Posted by Kevin on Friday, February 22, 2013 at 1:52pm.
A stuntman falls from a hot air balloon that is hovering at a constant altitude,100 feet above a lake. A television camera on shore, 200 feet from a point directly below the balloon,follows his descent. At what rate is the angle of elevation, theta, of the camera changing 2 seconds after the stuntman falls? ( neglect the height of the camera)
It seems like I'm missing a rate of change to solve this problem. I think I need to do something with the 2 seconds,but I don't know what. Any help would really be appreciated.
An isosceles triangle has equal sides of length 12m. If the angle theta between these sides is increased from 30 to 33 degrees, use differentials to approximate the change in the area of the triangle.
I know how to find the area, missing sides, and the height. My intial thought was to do a derivative of the area and multiply by 3 degrees. However, i don't have any given rates so I am lost on what to do. Any help would really be appreaciated.
- Calculus - Steve, Friday, February 22, 2013 at 2:40pm
the height of the stuntman at time t seconds after falling is
y = 100-16t^2
dy/dt = -32t
When the stuntman is at height y,
tanθ = y/200
sec^2 θ dθ/dt = 1/200 dy/dt
(1+tan^2θ) dθ/dt = 1/200 dy/dt
(1+(y/200)^2) dθ/dt = 1/200 dy/dt
at t=2, y=36, dy/dt = -64, so
(1+(36/200)^2) dθ/dt = 1/200 (-64)
dθ/dt = -.3 rad/s or -17.8°/s
- Calculus - Steve, Friday, February 22, 2013 at 2:46pm
The length of half the base is 12 sin(θ/2), so the area of the triangle is
a = 1/2 bh = 1/2 (24 sin(θ/2)) (12 cos(θ/2))
= 144 sinθ
da = 144 cosθ dθ
θ increases by 3θ, so dθ = π/60
da ~= 144 cos π/6 * π/60 = 144 * √3/2 * π/60 = 6π/5 √3
Answer This Question
More Related Questions
- Calculus - A hot air balloon headed due east at a speed of 20 miles per hour and...
- Tech Math 1b - Suppose that a hot air balloon moving due east at a speed of 15 ...
- math - A hot-air balloon, headed due east at an average speed of 15 miles per ...
- Physics - on a perfect fall day you are hovering at rest at low altitude in a ...
- Algebra - in that equation I have to solve for x. 5. x^3 + 2x^2 - 9x = 18 in ...
- Algebra - A balloon is secured to rope that is staked to the ground. A breeze ...
- Physical science - On a perfect day, you are hovering at low altitude in a hot-...
- pre cal - Tamara is flying in a hot air balloon. The hot air balloon is flying ...
- pre cal - Tamara is flying in a hot air balloon. The hot air balloons is flying ...
- physics - On a perfect fall day, you are hovering at low altitude in a hot-air ...