A hot air balloon is rising vertically upward from the ground. The crew of a boat from a nearby
lake notices this situation and looks upward at an angle of
10 degree
to see the balloon. If the boat is
400meters away from the balloon, and the angle of observation is changing at 0.2 rad/min, how
fast is the balloon rising?
Sketch a right angles triangle with a base of 400 m, a height of h m , (the balloon), and an angle of Ø opposite the height.
tan Ø = h/400
h = 400 tanØ
dh/dt = 400 sec^2 Ø (dØ/dt)
given: when Ø = 10° or 10(π/180) radians dØ/dt = .2
(you can simply take sec 10° and square it)
dh/dt = 400 sec^2 10° (.2)
= appr 82.5 m/min
check my arithmetic
f(x)=7
f'(x)=7x^0
=0
To find the rate at which the balloon is rising, we can use trigonometry and the given information. Let's break down the problem step by step.
First, let's draw a diagram to visualize the situation. On the diagram, label the balloon's position (B), the boat's position (A), and the angle of observation (θ).
Next, we can think of the problem as a right triangle. The vertical side of the triangle represents the height at which the balloon is rising, the horizontal side represents the distance between the boat and the balloon, and the hypotenuse represents the line of sight between the boat and the balloon.
Since we know the distance between the boat and the balloon (400 meters) and the angle of observation (10 degrees), we can use the trigonometric function tangent to relate these quantities.
The tangent of an angle is equal to the ratio of the length of the opposite side to the length of the adjacent side.
Therefore, we have tan(θ) = opposite/adjacent.
In this case, tan(10 degrees) = height/400 meters.
To find the height, we can rearrange the equation to solve for it:
height = tan(10 degrees) * 400 meters.
Now, to find the rate at which the balloon is rising, we need to differentiate the equation with respect to time. Since we are given that the angle of observation is changing at a rate of 0.2 rad/min, we can differentiate the equation with respect to time.
Differentiating both sides of the equation, we get:
d(height)/dt = d(tan(10 degrees) * 400 meters)/dt.
Now, we need to use the chain rule to differentiate the equation. The chain rule states that if we have a function f(g(t)), then the derivative of f(g(t)) with respect to t is given by f'(g(t)) * g'(t).
In this case, f(g(t)) = tan(g(t)), and g(t) = 10 degrees.
Using the chain rule, we have:
d(height)/dt = d(tan(g(t)))/dg(t) * dg(t)/dt.
The derivative of tan(g(t)) with respect to g(t) is sec^2(g(t)), and the derivative of g(t) with respect to t is given by the rate at which the angle is changing, which is 0.2 rad/min.
Therefore, we have:
d(height)/dt = sec^2(g(t)) * dg(t)/dt.
Plugging in the values, we have:
d(height)/dt = sec^2(10 degrees) * 0.2 rad/min.
Finally, we can calculate the value of d(height)/dt using a calculator or by approximating the value of sec^2(10 degrees).
So, the rate at which the balloon is rising is approximately equal to sec^2(10 degrees) * 0.2 rad/min.