Lighthouses Shapes
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Calculus Related Rates?
Can someone please tell me if i solved this correcty?
1. A trough is 10 ft long and its ends have the shape of isoscles triangles that are 3ft across at the top and have a height of 1ft. If the trough is being filled with water at a rate of 12 ft^3/min, how fast is the water level rising when the water is 6 inches deep?
dv/dt = 12 and h=6 inches=.5ft
Volume= (1/2 base * height)* legnth
Using similar triangles b/h = 3/1 so b=3h
So V=1/2bh*10
V=5(3h)(h)
V=15h^2
dv/dt=15h dh/dt
12=15(.5) dh/dt
dh/dt= 1.6
I was having trouble visualizing this one and coming up with the equation to solve it. If someone can help me find the equation I would appreciate it....
2. A lighthouse is located on a small island 3 km away from the nearest point P on a straight shoreline and its like makes four revolutions per minutes. How fast is the beam of light moving along the shore line when it is 1 km from point P?
Thank you in advance for your help!
1.
i think you were correct up to
v = 15h^2
then dv/dt is
dv/dt = 30h*dh/dt
dh/dt = (1/(30h))dv/dt
dh/dt = 12/(30*0.5)
dh/dt = 0.8 ft/min
2.
let A = angle of light from horizontal
x = distance of light beam from point P
tan(A) = 3/x
take derivatives with respect to time
sec^2(A)dA/dt = -3(x^-2)dx/dt
dx/dt = -(1/3)(x^2)sec^2(A)dA/dt
next, plug in known values
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