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A manufacturer of chemical glassware needs to purchase a certain amount of raw material. The profit, p, in dollars expected from purchasing t tons of raw material, where t is positive, is: p=50,000(2t-1)(t-5) What is the smallest amount of raw material in tons that the manufacturer can purchase to break even with a profit of 0 dollars?

A manufacturer of chemical glassware needs to purchase a certain amount of raw material. The profit, p p , in dollars expected from purchasing t t tons of raw material, where t t is positive, is:\newlinep=50,000(2t−1)(t−5) p=50,000(2 t-1)(t-5) \newlineWhat is the smallest amount of raw material in tons that the manufacturer can purchase to break even with a profit of 00 dollars?

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Q. A manufacturer of chemical glassware needs to purchase a certain amount of raw material. The profit, p p , in dollars expected from purchasing t t tons of raw material, where t t is positive, is:\newlinep=50,000(2t−1)(t−5) p=50,000(2 t-1)(t-5) \newlineWhat is the smallest amount of raw material in tons that the manufacturer can purchase to break even with a profit of 00 dollars?
  1. Set Profit Equation: To find the smallest amount of raw material in tons that the manufacturer can purchase to break even, we need to set the profit equation pp equal to 00 and solve for tt.
    p=50,000(2t−1)(t−5)p = 50,000(2t - 1)(t - 5)
    0=50,000(2t−1)(t−5)0 = 50,000(2t - 1)(t - 5)
  2. Divide by 5050,000000: Since the profit pp is multiplied by 50,00050,000, we can divide both sides of the equation by 50,00050,000 to simplify the equation without affecting the value of tt.0=(2t−1)(t−5)0 = (2t - 1)(t - 5)
  3. Solve Quadratic Equation: Now we have a quadratic equation in factored form. To find the values of tt that make the profit zero, we set each factor equal to zero and solve for tt.\newlineFirst, set the first factor equal to zero:\newline2t−1=02t - 1 = 0
  4. First Factor Zero: Solve for tt by adding 11 to both sides and then dividing by 22:2t=12t = 1t=12t = \frac{1}{2}
  5. Second Factor Zero: Now, set the second factor equal to zero:\newlinet−5=0t - 5 = 0
  6. Valid Solutions: Solve for tt by adding 55 to both sides:\newlinet=5t = 5
  7. Compare Values: We have found two values of tt, 12\frac{1}{2} and 55. Since tt represents the tons of raw material and must be positive, both are valid solutions. However, we are looking for the smallest amount of raw material to break even.
  8. Smallest Amount: Comparing the two values, 12\frac{1}{2} ton and 55 tons, we see that 12\frac{1}{2} ton is the smaller amount. Therefore, the smallest amount of raw material the manufacturer can purchase to break even is 12\frac{1}{2} ton.

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