Question
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A spherical snowball melts so that the surface area decreases at the rate of 1 cm 2 / min. Find the rate at which the radius is changing when r = 10 cm. (Note: Surface area S = 4 pi r 2). Two cars start moving from the same point(say the origin). One travels south at 60 mph and the other travels west at 25 mph. How fast is the distance between then changing after 2 hours ? A spotlight on the ground shines on a wall 12m away. If a 2m tall man walks away from the spotlight towards wall at a speed of 1.6 m / s, how fast is the length of the shadow changing when he is 4m from the wall ? Consider the function f (x) = x 4 - 6 x 2 + 8 x + 10 What is the domain of the function ? Express as an interval. Find the derivative of the function. Find x where f' (x) = 0 or f' (x) does not exist. Determine at each of the points in part (c), whether the function has a local minimum or maximum Find any absolute maximum and / or minimum values of the function and where it occurs. Consider the function f (x) = x 2 What is the domain of the function ? Express as an interval. Show that the derivative f' (x) = x (5x + 8) /2 Find x where f' (x) = 0 or f' (x) does not exist. Determine at each of the points in part (c), whether the function has a local minimum or maximum
Explanation / Answer
3.)
Let y = height of shadow on the building
Use similar triangles:
y/2 = 12/x, where x = distance of man from the spotlight
y = 2(12)/x
y = 24/x
Take the derivative of y with respect to time t:
dy/dt = - 24(dx/dt)/x^2
Parameters from Problem:
dx/dt = 1.6 m/s
x = 12m - 4m = 8m.
Substitute:
dy/dt = -24(1.6 m/s)/(8 m)^2
dy/dt = -0.6 m/s
