Mathematics 10 — Exercise 2.7 · Q1

Full question

A town’s population is P(t) = −2t² + 40t + 800, where t is years since 2020. In which years is population at least 1000?

Step-by-step solution

1. Set P(t) ≥ 1000 and subtract 1000.

2. Substitute the function expressions, keeping the brackets.

   $$-(2\times t^2-40\times t+200)-0$$

3. Remove the brackets. A preceding minus sign reverses every sign inside that bracket.

   $$-2\times t^2+40\times t-200$$

4. Write EVERY distributed product before collecting terms.

   $$-200-2\times t^2+40\times t-0$$

5. Calculate this individual product, including its sign and powers.

   $$-200=-200$$

6. Calculate this individual product, including its sign and powers.

   $$-2\times t^2=-2\times t^2$$

7. Calculate this individual product, including its sign and powers.

   $$40\times t=40\times t$$

8. Calculate this individual product, including its sign and powers.

   $$\left(-1\right)\times 0=0$$

9. These terms have the same variable part. Add ONLY their coefficients.

   $$-200+0=-200$$

10. These terms have the same variable part. Add ONLY their coefficients.

    $$-2=-2$$

11. These terms have the same variable part. Add ONLY their coefficients.

    $$40=40$$

12. Put the simplified groups together.

    $$-2\times t^2+40\times t-200$$

13. Collect terms with the same power and simplify their coefficients.

    $$-2\times t^2+40\times t-200$$

14. Expand the brackets and subtract the right-hand terms from both sides. Collect powers of the variable.

    $$2\times t^2-40\times t+200=0$$

15. Split the middle coefficient into two numbers whose sum is b and whose product is a·c.

    $$(-20-20=-40,\left(-20\right)\times \left(-20\right)=400)$$

16. Rewrite the middle term using those two parts.

    $$2\times t^2-20\times t-20\times t+200=0$$

17. Group the first two terms and the last two. Take out the common factor from each group.

    $$\left(2\times t\right)\times \left(t-10\right)-20\times \left(t-10\right)=0$$

18. The common bracket can now be factored out.

    $$2\times \left(t-10\right)\left(t-10\right)=0$$

19. A product is zero only if at least one factor is zero. Set each linear factor equal to zero.

    $$(t-10=0,t-10=0)$$

20. Take this factor equal to zero.

    $$t-10=0$$

21. Subtract the constant from both sides.

    $$t-10+10=0+10$$

22. Simplify both sides.

    $$t=10$$

23. Divide both sides by the coefficient of the variable.

    $$t=10\times \frac{1}{1}$$

24. Reduce the fraction to obtain this root.

    $$t=10$$

25. Take this factor equal to zero.

    $$t-10=0$$

26. Subtract the constant from both sides.

    $$t-10+10=0+10$$

27. Simplify both sides.

    $$t=10$$

28. Divide both sides by the coefficient of the variable.

    $$t=10\times \frac{1}{1}$$

29. Reduce the fraction to obtain this root.

    $$t=10$$

30. Substitute this root into the original equation before simplifying.

    $$0=0$$

31. Calculate both sides; their equal values confirm the root.

    $$0=0$$

32. First solve the associated equation to locate the critical values.

    $$-2\times {\left(t-10\right)}^2=0$$

33. The critical values split the number line into intervals. Test one value in each interval.

    $$\left\{10\right\}$$

34. Calculate this operation before combining the remaining terms.

    $$2\times 81=162$$

35. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 162=-162$$

36. Substitute the test value and evaluate the expression; its sign is constant within this interval.

    $$-200-2\times 9^2+40\times 9$$

37. Calculate this operation before combining the remaining terms.

    $$9^2=81$$

38. Calculate this operation before combining the remaining terms.

    $$\left(-2\right)\times 81=-162$$

39. Calculate this operation before combining the remaining terms.

    $$40\times 9=360$$

40. Calculate this operation before combining the remaining terms.

    $$-200-162+360=-2$$

41. Compare the calculated value with zero using the required inequality.

    $$\mathrm{test\; value}=-2$$

42. Calculate this operation before combining the remaining terms.

    $$2\times 121=242$$

43. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 242=-242$$

44. Substitute the test value and evaluate the expression; its sign is constant within this interval.

    $$-200-2\times {11}^2+40\times 11$$

45. Calculate this operation before combining the remaining terms.

    $${11}^2=121$$

46. Calculate this operation before combining the remaining terms.

    $$\left(-2\right)\times 121=-242$$

47. Calculate this operation before combining the remaining terms.

    $$40\times 11=440$$

48. Calculate this operation before combining the remaining terms.

    $$-200-242+440=-2$$

49. Compare the calculated value with zero using the required inequality.

    $$\mathrm{test\; value}=-2$$

50. Keep the intervals with the required sign. Include roots only for ≤ or ≥; strict < or > excludes roots.

    $$\left\{10\right\}$$

51. Only t = 10 works. Add 10 years to 2020.

Answer

Only the year 2030.