4.6: Related Rates

A square with sides x has an area If a 2 X 2 square has it’s sides increase by 0.1, use differentials to approximate how much its area will increase. Remember this problem?

cm/sec A square with sides x has an area What if we were to observe the square getting bigger and each side were increasing at a rate of 0.1 cm/sec This would no longer simply be called dx because it is a rate of change. So what should we call it? Hint: It’s a change in length with respect to time. We did this before when talking about units of velocity…

cm 2 /sec Before we work on that, what would the units for this rate of change be? A square with sides x has an area What if we were to observe the square getting bigger and each side were increasing at a rate of 0.1 cm/sec cm/sec How we label this will always be decided by… UNITS! At what rate would the area of the square be changing?

Try taking the derivative of A with respect to… cm 2 /sec Given what we’ve just seen, how would we label this change? How would we find ? t

Remember the Chain Rule… But how? Try taking the derivative of A with respect to… t Now let’s find the answer we were trying to find. …this is an exact answer, not an approximation. Because this is an instantaneous rate of change...

Suppose that the radius of a sphere is changing at an instantaneous rate of 0.1 cm/sec. How fast is its volume changing when the radius has reached 10 cm? First, we need the volume formula: Next, what are we looking for? (Remember: How fast is its volume changing?) Then, what do we know? r = 10 cm Now what do we do? and…

The sphere is growing at a rate of. Suppose that the radius of a sphere is changing at an instantaneous rate of 0.1 cm/sec. How fast is its volume changing when the radius has reached 10 cm? r = 10 cm

“How fast is the water level dropping” means that we need to solve for… Water is draining from a cylindrical tank of radius 20 cm at 3 liters/second. How fast is the water level dropping? Find 1 liter = 1000 cm 3

Water is draining from a cylindrical tank of radius 20 cm at 3 liters/second. How fast is the water level dropping? 1 liter = 1000 cm 3

Steps for Related Rates Problems: 1. Draw a picture (sketch). 2. Write down known information. 3. Write down what you are looking for. 4. Write an equation to relate the variables. 5. Differentiate both sides with respect to t. 6. Evaluate.

B A Truck Problem: Truck A travels east at 40 mi/hr. How fast is the distance between the trucks changing 6 minutes later? 6 minutes =miles Since x, y, and z are always changing, they are all variables. So the equation we will have to begin with will be… Truck B travels north at 30 mi/hr

B A Truck Problem: How fast is the distance between the trucks changing 6 minutes later? ( ) Truck A travels east at 40 mi/hr. Truck B travels north at 30 mi/hr.

8 cm 9 cm But Wait! What do we know besides the dimensions of the cone? We have three variables Whatever shall we do? That’s one variable too many A cone filter of diameter 8 cm and height 9 cm is draining at a rate of 2 cm 3 /min. Find the rate at which the fluid depth h decreases when h = 5 cm.

8 cm But Wait! What do we know besides the dimensions of the cone? As we’ve done in the past, let’s see if one variable can be substituted for the other. That’s one variable too many A cone filter of diameter 8 cm and height 9 cm is draining at a rate of 2 cm 3 /min. Find the rate at which the fluid depth h decreases when h = 5 cm.

8 cm 4 cm 9 cm r h Similar triangles Now we have two variables! A cone filter of diameter 8 cm and height 9 cm is draining at a rate of 2 cm 3 /min. Find the rate at which the fluid depth h decreases when h = 5 cm.

8 cm Will this rate increase or decrease as h gets lower? A cone filter of diameter 8 cm and height 9 cm is draining at a rate of 2 cm 3 /min. Find the rate at which the fluid depth h decreases when h = 5 cm.

8 cm Show that this is true by comparing both when h = 5 cm and when h = 3 cm. As h gets smaller, gets faster because h is in the denominator.

8 cm Show that this is true by comparing both when h = 5 cm and when h = 3 cm. Problems like this surface often so remember your geometric relationships such as similar triangles, etc.

Hot Air Balloon Problem: Given: How fast is the balloon rising? Find

 Hot Air Balloon Problem: Given: How fast is the balloon rising? Find