Solve the problem.
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).
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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 that makes the expression P² - 36t + 320 positive.
To start, we have the expression P² - 36t + 320. Let's set it greater than zero:
P² - 36t + 320 > 0
Now, we can use different methods to solve this quadratic inequality. One approach is to factorize the expression or complete the square. However, in this case, we can solve it by finding the roots of the quadratic equation.
To find the roots, we set the expression equal to zero:
P² - 36t + 320 = 0
Now, we can solve for t by factoring or using the quadratic formula:
t = (-(-36) ± √((-36)² - 4(1)(320))) / (2(1))
Simplifying,
t = (36 ± √(1296 - 1280)) / 2
t = (36 ± √16) / 2
t = (36 ± 4) / 2
This gives us two potential solutions for t:
t₁ = (36 + 4) / 2 = 20
t₂ = (36 - 4) / 2 = 16
Now, we need to check which value of t satisfies the condition P > 0. We substitute each value of t back into the expression P² - 36t + 320:
For t = 20:
P² - 36(20) + 320 = 400 - 720 + 320 = 0
For t = 16:
P² - 36(16) + 320 = 256 - 576 + 320 = 0
Neither value of t yields a positive profit. However, since our original quadratic equation P² - 36t + 320 has a positive coefficient for the squared term (P²), we can conclude that there is no solution for P > 0. In other words, it is not possible to make a profit with this equation.