The total world population is forecast to be P(t) = .00079t^3 -.0701t^2 +.96t +

Question

The total world population is forecast to be P(t) = .00079t^3 -.0701t^2 +.96t + 6.04 (t may be greater than or equal to 0 or less than or equal to 10) in year t, where t is measured in decades, with t = 0 corresponding to 2000 and P(t) is measured in billions. (a) World population is forecast to peak in what year? Hint: Use the quadratic formula. (Remember that t is in decades and not in years. Round your answer down to the nearest year.) (b) At what number will the population peak? (Be sure to use the value of t found in part (a). Round your answer to two decimal places.) billion

Explanation / Answer

P(t) = .00079t^3 -.0701t^2 +.96t + 6.04 P'(t) = 3(.00079)t^2 - 2(0.0701)t +.96 = 0 .00237t^2-0.1402t+.96 = 0 This equation is of form ax^2+bx+c=0 a = 0.00237 b = -0.1402 c = 0.96 x=[-b+/-sqrt(b^2-4ac)]/2a] x=[0.1402 +/-sqrt(-0.1402^2-4(0.00237)(0.96)]/(2)(0.00237) discriminant is b^2-4ac =0.010555 x=[0.1402 +v(0.010555)] / (2)(0.00237) x=[0.1402 -v(0.010555)] / (2)(0.00237) x=[0.1402+0.1027375297] / 0.00474 x=[0.1402-0.1027375297] / 0.00474 The roots are 51.5142 and 6.2368 t= 51.25 or 7.90 P''(t) =.00474t-0.1402 At t=7.9 , P''(t) =-.1027 < 0 ,so P is maximum 0 < 7.9 < 10 In 2007 , World population is forecast to peak b) P(7.9) = 9.63 billion

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