Question
cos(145) = (10000 - 200b + d^2) / (200d - 2bd)
cos(225) = (200b - 10000 + d^2) / (2bd)
Solve simultaneously
cos(225) = (200b - 10000 + d^2) / (2bd)
Solve simultaneously
Answers
GPT 3.5
To solve the given system of equations simultaneously, we can set the two expressions for cos(145) and cos(225) equal to each other and then solve for the variables:
(10000 - 200b + d^2) / (200d - 2bd) = (200b - 10000 + d^2) / (2bd)
Cross multiplying, we get:
(10000 - 200b + d^2)(2bd) = (200b - 10000 + d^2)(200d - 2bd)
Expanding both sides:
20000bd - 400b^2 + 2bd^2 = 40000d - 400bd + 200b^2 - 200000 + 200d^2 - 200bd
Rearranging the terms:
2bd^2 + 200b^2 + 200bd + 200000 + 400bd + 200000 - 20000bd + 40000d - 200d^2 = 0
Rearranging and simplifying further:
-2bd^2 + 200b^2 + 400bd + 40000d - 200d^2 = -400000
Now we can simplify by dividing everything by 2 to get:
-bd^2 + 100b^2 + 200bd + 20000d - 100d^2 = -200000
We have now extracted a single equation relating b and d. However, there may be multiple solutions to this equation, so it is a non-linear system and may require further analysis or the use of numerical methods to find the exact values of b and d.
(10000 - 200b + d^2) / (200d - 2bd) = (200b - 10000 + d^2) / (2bd)
Cross multiplying, we get:
(10000 - 200b + d^2)(2bd) = (200b - 10000 + d^2)(200d - 2bd)
Expanding both sides:
20000bd - 400b^2 + 2bd^2 = 40000d - 400bd + 200b^2 - 200000 + 200d^2 - 200bd
Rearranging the terms:
2bd^2 + 200b^2 + 200bd + 200000 + 400bd + 200000 - 20000bd + 40000d - 200d^2 = 0
Rearranging and simplifying further:
-2bd^2 + 200b^2 + 400bd + 40000d - 200d^2 = -400000
Now we can simplify by dividing everything by 2 to get:
-bd^2 + 100b^2 + 200bd + 20000d - 100d^2 = -200000
We have now extracted a single equation relating b and d. However, there may be multiple solutions to this equation, so it is a non-linear system and may require further analysis or the use of numerical methods to find the exact values of b and d.