Section 44

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Harvard University *

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245

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Mathematics

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Jan 9, 2024

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docx

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400. Solve the system of equations: {6 −3 =184 +5 =32 { 6 x −3 y =184 x +5 y =32 . - Answer : =5, =4 x =5, y =4 . 401. Calculate the value of x in the equation log2( +35)−log2( −30)=5log 2 ( x +35)−log 2 ( x −30)=5 . - Answer : =31 x =31 . 402. Find the area of a regular 420-gon inscribed in a circle with a radius of 1515 units. - Answer : Area = 420×12×15×15×sin(360 420)420× 21 ×15×15×sin ( 420360 ) . 403. Determine the value of x in the equation 20 −2 =104857520 x −2 x =1048575 . - Answer : =10 x =10 . 404. Find the sum of the first 330330 terms of an arithmetic sequence where the first term is 280280 and the common difference is 160160 . - Answer : Sum = 2(2 +( −1) ) 2 n (2 a +( n −1) d ) with =330 n =330 , =280 a =280 , =160 d =160 . 405. Solve the equation 56 −2+59 +3=62 2− −6 x −256 + x +359 = x 2 x −662 . - Answer : =62 x =62 . 406. Calculate the area enclosed by the curve =sin(42 ) y =sin(42 x ) , the x-axis, and the lines =0 x =0 and = 84 x = 84 π . - Answer : Area = ∫0 84sin(42 ) ��∫ 0 84 π sin(42 x ) dx . 407. Solve the inequality 2−86 +1849<0 x 2 −86 x +1849<0 . - Answer : =43 x =43 . 408. Find the derivative of ( )=58 2+44 +58 f ( x )= x 2 +44 x +5858 . - Answer : ′( ) f ( x ) . 409. Determine the value of x if log140( −34)+log140( )=2log 140 ( x −34)+log 140 ( x )=2 . - Answer : =280 x =280 .
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