A biologist claims that the average height , h meters, of trees of a certain species after t months is given by h = 1 5 t 2 3 + 1 3 t 1 3 . Find , to 3 significant figures , the number of months that the trees have been growing when the model would predict an average height of 10 metres.

Given, h = 1 5 t 2 3 + 1 3 t 1 3 Let t 1 3 = x So, 10 = 1 5 x 2 + 1 3 x 10 = 3 x 2 + 5 x 15 150 = 3 x 2 + 5 x 3 x 2 + 5 x - 150 = 0 Quadratic Formula x = - b ± b 2 - 4 a c 2 a From equation a = 3 , b = 5 a n d c = - 150 x = - 5 ± 5 2 - 4 × 3 × - 150 2 × 3 x = - 5 ± 25 + 1800 6 x = - 5 ± 1825 6 So, x = - 5 ± 1825 6 = - 5 + 42 . 72001873 6 = 6 . 283 and x = - 5 ± 1825 6 = - 5 - 42 . 72001873 6 = - 7 . 953 Months cannot be negative So, p = 6 . 283 t = 6 . 283 3 = 248 Thus, the number of months that tree have been growing when the model would predict an average height of 10 meters is 248 months

A biologist claims that the average height , h meters, of trees of a certain species after t months is given by h = 1 5 t 2 3 + 1 3 t 1 3 . Find , to 3 significant figures , the number of months that the trees have been growing when the model would predict an average height of 10 metres.

Given, h = 1 5 t 2 3 + 1 3 t 1 3 Let t 1 3 = x So, 10 = 1 5 x 2 + 1 3 x 10 = 3 x 2 + 5 x 15 150 = 3 x 2 + 5 x 3 x 2 + 5 x - 150 = 0 Quadratic Formula x = - b ± b 2 - 4 a c 2 a From equation a = 3 , b = 5 a n d c = - 150 x = - 5 ± 5 2 - 4 × 3 × - 150 2 × 3 x = - 5 ± 25 + 1800 6 x = - 5 ± 1825 6 So, x = - 5 ± 1825 6 = - 5 + 42 . 72001873 6 = 6 . 283 and x = - 5 ± 1825 6 = - 5 - 42 . 72001873 6 = - 7 . 953 Months cannot be negative So, p = 6 . 283 t = 6 . 283 3 = 248 Thus, the number of months that tree have been growing when the model would predict an average height of 10 meters is 248 months