Related Rates
The radius r and area A of a circle are related by the equation A = πr2. Write an equation that relates dA/dt to dr/dt
dA/dt = (2πr)(dr/dt)
What is the smallest perimeter possible for a rectangle whose area is 16in2, and what are its dimensions?
Smallest perimeter = 16in
width = 4in
length = 4in
IF f-1(x) = g(x), THEN f(a) = b and g(b) = a, AND f'(a) = ___________ (fill in the blank)
dy/dx=12x2+(5/x) , find y=f(x) if y=9 when x=1
y=4x3+5lnx+5
A particle is moving with a velocity modeled by v(t)=5t3+6t-3 find its acceleration at t=2.
a(2)=66 m/s2
The mechanics at Lincoln Automotive are reboring a 6-in. deep cylinder to fit a new piston. The machine they are using increases the cylinder's radius (0.001/3) in/min. How rapidly is the cylinder volume increasing when the bore (diameter) is 3.8 in.?
dV/dt = 0.0239in3/min
(Calculator) A 1125-ft^3 open-top rectangular tank with a sqaure base x ft on a side and y ft deep is so be built with its top flush with the ground to catch runoff water.
If total cost is c = 5(x2 + 4xy) + 10xy, what values of x and y will minimize it?
x=15
y=5
Find the inverse for h(x)=(1+9x)/(4−x).
h-1(x)= (4x-1)/(9+x)
find y=f(x) if dx+e3xdy=0
y = (1/3)e-3x+C
A particle has a velocity of v(t)=2t4+5t1/2 find the function for position at any time , t, given that s(1)=0.
s(t)=(2/5)t5+(10/3)t3/2-(56/15)
A spherical balloon is inflated with helium at the rate of 100π ft3/min.
a. How fast is the balloon's radius increasing at the instant the radius is 5 ft?
b. How fast is the surface area increasing at that instant?
a. 1 ft/min
b. 40(pi) ft2/min
(Calculator) We want to build a box whose base length is 6 times the base width and the box will enclose 20 in3. The cost of the material of the sides is $3/in2 and the cost of the top and bottom is $15/in2. Determine the dimensions of the box that will minimize the cost.
L=4.3794
w=0.7299
h=6.2568
g'(3) = 1/5
Find y=f(x) if dy/dx=(xy)2 if f(1)=2
y=(-6/2x3-5)
(Calculator)
The particle is moving in the (right) positive direction on the intervals (0, 0.6339) and (2.366, ∞)
Sand falls from a conveyer belt at the rate of 10m3/min onto the top of the conical pile. The height of the pile is always three-eights of the base of the diameter. How fast is the height changing when the pile is 4 meters high?
(Calculator) We want to construct a cylindrical can with a bottom but no top that will have a volume of 30 cm3. Determine the dimensions (radius and height) of the can that will minimize the amount of material needed to construct the can.
r= 2.1216
h=2.1215
f(x) = 3x2-x, x>1 and g(x)= f-1(x), find g'(10)
g'(10) = 1/11
dy/dx=(3-y)cos(x) Let y=f(x) be the particular solution to the differential equation given that f(0)=1.
y=-2esinx+3
A particle is moving with acceleration modeled by a(t)=e2t-t1/2+14t3 given that v(0)=1/2 find the acceleration and velocity of the particle at t=2 and state whether it is speeding up or slowing down.
(Use calculator to plug in 2)
a(2)=163.770
v(2)=921.413
Speeding up
Water drains from a conical tank with a radius of r=4 and height h=10. The water is draining at 5 ft3/min.
a. What is the relation between new variables h and r of the water inside.
b. How fast is the water level dropping when h=6 ft
b) dh/dt = -0.276 ft/min
You operate a tour service, the rates are...
$200 per person if 50 people (the minimum # to book the tour) go on the tour. For each additional person, up to a maximum of 80 people total, the rate per person is reduced by $2. It costs $6000 (fixed cost) plus $32 per person to conduct the tour, How many people does it take to maximize profit?
x=67 people
Find the equation of the tangent line to the inverse at the given point.
f(x) = e-2x–9x3+4 @ (5, 0)
y = -½(x – 5)
Find the particular solution, y=f(x) for dy/dx=4x(y+1) when f(2)=4.
y=5e2x^2-8-1
A particle is traveling with an acceleration modeled by a(t)=6x+5 for [0,2] and a(t)=3x3+4 for [2,5]. Do not consider "c". Find the total distance traveled on the interval [0,5].
(Calculator)
Total Distance= 505.95