United Express, a national package delivery service, charges a base price for overnight delivery of packages weighing 1 pound or less and a surcharge for each additional pound (or fraction thereof). A customer is billed $29.00 for shipping a 7-pound package and $64.00 for a 27-pound package. Find the base price and the surcharge for each additional pound.

The base charge = $18.5 and surcharge for each pound is = $1.75

Step-by-step explanation:

Solution:

- Denote:

     Base charge = $x   For for Weight ≤ 1 lbs

     Additional charge = $y / unit pounds

- We are given that the customer is billed $29.00 for shipping a 7-pound package. We can express this statement in terms of "x" and "y".

- The 1-pound out of 7 pounds will be charged with $x and remaining 6 additional pounds of weight will be charged "$y" for each unit of weight.

                         $x + ( 7 - 1 )*$y = $29.00     Eq1

- Similarly for a receipt of $64.00 for a 27-pound package. The 1-pound out of 27 pounds will be charged with $x and remaining 26 additional pounds of weight will be charged "$y" for each unit of weight.      

                         $x + ( 27 - 1 )*$y = $64.00   Eq2

- Now solve the two equations simultaneously by subtracting Eq1 from Eq2:

                        (26 - 6 )*$y = $(64 - 29)

                        20*$y = $35

                         y = 1.75

                        $x = 29 - 6*1.75

                        x = 18.5    

- The base charge = $18.5 and surcharge for each pound is = $1.75

(sin 42)/31. = sinx)/37

sinx = 37 (sin42)/31

= 37× 0.66/31 = 0.787

x= 51.9

other angle opposite to y = 180-51.9-42

= 180- 93.9 =86.1

so

sin42)/31 = sin 86.1)/y

0.021 = 0.997 /y

y= 0.997/0.021 =47.47