Name__________________________________
Related Rates 1
7) The range R of a projectile is related to the
SHORT ANSWER. Show all work clearly on a separate
initial velocity v and projection angle θ by the
sheet of paper. Circle you answer. Algebraic solutions
only. Attach this worksheet to your work.
v2 sin 2θ
equation R = , where g is a constant.
g
Solve the problem.
How is dR/dt related to dθ/dt if v is constant?
1) Suppose that the radius r and the
circumference C = 2πr of a circle are
dR 2v2 cos 2θ dθ
Answer:
= differentiable functions of t. Write an equation
dt
g
dt
that relates dC/dt to dr/dt.
Answer:
dC
dr
= 2π
dt
dt
8) The range R of a projectile is related to the
initial velocity v and projection angle θ by the
v2 sin 2θ
equation R = , where g is a constant.
g
4
2) Suppose that the radius r and volume V = π
3
How is dR/dt related to dv/dt if θ is constant?
r3 of a sphere are differentiable functions of t.
Write an equation that relates dV/dt to dr/dt.
Answer:
Answer:
dV
dr
= 4πr2
dt
dt
9) The range R of a projectile is related to the
initial velocity v and projection angle θ by the
v2 sin 2θ
equation R = , where g is a constant.
g
3) The area of the base B and the height h of a
pyramid are related to the pyramidʹs volume
1
V by the formula V = Bh. How is dV/dt
3
How is dR/dt related to dv/dt and dθ/dt if
neither v nor θ is constant?
related to dh/dt if B is constant?
Answer:
dV B dh
Answer:
= dt
3 dt
4) The volume of a square pyramid is related to
the length of a side of the base s and the height
1
h by the formula V = s2 h. How is dV/dt
3
related to ds/dt if h is constant?
Answer:
dV 2hs ds
= dt
3 dt
5) If a and b are the lengths of the legs of a right
triangle and c is the length of the hypotenuse,
c2 = a 2 + b2 . How is dc/dt related to da/dt and
db/dt?
Answer:
db
dc 1 da
= a + b
dt
dt c dt
6) The kinetic energy K of an object with mass m
1
and velocity v is K = mv2 . How is dm/dt
2
related to dv/dt if K is constant?
Answer:
dR 2v sin 2θ dv
= dt
g
dt
2m dv
dm
= - v dt
dt
1
dR 2v
dθ
dv
= v cos 2θ
+ sin 2θ
dt
g
dt
dt