Water is poured into a conical paper cup at the rate of 3 cubic centimeters (cm) per second (s). The paper cup measures 8 cm tall and the top has a
radius of 4 cm.
a.) Use similar triangles to express the radius r in terms of the height h.
Be sure to solve your equation for r.
ANSWER: r =
<
b.) Determine how fast the water level is rising at the instant the water is 4
cm deep.(The volume of a cone is V = r2h.) Be sure to round your
answer to 3 Decimal Places.
ANSWER:
dh
dt
m/s

The radius in terms of height is , while the water level increases at 0.239 cm per second when the height is 4 cm

(a) Radius in terms of height

The given parameters are:

--- height

--- radius

The above parameters can be written as:

Express as fractions

Multiply both sides by h

(b) How fast the water is rising

The volume of a cone is:

Differentiate with respect to time

When the water is 4cm deep, then the radius is 2cm using

So, we have:

Also, we have:

Differentiate with respect to time

So, we have:

Expand

The volume increases at 3 cm^3 per second.

So, we have:

Divide both sides by 4

Divide both sides by pi

Rewrite as:

Hence, the water level increases at 0.239 cm per second when the height is 4 cm

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The water level is rising at the rate 0.238 cm per second

Step-by-step explanation:

Let the radius and height of water at time t be r and h respectively.

Height of cup =H= 8 cm

Radius of cup = R = 4 cm

By symmetry

Volume of water at time t ,

With 1

So,

We are given that Water is poured into a conical paper cup at the rate of 3 cubic centimeters  (cm) per second (s).

We are supposed to find how fast the water level is rising at the instant the water is 4 cm deep.

So,

Hence the water level is rising at the rate 0.238 cm per second

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