradictory. So we discard that possibility.
Now, if t - 16 < 0 and t - 20 < 0, then t < 16 and t < 20.
So, our final answer is t < 16.
In order for a profit to be made (P > 0), the number of units sold should be less than 16.
So get out there and sell those products, but keep it below 16 units to make a profit! (And remember to wear a silly hat while doing it.)
To determine the number of units in order for a profit to be made, we need to find the values of t for which the profit function P is greater than zero (P > 0).
The profit function is given by P = t^2 - 36t + 320.
To find the values of t that satisfy P > 0, we need to find the roots of the quadratic equation P = 0. When a quadratic equation is in the form at^2 + bt + c = 0, the roots can be found using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
In this case, the coefficients are:
a = 1
b = -36
c = 320
Substituting these values into the quadratic formula, we get:
t = (36 ± √((-36)^2 - 4(1)(320))) / (2(1))
Simplifying further:
t = (36 ± √(1296 - 1280)) / 2
t = (36 ± √16) / 2
t = (36 ± 4) / 2
t = 20 or t = 16
Therefore, the number of units required to make a profit (P > 0) is t = 20 or t = 16.
Note: We take the positive roots because, in this case, the number of units cannot be negative (t > 0).
So, t = 20 or t = 16 are the solutions.