Solve the trigonometric equation for x: tan3x=1/tan2x.
Since 1/tanA = tan (π/2 − A), we can write the above equation as tan3x = tan(π/2 − 2x), 3x = (π/2 − 2x) + Kπ, since if tanA = tanB, then A = B + Kπ, where K= some integer 3x − (π/2 − 2x) = Kπ, 5x − π/2 = κπ, 5x = Kπ + π/2, x = Kπ/5 + π/10
A rectangular swimming pool has a length of 30 meters, a width of 10 meters, and an average depth of 2 meters. If a hose can fill the pool at a rate of 0.5 cubic meters per minute, how many hours will it take the hose to fill the pool?
Since the swimming pool is rectangular, we can find its volume as the length, width and height (depth) are already given. So, volume = length X width X height = 30 X 10 X 2 = 600 cubic meters Rate of filling the pool by the hose = 0.5 cubic meters per minute. Therefore, hours taken by the hose to fill the pool =(600 cubic meter)/ (0.5 cubic meters per minute) =600/0.5 minutes =6000/5 minutes =1200 minutes =1200/60 hours =20 hours
Two people are 14 miles apart and begin walking towards each other. Person A walks 3 miles per hour and Person B walks 4 miles per hour. How long will it take them to reach each other?
Total distance to be covered by walking= 14 miles Distance covered by A = lets say, d miles Then, distance covered by B = 14 - d miles. Both use the same amount of time to meet each other and lets refer that time as t hours. By the equation, speed X time = distance, we get: since speed of A is 3 miles/hour, equation for A: 3t = d...............................(1) Similarly, equation for B: 4t = 14 - d........................(2) Substituting the value of d from (1) to (2), 4t = 14 - 3t 4t + 3t = 14 7t = 14 t= 2 Therefore, it will take 2 hours for A and B to reach each other.