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Difficult Problems | ![]() |
A walk is perpendicular to a long wall, and a man strolls along it, away from the wall, at the rate of 4 feet per second. There is a light 10 feet from the walk and 25 feet from the wall. How fast is his shadow moving along the wall when he is 15 feet from the wall?
x/25=s/(10+s) 15/25=sd/(10+s) 25s+10x+xs 150+15s=25s (25)ds/dt=(10)dx/dt+dx/dt(s)+(x)ds/dt 150=10s (25)ds/dt=(10)(4)+(4)(15)+(15)ds/dt 15=s (10)ds/dt=10 ds/dt=10Answer: 10ft/sec
A walk is perpendicular to a long wall, and a
man strolls along it away from the wall at the rate of 3 ft per second.
There is a light 8 feet from the walk and 24 feet from the wall. How fast
is his shadow moving along the wall when he is 20 feet from the wall?
x/24=s/(8+s) 20/24=s/(8+s) 24s=8x+xs 24s=160+20s (24)ds/dt=(8)dx/dt+(s)dx/dt+(x)ds/dt 4s=160 (24)ds/dt=(8)(3)+(3)(40)+20ds/dt s=40 (4)ds/dt=144 ds/dt=36ft/secAnswer: 36 feet/second
A man, 6 feet tall, is walking away from a light, 14 feet above the ground, at the rate of 5 feet per second. How fast is his shadow changing in length?
(In general, shadow problems are set up as proportions!)
6/14=s/(x+s) 14s=6x+6s 8s=6x (8)ds/dt=(6)dx/dt (8)dx/dt=(6)(5) dx/dt=30/8=15/4Answer: 15/4 feet/second
Solve. Support your answer!
abs(x+2)+abs(x-5)@10 a) x+2+x-5 when x+2<0 and x-5<0="2x-3" x<-2 and x<5 x<5 b) x+2-(x-5) when x+2<0 and x@5="x+2-x+5" x<-2 and x@5="7" 2*x@5 c) (x+2)-(x-5) when x+2@0 and x-5<0 dne x<-2 and x<5 impossible d) (x+2)-(x-5) when x+2<0 and x-5<0="-x-2-x+5" x<-2 and x<5="-2x+3" x<-2 a)2x-3<10 b)7<10 d)-2x+3<10 2x<13 r 2x<7 x<13/2 x>-7/2 Solution: -7/2These inequalities do not govern the solution set. They determine if a solution is possible or not for each equation.
A weight W is attached to a rope that is 50 feet long which passes over a pulley at P, 20 feet above the ground. The other end of the rope is attached to a truck at point A, 2 feet above the ground. If the truck moves off at a rate of 9 ft./sec., how fast is the weight rising when it is 6 feet above the ground?
A 10 pound monkey hangs at the end of a 20 foot chain that weighs .5 pounds per foot. How much work does it do in climbing the chain to the top? Assume that the end of the chain is attached to the monkey.
A point moves along the graph of y = 3x2 - 4 such that the rate of change of its x-coordinate is 8 inches per second. How fast is the y-coordinate changing when the point is at (2,8)?
givens:
These are problems that for some reason or another, we never quite decided what to do with. Maybe you'll have better luck!
a) At what rate is the distance between the two planes changing after 15 minutes?
b) When do the planes cease to approach one another?
Technically,
the lengths of the sides of the triangle should be expressed as a rate
times a time, so A would be 600t, and B, 300-500t. However, when you differentiate,
you get dt/dt. What do you do with that?
b. The rate of change of x would be negative--or would it?
One
ship A is sailing due south at 16 mph and a second ship B, 32 miles south
of A, is sailing due east at 12 mph.
a) At what rate are they approaching or separating at the end of 1 hour?
b) When do they cease to approach each other and how far apart are they at that time?
This
one suffers from the same problems as the previous question.
A=2xy A=2x(18-x2) [plug in for y from eqn of parabola] A=36x-2x3 A'=36-6x2 [take derivative] 0=36-6x2 -36=-6x2 6=x2 x=A rectangle has two of its vertices as the x-axis. The other two vertices are on the parabola whose equation is y=18-x^2. What are the dimensions of the rectangle if its area is to be a maximum?
A
wire 14 feet in length is cut into two pieces. One part is used to make
a square. The other part is bent into a rectangle such that L=3w. How should
the wire be cut if the sum of the areas is to be a minimum?
P=4x+8w [x is side of square, w is short side of rectangle] 14=4x=8w 14-8w=4x 7/2 -2w=x A=x2+3w2 A=(7/2 -2w)2+3w2 A=49/4 -14w+4w2+3w2 A=49/4 -14w+7w2 A'=-14+14w 0=-14+14w 14=14w 1=w w=[0,7/2] [Set domain. w can be no bigger than 7/4, because when w=7/4, the perimeter of the rectangle is 14.] A(0)=49/4 [test limits of domain and answer] A(1)=21/4=min A(7/4)=147/16 The piece for the rectangle is 8 ft long. The piece for the square is 6 ft long.
The
area bounded by the curve y=x2 and the line y=4 generated various solids
of revolution when rotated
a) about the y-axis b) about the line y=4 c) about the x-axis d) about the line y=-1 e) about the line x=2 Find the volume generated in each case.a)
b)
[In this case, the graph was translated so y=4
is on y=0. There are other ways to approach it. For ease, half the graph
was integrated and the result will be doubled.]
c)
d)
[In this case, the shell method was used. The
average radius is y+1, since the graph was moved 1 down.]
e)
[Shell method used. Average radius is x-2 since
the graph was moved 2 to the right. In this case, the function must be
integrated from -2 to 2.]