Problem 1 Statement: A 20-foot-wide swimming pool with cross-section (drawn belo

Question

Problem 1 Statement: A 20-foot-wide swimming pool with cross-section (drawn below) has been cleaned and is ready to be filled with water. If water is entering the pool at a rate kf 5ft^3/sec, how fast is the depth (h) of the pool changing when the volume is (a) 5000ft^3 and (b) 13,500 ft^3?
Problem 2 Statement: Consider the function f(x) = (9x - 6)/(9x^2 - 12x + 13). (a) What is the only root of f? (b) What happens if we use Newton’s Method starting at x0 = 1? Do we get a good approximation of the root? (c) What happens if we use Newton’s Method starting at x0 = 2? Do we get a good approximation of the root? (d) Graph of f on the interval [0, 15]. Using the graph, explain why Newton’s Method didn’t find the root when we started with x0 = 2.
Problem 3 Statement: Let f be the function whose graph is sketched below. (a) What root will Newton’s Method find if we use x0 = 0? Explain your reasoning. (b) What root will Newton’s Method find if we use x0 = 1? Explain your reasoning. (c) What root will Newton’s Method find if we use x0 = 3? Explain your reasoning. (d) What root will be Newton’s Method find if we use x0 = 4? Explain your reasoning. (e) What root will be Newton’s Method find if we use x0 = 5? Explain your reasoning.

Workshop 8 Problem 1 Statement: A 20-foot-wide swimming pool with cross-section (drawn below) 4 tt 25 ft 20 ft has been cleaned and is ready to be filed with water. If water is entering the pool at a rate of 5 ft/see, how fast is the depth (h) of the pool changing when the volume is (a) 5000 ft3 and (b) 13,500 ft? Problem 2 Statement: Consider the function( 13 (a) What is the only root of ? root? (b) What happens if we use Newton's Method starting at zo 1? Do we get a good approxcimation of the (e) What happens if we use Newton's Method starting at ro 2 Do we get a good approximation of the root? (d) Graph f on the interval (o, 15). Using this graph, explain why Newton's Method didn't find the root when we started with zo = 2. Problem 3 Statement: Let f be the function whose graph is sketched below. -21 -1 (a) what root will Newton's Method find if we use zo 07 Explain your reasoning. (b) What root will Newton's Method find if we use zo = 1? Explain your reasoning (e) What root will Newton's Method find if we use zo 3? Explain your reasoning (d) What root will Newton's Method find if we use zo 47 Explain your reasoning. (e) what root will Newton's Method find if we use zo = 5? Explain your reasoning.

Explanation / Answer

Since h is height of the water from the bottom of the pool, there are two separate cross sections to consider.

1) The section from h=0 to h=16.
V = 20[25*h +(1/2)(h)*(25h/16)] ----(i)

2.) When water is above h=16ft ( h=16 to h=20 only).
V = 20[25*16 +(1/2)(25)(16) +(75)(h-16)] ---(ii)

-----------
if V = 20[25*h +(1/2)(h)*(25h/16)] ----(i)
V = 20[25h +(25h^2)/32]
V = 500h +(500h^2)/32
V = 500h +(125/8)(h^2) -----(ia)
Differentiate both sides with respect to time t,
dV/dt = 500(dh/dt) +(125/8)[2h*(dh/dt)]
dV/dt = [500 +(125h/4)](dh/dt) -----------------------(1)

if V = 20[25*16 +(1/2)(25)(16) +(75)(h-16)] ---(ii)
V = 20[400 +200 +75h -1200]
V = 20[75h -600]
V = 1500h -12,000 -----(iia)
Differentiate both sides with respect to time t,
dV/dt = 1500(dh/dt) ----------------------------------(2)

From equation (i) if h=16

V = 20[25*16 +(1/2)(16)*(25*16/16)] = 12000 ft^3

So for part a)

The depth corresponding to 5000 ft^3 volume

5000 = 20[25*h +(1/2)(h)*(25h/16)]

solving h=8 ft

we will use dV/dt = [500 +(125h/4)](dh/dt) -----------------------(1) as 5000 ft^3 <12000 ft^3

dV/dt = [500 +(125h/4)](dh/dt)

dh/dt = ( dV/dt ) / [500 +(125h/4)] =5 / [500 +(125*8/4)] = 1/150 ft/sec

(b) For part b

The depth corresponding to 13500 ft^3 volume

13500 = 20[25*16 +(1/2)(25)(16) +(75)(h-16)]

Solving h=17 ft

we will use dV/dt = 1500(dh/dt) -----------------------(2) as 13500 ft^3 >12000 ft^3

dh/dt =(dV/dt) / 1500 = 5/1500 ft/sec

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