A plane is flying at 240mph and is climbing at 22 degrees. Find the rate at which it is gaining altitude when y=3miles
240 sin 22
To find the rate at which the plane is gaining altitude when y = 3 miles, we need to use the concepts of trigonometry and calculus.
Let's break down the problem step by step.
Step 1: Draw a diagram or visualize the situation. In this case, imagine a right triangle where one angle is 22 degrees, and the hypotenuse represents the plane's velocity, which is 240 mph.
Step 2: Identify the relevant quantities. The rate at which the plane is gaining altitude is the derivative of the altitude (y) with respect to time (t). We will be looking for dy/dt.
Step 3: Set up the equation using trigonometry. In the right triangle, the opposite side corresponds to the change in altitude (dy), and the adjacent side corresponds to the horizontal distance traveled (dx). Since we are given the angle and the hypotenuse, we can express dy and dx in terms of the given information.
dy = sin(22) * dx
Step 4: Express dx in terms of dt. Recall that velocity (v) is defined as the rate of change of distance with respect to time. In this case, dx/dt represents the horizontal velocity of the plane, which is given as 240 mph.
dx/dt = 240 mph
Step 5: Differentiate both sides of the equation in Step 3 with respect to t using the chain rule. This will allow us to find dy/dt.
dy/dt = sin(22) * (dx/dt)
Step 6: Substitute the values from Step 4 into the equation in Step 5.
dy/dt = sin(22) * 240 mph
Step 7: Convert the units if necessary. If the speed (240 mph) is not in the desired unit (miles per hour), you may need to convert it accordingly.
dy/dt = sin(22) * 240 mph
Finally, evaluate the equation to get the rate at which the plane is gaining altitude when y = 3 miles.
dy/dt ≈ sin(22) * 240 mph