This document contains instructions for solving four calculus word problems involving rates of change. The problems involve finding the rate of change of a sphere's radius with respect to time, the rate of change of a circle's circumference with respect to time, finding the rate at which a shadow's length is increasing, and finding the rate at which the distance between two girls is increasing. The instructions provide steps such as listing given information, identifying relationships between variables, differentiating equations with respect to time, and solving for the desired rate of change.

1. Content!

2. Question 1.

3. • We start by listing all of the
given information

4. • We divide diameter by 2 to
get the radius
• Notice that the units for are
different for each piece of
information (feet and inches)
• We divide the number of
inches by 12 to convert the
radius into units of feet

5. • Now with all of our
variables listed, we find the
relationship between them
according to that of a
sphere

6. • Next we differentiate
implicitly with respect to time
followed by inputting the given
information

7. • Solve for dr/dt and
complete with a
sentence answer
• Include units of
measure

8. *Side note: Some general volume equations for
solids that should be known

9. Question 2.

10. • We start again by listing
the information given

11. • Find the relationship
between the variables. In
this case, it is the
circumference formula

12. • Now we differentiate implicitly with chain rule
with respect to time and input the variables listed
• Solve for dC/dt

13. • Finish with a
sentence answer and
include appropriate
units of measure

14. Question 3.

15. • Start by drawing out a diagram
and listing all given information

16. • Note that the two
triangles are similar and
therefore, we can find
an expression for ds/dt
using the idea of
proportionality
• Solve for ds/dt, the
rate at which the
shadow is increasing in
length

17. • Now, with an
expression relating
the distances x and
s, we differentiate
implicitly with
respect to time and
solve for the rate
at which the tip of
the shadow is
moving

18. Bonus Question!
Just some extra practice =)

19. • List all given information
• Note that to find the
distance travelled at 1
minute, multiply the rate by
time (in seconds)
• Now, find the relationship
between the distances

20. • Next we differentiate
implicitly using chain rule
with respect to time and
solve for dd/dt, the rate at
which the distance between
the girls is increasing

21. • Finish with a
sentence answer and
include the
appropriate units! =)

22. DONEDONEDONE! =)