Related rates. What are related rates?. Related rates are found when there are two or more variables that all depend on another variable, usually time Two or more quantities change as time changes
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1. Let A be the Area of a circle of radius r.
How is dA/dt related to dr/dt?
2. Let V be the Volume of a cube of side length x.
How is dV/dt related to dx/dt?
3. Let V be the Volume of a sphere of radius r.
How is dV/dt related to dr/dt?
A pipe is filling a cylindrical tank at the rate of 2500 cm3 per minute. If the radius of the tank is 25cm, how fast is the height of the water in the tank changing?
What is changing?
What is constant?
Find per minute. If the radius of the tank is 25cm, how fast is the height of the water in the tank changing?
Water is draining from a cylindrical tank at 3 liters/second. How fast is the surface dropping when the cylinder’s radius is 20 cm?
Differentiate both sides with respect to t
Need to get rid of the variable r some how, can we rewrite r in terms of h? We can if we use similar triangles.
We want to find d/dt (both derivatives with respect to t) of both x and h, so…
We were told that dx/dt was 3 ft/s. Plug it in, we also know that x(0)=5 (how far the ladder is away from the base of the wall at time 0, so to see where it is at t=1 we would take 5+3=8. Now we know x @ t=1, find h @ t=1 by the Pythagorean theorem. h(1) = 13.86
NOW WE NEED TO GET RID OF W (the extra variable) ladder is 5 ft. from the wall at time t=0 and slides away from the wall at a rate of 3 ft/s. Find the velocity of the top of the ladder at time t=1.
A light shines from the top of a pole 50 ft high. A ball is dropped from the same height from a point 30 ft away from the light as shown in the figure. How fast is the shadow of the ball moving along the ground ½ second later?
da/dt = 5
db/dt = 8
We want to find the rate of change of the hypotenuse which would be dd/dt.
The Pythagorean theorem relates the variables.
Solve now for dh/dt
To think about the second part of the question, one would think about the volume of a cross section close to the tip of the cone, and a cross section at the base of the cone. Which is a greater volume?
Or you could think about letting t=1 (close to when you first start filling up) and compare that to t=5.
We want to find dy/dt, the velocity (distance/time) of the rocket
We know dӨ/dt
A relationship between theta and y would be…
Find dx/dt when Ө=π/3
The relationship to use is law of cosines.