Question
A conical tank (with vertex down) is 28 ft across the top and 7 ft ft
deep. If water is flowing into the tank at the rate of
, find
the rate of change of the depth of the water at the instant when it is 2
deep.
deep. If water is flowing into the tank at the rate of
the rate of change of the depth of the water at the instant when it is 2
Ask by Boone Munoz. in Puerto Rico
Nov 01,2024
Upstudy AI Solution
Tutor-Verified Answer
Answer
The rate of change of the depth of the water when it is 2 ft deep is
.
Solution

Sign in to Unlock Answers for Free!
A Learning Platform Trusted by Millions of Real Students and Teachers.

Answered by UpStudy AI and reviewed by a Professional Tutor
Like
Bonus Knowledge
To solve the problem, we first need to determine the relationship between the volume of water in the tank and its height. The tank forms a cone, and the volume
of a cone is given by the formula
.
Given that the tank is 28 ft across the top, the radius
at the top is
ft, and it is
ft deep. Using the depths and corresponding radius, we find the relationship between the radius
of the water’s surface and the height
of the water.
Using similar triangles, we get:
Substituting
into the volume formula gives:
Now, we need to differentiate this volume with respect to time
:
We know that water is flowing into the tank at a rate of
, and we want to find
when
.
Substituting
:
Evaluating this gives us the rate of change of the depth of the water at that moment:
So, the rate of change of the depth of the water when it is 2 ft deep is approximately
.