AboutAlon Mandes Expertise Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems.
Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .
Experience 1. I'm a team member of mathnerds (math site for answering questions)
2. I'm a team member in the Student's Union of the Technion, helping
students who have problems in mathematics.
3. 2 years of experience as a math teacher in college.
4. I give free homework help for high school students in
Mathematics & Physics.
5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" ,
"Complex Functions".
Question A horozontal trough is 16 meters long, and its ends are isosceles trapezoids with an altitude of 4 meters, a lower base of 4 meters, and an upper base of 6 meters. Water is being poured into the trough at the rate of 10 cubic meter per minute. How fast is the water level rising when the water is 2 meter deep?
Answer OK, we will solve this question step by step. First let's review
some of the facts & define some variables :
1. Let's define h as the depth of the water
2. The volume occupied by water in time t is 10t m^3 (t in min)
3. The water reaches depth h in t minutes .When the water gets to
depth h the volume occupied is function of t, we will call it
W(h). It's clear thought that W(h)=10t.
4. Now we have to find the function W(h). To do so we need to use
some geometry !! . First, notice that the volume is equal to the
vertical cross section * 16. (why?)
The area of the vertical cross section trapezoid when the level
is h , is :( h/2 + 4 )*h/2. (I will leave it for you as an
exercise to prove it ! )
So, now we can claim that W(h)=16*( h/2 + 4 )*h/2.
that means W(h)=4h^2+32h.
5. As we claimed in 3 , W(h)=10t. So now we can find h as a
function of time, & that is what we are looking for.
4h^2+32h=10t. From here you can calculate h(t), I will leave
it to you.
6. After you found h(t) which is the depth as function of time,
deriving it, will give you the speed of the rising.
7. In order to know what value of t you need to put in the derivative
of h(t), you must know, when does the water reaches a depth of
2 meters. To do so, solve the equation 10t=16*(2/2 + 4)*2/2.
The t that you will get, is the t that you will put in h'(t).
