The 20 character minimum is stupid but the answer is 68 Km/h
![I=5\int \frac{cos^{4}\theta }{sin\theta }\times cos\theta d\theta \\\\I=5\int \left ( 1-sin^{2}\theta \right )^{2}\times \frac{cos\theta }{sin\theta }d\theta \\put\ \sin\theta =t\\\\dt=cos\theta d\theta \\\\I=5\int\frac{t^{4}+1-2t^{2}}{t}dt\ \ \ \ \ \ \ \ \ \ \because (a-b)^2=a^2+b^2-2ab\\\\I=5\left ( \int t^{3}dt + \int \frac{1}{t} -2\int t \right )dt](https://tex.z-dn.net/?f=I%3D5%5Cint%20%5Cfrac%7Bcos%5E%7B4%7D%5Ctheta%20%7D%7Bsin%5Ctheta%20%7D%5Ctimes%20cos%5Ctheta%20d%5Ctheta%20%5C%5C%5C%5CI%3D5%5Cint%20%5Cleft%20%28%201-sin%5E%7B2%7D%5Ctheta%20%20%5Cright%20%29%5E%7B2%7D%5Ctimes%20%5Cfrac%7Bcos%5Ctheta%20%7D%7Bsin%5Ctheta%20%7Dd%5Ctheta%20%5C%5Cput%5C%20%5Csin%5Ctheta%20%3Dt%5C%5C%5C%5Cdt%3Dcos%5Ctheta%20d%5Ctheta%20%5C%5C%5C%5CI%3D5%5Cint%5Cfrac%7Bt%5E%7B4%7D%2B1-2t%5E%7B2%7D%7D%7Bt%7Ddt%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5Cbecause%20%28a-b%29%5E2%3Da%5E2%2Bb%5E2-2ab%5C%5C%5C%5CI%3D5%5Cleft%20%28%20%5Cint%20t%5E%7B3%7Ddt%20%2B%20%5Cint%20%5Cfrac%7B1%7D%7Bt%7D%20-2%5Cint%20t%20%5Cright%20%29dt)
by using the integration formula
we get,
![\\I=5\left ( \frac{t^{4}}{4} +logt -t^{2}\right )\\\\I=\frac{5}{4}t^{4}+5\log t-5t^{2}+c](https://tex.z-dn.net/?f=%5C%5CI%3D5%5Cleft%20%28%20%5Cfrac%7Bt%5E%7B4%7D%7D%7B4%7D%20%2Blogt%20-t%5E%7B2%7D%5Cright%20%29%5C%5C%5C%5CI%3D%5Cfrac%7B5%7D%7B4%7Dt%5E%7B4%7D%2B5%5Clog%20t-5t%5E%7B2%7D%2Bc)
now put the value of t=\sin\theta in the above equation
we get,
![\int 5\cot^5\theta \sin^4\theta d\theta=\frac{5}{4}sin^{4}\theta+5\log \sin\theta - 5sin^{2} \theta+c](https://tex.z-dn.net/?f=%5Cint%205%5Ccot%5E5%5Ctheta%20%5Csin%5E4%5Ctheta%20d%5Ctheta%3D%5Cfrac%7B5%7D%7B4%7Dsin%5E%7B4%7D%5Ctheta%2B5%5Clog%20%5Csin%5Ctheta%20-%205sin%5E%7B2%7D%20%5Ctheta%2Bc)
hence proved
Step-by-step explanation:
spring water is 2.08/16= 13 cents per ounce
fresh water is 2.40/20= 12.5 cents per ounce
fresh water is .5 cents per ounce less expensive
Tom needs to earn $77.82 in order to buy the game system.