Q. 150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on second day, 4 more workers dropped out on the third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was completed.
Sol: Suppose the work is completed in n days. Since 4 workers went away on every day except the first day. ∴ Total number of worker who worked all the n days is the sum of n terms of A.P. with first term 150 and common difference – 4. Total number of worker who worked all the n days = n/2[2 x 150 + (n-1) x -4 ] = n (152 – 2n) If the workers would not have went away, then the work would have finished in (n – 8) days with 150 workers working on every day. ∴ Total number of workers who would have worked all n days = 150 (n – 8) ∴ n (152 – 2n) = 150 (n – 8) ⇒ 152n – 2n2 = 150n – 1200 ⇒ 2n2 – 2n – 1200 = 0 ⇒ n2 – n – 600 = 0 ⇒ n2 – 25n + 24n – 600 = 0 ⇒ n(n – 25) + 24 (n + 25) = 0 ⇒ (n – 25) (n + 24) = 0 ⇒ n – 25 = 0 or n + 24 = 0 ⇒ n = 25 or n = – 24 ⇒ n = 25 ( Number of days cannot be negative) Thus, the work is completed in 25 days. Q. Divide 32 into 4 parts which are in A.P. such that the product the product of means is 7:15. Let four numbers in A.P. be a – 3d, a – d, a + d and a + 3d. (a – 2d) + (a – d) + (a + d) + (a + 2d) = 32 ⇒ 4a = 32 ⇒ a = 8 Product of first and fourth term(extremes) = (a – 3d) (a + 3d) = (8 – 3d) (8 +32d) = 64 – 9d2 Product of second and third term (means) = (a – d) (a + d) = (8 – d) (8 + d) = 64 – d2 Product of extremes: product of means = 7:15 (Given) ( 64 – 9d2) / (64 – d2) = 7 : 15
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Ans: Melanin
The actual skin color of different humans is affected by many substances, although the single most important substance is the pigment melanin. Melanin is produced within the skin in cells called melanocytes and it is the main determinant of the skin color of darker-skinned humans. |