How to Solve Problems in Calculus Solution

8/10/2019 How to Solve Problems in Calculus Solution

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1. SolutionThe volume of a sphere of radius r is V(r ) =

V(r ) = 4

Ans. V(5) = 100

2. Solution(a) Let s represent the bicycles positionrelative to the end of the road.

4. Solutionince the position of the bicycle is !no"n atevery point in time# the instantaneousvelocity can be computed.

Let s(t) =

Then v(t) = s(t) = 2t

$t % ocloc!# t = % so v (%) = s(%) = 6 mi/h.

&.. Solution

'hen t = %# its positionx = 1. Theparticles instantaneous velocity isdetermined usin the derivative.

v = dx/dt = 4*t + %

'hen t = %, v = 117 in/sec.

,. Solution

v = dh/dt = -%2t 2&,.

'hen t = 1/, v = 64 ft/sec. The neativevelocity indicates that the pro0ectile ismovin in the neative direction (do"n"ard)at a speed of ,4 ftsec.

. . Solution'e let m(x) represent the mass of thesection of the rod of lenthx measured fromits leftmost end. The description of the

problem tells us that m(x) = k . ince the

total mass of the rod is 24 slus# m() = 24.

m() = 24

k = 24

k = *'e represent the density function by (x).(x) = m(x).

m(x) = * = *

(x) = m(x) = 4

(4) = 2 slugs/ft

*. Solution

I(t) = Q(t) = % - 4t &

I(2) = %(2)2- 4(2) & = 9 ampees

. Solutiondn/dt = ,t2 1/t 1

'hen t = %, dn/dt = !5

1/. Solution

'hen t = 2/

'ater is drainin at the rate of 4/ allonsper minute. The neative sin indicates thatthe volume of "ater is ettin smaller.

12.Solution3or comparison# the eact cost to producethe &1st item is

C(&1) - C(&/) = 5/.//1(&)% /./2&(&1)2%(&1) &6 - 5/.//1(&/)% /./2&(&/)2%(&/) &6

= %&&.,, - %42.&/

= "1#.176

1%. SolutionThe rate of depreciation is the rate at "hichvalue is lost.7f V(t) represents the value ofthe machine after t years# V(t)representsthe rate at "hich its value chanes.

V(t) = 2//t + %///V(&) = 2000

The machine depreciates at the rate of

82/// per year after & years.

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The surface area of the bois the sum of theareas S = 2x2 4xyince its volume is 1//in%#x2y = 1// and y =1///x2 . 7t follo"s that

S(x) = 4x - 4//

The rate of chane is determined by lettinx= &.

1&. Solutionince revenue = (number of items sold)(priceper item)#

R(x) = (&// - 2x - /.1x2)x = &//x + 2x2- /.1x3

R(x) = &// - 4x - /.%x2

R(1/) = &// - 4/ + %/ = 4#0

1,. Solution

9bservin a riht trianle# "e use thetheorem of :ythaoras to obtain arelationship bet"eenx andy;

x2y2 = 2/2uestion.?efore "e can compute dy/dt# ho"ever# "eneed to @ndy at this instant. This isaccomplished by usin the e>uationobtained in step %.

The neative derivative means thaty isdecreasin. The top of the ladder is fallin atthe rate of #/4 ft/sec.

1. olution

The distance bet"een the cars is increasinat the rate of 2 mi/h.

1*. olution

ince the plane travels 4*/ mih# it "ill have

Ao"n * miles in 1,/ hour (4*/ B 1,/ = *).incex = *# the value of! is easilydetermined by the :ythaorean theorem.

$ns is 465.67 mi/h

1. olution


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2/. olution

ince dAdtis neative# the area is

d"#r"asin at the rate of 16 in2/sec.

21. olution

The crossCsectional area of the "ater isA =(h2) (x %/). The volume of "ater V = l B

A. l is the lenth of the trouh soV = l (h/2)(x %/). ince the units in the problem mustbe consistent#"e ta!e l = 2// cm. Thus V =1//h(x %/). incex is not mentioned ineither the DEivenF or the D3indF in step 2# "eshould eliminatex from this e>uation. Toaccomplish this# "e observe similartrianles.

7n this fiure GA$C is similar to GA%&. incetheir correspondin sides are proportional#

'e substitute ddt = -2/// and h = 2/ intothe result

The minus sin indicates that the "ater levelis fallin at the rate of 0.24 cm/min.22. olution

?y observin that GA$C is similar to GA%& inthe diaram to the riht# "e see that

The "ater level is risin at the rate of 5.09ft/min.

2%. olution

?ecause drdtis neative# r is shrinkin at therate of0.01## in/min.

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The volume is shrin!in at the rate of 1.5in#/min.2&. olution

T"o hours after the cars leave their initialpoint#x = */ miles(2 B 4/ = */) andy = ,/miles (2 B %/ = ,/). 7t follo"s that

The distance bet"een the cars is increasin atthe rate of 50 mi/h.

2,. olution

$ns"er is 62.61 $m/h.

2. olution

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incey = 1///x#y = 1/ "henx = 1/. Theminimum perimeter = 2x 2y = 40.

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%he ma&imum 'olume of 4!6in#occurs"henx = %.

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How to Solve Problems in Calculus Solution

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