Question

Two sides of a triangle have lengths 9 m and 15 m. The angle between them is increasing at a rate of 2°/min. How fast is the length of the third side increasing when the angle between the sides of fixed length is 60°? (Round your answer to three decimal places.)

Answer 1

guess you need the law of cosines for this relationship
do you know it?

Answer 2

yes

Answer 3

9^2+15^2-2(9)(15)cos

Answer 4

if you call the side you are interested n \(x\) you have
\[x^2=15^2+9^2-2\times 15\times 9\cos(\theta)\]

Answer 5

ok then take the derivative wrt time and get \[2xx'=270\sin(\theta)\theta'\]

Answer 6

you are told \(\theta'=2\) and that \(\theta=60\) plug them in and solve

Answer 7

actually there is something seriously wrong with this problem, but i won't bother you about it
i will just add it to my "bad math" pile

Answer 8

i would love to see what answer you get
i can guarantee that it will not be correct

Answer 9

270 sqrt(3)

Answer 10

i am wondering who wrote this problem

Answer 11

it has a big flaw in it, or else we need to do something much more complicated

Answer 12

ok

Answer 13

do you happen to know what the right answer is supposed to be?

Answer 14

no

Answer 15

i got 0.416

Answer 16

ok well lets pretend that there is no problem here, i will get back to that
you have \[2xx'=270\sin(\theta)\theta'\] which becomes
\[2xx'=270\times \frac{\sqrt{3}}{2}\times 2\] but you still need \(x\) in order to solve for \(x'\)

Answer 17

okay

Answer 18

solve for \(x\) here
\[x^2=15^2+9^2-2\times 15\times 9\cos(60)\]

Answer 19

i get \(x^2=171\) making \(x=\sqrt{171}\) or \(3\sqrt{19}\) if you prefer

Answer 20

yes, i got that too

Answer 21

so maybe the answer is supposed to be
\[x'=\frac{270\sqrt{3}}{3\sqrt{19}}\] but i am sure that this is all nonsense for one simple reason

Answer 22

sine and cosine are functions
as function they are functions of numbers, not angles
as functions of numbers they correspond to angles, only if the angles are measured in radians, not degrees
as a function of degrees, the derivative of cosine is NOT sine
so this is all junk

Answer 23

2= pi/90

Answer 24

convert it into radians.

Answer 25

ok then it should work
of course you get a completely different answer

Answer 26

it is a really really bad idea to use calculus and degrees in the same problem
they do not match up

Answer 27

so what is the answer?

Answer 28

i keep getting 0.416

Answer 29

replace the \(2\) here by \(\frac{\pi}{90}\) as you said