M_Bank\YR12-2U\ApplicationsOfCalculus.CAT
Applications of Calculus to the Physical World
1)! ¥2U84-4iii¥
The rate at which water runs out of a tank is proportional to the volume of water in the tank, ie. =kV. The tank is full to start with and has a capacity of 36000litres.
a. Show that V=V0ekt satisfies this equation where V0is the volume of water in the tank initially.
b. If of the water in the tank runs out in 30minutes, find the volume of water remaining in the tank after 60minutes.†
«® a)Proof b)20250litres »
2)! ¥2U84-8ii¥
A particle moves in a straight line such that its displacement, xcm, after tseconds is given by: x=t3-6t2+9t+4.
a. When and where does the particle first change direction?
b. What is the average speed of the particle in the first second?†
«® a)1second, 8cm b)4cm/s »
3)! ¥2U84-9ii¥
If the area of this sector is 625m2, find the values of Randq (to the nearest degree) so that the sector has minimum perimeter.†
«® 25m, 2radians »
4)! ¥2U85-8i¥
A train runs between 2stations stopping at each. Its velocity, vkm/min, tminutes after leaving the first station is given by: . Find:
a. the time taken to travel between the two stations;
b. the maximum velocity attained;
c. the distance between the two stations.†
«® a)4minutes b) c) »
5)! ¥2U85-9i¥
The area of a rectangular sheet of paper is 300cm2. A margin of 2cm is allowed at the top, 1cm at the bottom and 2·5cm on each edge. What are the exact dimensions of the printed area if it is to be a maximum?†
«® »
6)! ¥2U85-10i¥
The rate at which liquid is flowing into a vessel after t minutes is given by . If (loge2)m3 of liquid flows into the vessel after 3minutes, how much liquid flows in after 8minutes? Give your answer to 3significant figures.†
«® 1·50m3 »
7)! ¥2U86-8iv¥
A farmer needs to construct 2holding paddocks, one rectangular and the other triangular, for goats and chickens. The figure shows that he uses an existing long fence as part of the boundary.
If he has only 440metres of fencing, find the maximum total area of holding paddocks that he can construct if AB=CD=DE and ÐCDE=30°.†
«® 17600m2 »
8)! ¥2U86-9ii¥
A stone is projected vertically upwards from the top of a tower. If the height of the stone above ground level is given by: h=25+20t-5t2 where his in metres and tis the time in seconds.
a. Find the initial velocity of projection.
b. Calculate the time taken for the stone to hit the ground.
c. Show the acceleration of the stone is constant. Explain.†
«® a)20m/s b)5·74seconds c), which is independent ofx. »
9)! ¥2U87-9i¥
The size of a colony of insects is given by the equation: P=5000 ekt where Pis the population after tdays.
a. If there are 8500 insects after 1day, find the value of kcorrect to 1decimal place.
b. When will the colony triple in size? (Answer to the nearest day)
c. What is the growth rate of the population after 2days?†
«® a)0·5 b)2days c)6796insects/day »
10)! ¥2U87-9ii¥
A particle moves in such a way that its distance, xmetres, from the origin after tseconds is given by: x=2-2sin2t for 0£t£2p.
a. Find an equation for its velocity after tseconds.
b. Where is the particle initially and what is its velocity then?
c. Describe the motion of the particle.†
«® a)=-4cos2t b)2m, -4m/s c)SHM, Amplitude=2, centre=2, Angular velocity=2, Phase=0 »
11)! ¥2U87-10¥
The diagram shows a sector OAB of a circle, centreO and radius xmetres. Arc AB subtends an angle qradians atO. An equilateral triangle BCO adjoins the sector.†
i. Write down expressions for the:
a. area of the sector OAB;
b. area of the triangle BCO;
c. length of the arc AB.
ii. Hence write down an expression for the:
a. area
b. perimeter
of the figure bounded by ABCO.
iii. The perimeter of this figure is (12-2)metres.
a. Express its area in terms ofq.
b. For what value of q is the area a maximum?
c. Show that the maximum area is (6-)m2.†
«® i)a) b) c)xq ii)a) b)x(q+3) iii)a) b) c)Proof »
12)! ¥2U88-9i¥
A particle moving with a constant acceleration of 4ms–2 starts from rest. Find the:
a. time taken for the particle to attain a velocity of 22ms–1;
b. distance travelled by the particle in this time.†
«® a)5·5seconds b)60·5m »
13)! ¥2U88-9ii¥
The diagram below is an artists entry in the ‘Designing a Bicentennial Flag’ competition. It is to consist of two green rectangular regions with a yellow vertical stripe between these regions.
The perimeter of the entire flag is 376cm and the green regions cover an area of 6561cm2. Let the width of the vertical stripe be ycm and the dimensions of the green regions be as shown in the diagram.
a. Show that a+b=188-x-y and hence show that the width of the yellow vertical stripe is given by y=(188-x-6561x-1)cm.
b. Find the dimensions of this flag if the width of the yellow vertical stripe is to be a maximum.†
«® a)Proof b)81cm´26cm »
14)! ¥2U88-10i¥
The diameter of a tree (Dcm), tyears after the start of a particular growth period is given by D=70ekt.
a. Show that wherek is a constant.
b. If k=0·15, how long will it take for the diameter of the tree to measure 74cm.? (Answer to the nearest integer.)†
«® a)Proof b)0·37years »
15)! ¥2U89-8a¥
A stone dropped into a still pond causes circular ripples on the water surface. The area of the disturbed water, tseconds after the stone hits the surface of the water is given by: A(t)=4pt2 where the area is measured in m2.
i. Find an expression to represent the rate of change of the area of the disturbed water.
ii. Find, in terms ofp, the rate of change of the area of the disturbed water after one second.†
«® i)8pt ii)8p »
16)! ¥2U89-8b¥
Two particles, AandB, move along a straight line so that A’s displacement, in metres, from the origin after tseconds, is given by xA=2-2sin2t and B’s acceleration, in metres per second per second, is given by .
i. What is the acceleration of particle Aafter 3seconds?
ii. Which particle is moving faster after one second if particle Bis initially at rest at xB=6.
iii. Find the maximum displacement of particleA.†
«® i)2·24m/s ii)B iii)2m »
17)! ¥2U89-9b¥
A railway enthusiast designs a miniature railway of length 1000metres. The route consists of two semicircles at opposite ends of a rectangle.
i. If the rectangle has a length of ymetres and its width is xmetres, show that:
y=500-.
ii. Show that the area,A, enclosed by the railway track is given by A=.
iii. Find the maximum area, to the nearest hectare, enclosed by the railway track.†
«® i)ii)Proof iii)8hectares »
18)! ¥2U90-7b¥
A particle moves along a straight line so that it’s position, s(t)cm, at time, tmins, is given by: s(t)=t3-12t2+36t.
i. Find an expression for the velocity, v(t), and acceleration, a(t), of the particle at any timet.
ii. When is the particle at rest?
iii. Copy this table into your answer sheet and complete it.
t / 0 / 2 / 4 / 6 / 8s(t) / 0 / 32 / 16 / 0 / 32
V(t)
A(t)
iv. Describe the motion of the particle.†
«® i)v(t)=3t2-24t+36, a(t)=6t-24 ii)t=2, t=6 iii)
t / 0 / 2 / 4 / 6 / 8s(t) / 0 / 32 / 16 / 0 / 32
v(t) / 36 / 0 / -12 / 0 / 36
a(t) / -24 / -12 / 0 / 12 / 24
iv)The particle moves away from the origin for 2minutes, stops and returns to the origin. It then moves away from the origin with an increasing acceleration. »
19)! ¥2U90-9b¥
A wire of length 30cm is cut into two pieces. One piece is bent to form an equilateral triangle and the other to form a circle.
If the piece of wire used to form the triangle is of length 3xcm:
i. show that the sum of the areas of the two figures, Acm2, is given by:
;
ii. find the length of wire that must be used to form the triangle so that the sum of the areas of the two figures is a minimum.†
«® i)Proof ii)18·7cm (to 1 d.p.) »
20)! ¥2U91-9a¥
A particle moves in a straight line so that its acceleration, fm/s2 at time tseconds is given by f=3(4+t)2. Initially the particle moves with a velocity of 64m/s from a position 3metres to the right ofO.
i. Show that the velocity of the particle at any time t seconds is given by: v=(4+t)3.
ii. Hence find the displacement, xmetres, of the particle as a function of time.
iii. Find the distance travelled by the particle during the first 2seconds.†
«® i)Proof ii) iii)260 »
21)! ¥2U91-9b¥
Max brews his own beer in a barrel. From past experience, he knows that the amount of sugar, Mkg, that is present in a liquid kept at a constant temperature after tminutes, satisfies the following equation: M=M0e-kt where kis a constant.
i. He places 10kg of sugar in the barrel for his brew. After 5minutes he finds that only half of this sugar is present in the barrel. Find the value of kcorrect to 2decimal places.
ii. How long will it take before 1kg of this sugar is present in this barrel? Give your answer correct to the nearest minute.†
«® i)k=0·14 ii)16minutes »
22)! ¥2U91-10b¥
A farmer, who wishes to keep his animals separate, sets up his field so that fences exist at FC, CD and BE, as shown in the diagram below. The side FD is beside a river and no fence is needed there.
Bis in the middle of FCandCD is twice the length ofBE.
i. If FB=x metres and BE=z metres, write down the expressions in terms of xandz for:
a. the area,A, of the field FCD.
b. the amount of fencing,L, that the farmer would need.
ii. If the area of the field is 1200m2, show that the length of fencing required is given by: .
iii. Hence find the values of xandz so that the farmer uses the minimum amount of fencing.†
«® i)a)A=2xz b)L=2x+3z ii)Proof iii)x=30, z=20 »
23)! ¥2U92-8b¥
A particle moves so that its velocity, vmetres per second, at any timet is given by: v=e-2t. Initially the particle is at x=2.
i. Find the acceleration,a, of the particle as a function of time.
ii. What is the acceleration of the particle after 1second?
iii. Find an expression for the displacement, xmetres, of the particle in terms oft.
iv. Find the distance the particle travelled during the first two seconds.
v. Describe what happens to the velocity of the particle for large values oft.†
«® i)a=-2e–2t ii)-2e–2 iii) iv)2·5 m (to 1 d.p.) v) It approaches zero. »
24)! ¥2U92-10b¥
A circular stained glass window of radiusmetres requires metal strips for support along AB,DCandFG. Ois the centre of the window.
i. Copy the diagram and information onto your answer sheet.
ii. If OF=OG=y metres and FB=xmetres, find an expression for y in terms ofx.
iii. The total length of the strips of metal used for support (i.e.AB+DC+FG) isL metres. Show that: L=4x+.
iv. The window will have a maximum strength when the length of the supports is a maximum. Show that when FB=2metres, the window will have a maximum strength.†
«® ii) iii)iv)Proof »
25)! ¥2U93-7d¥
The graph represents the velocity (vm/s) of a particle after t seconds. The particle is moving in a straight line starting from rest.
i. What is the velocity of the particle after 1second?
ii. What is the acceleration of the particle after 3seconds?
iii. When does the particle change directions?
iv. Explain what is represented by the area of the shaded region in the diagram.
«® i)20m/s ii)0m/s iii)2seconds iv)The distance travelled in the first 2seconds. »
26)! ¥2U93-8c¥
In a chemical experiment, the amount of crystals, xgrams, that dissolved in a solution after tminutes was given by: x=20(1-e-kt).
i. After 3 minutes it was found that 10grams of the crystals had dissolved. Show the value of kcorrect to 3significant figures was 0·231.
ii. At what rate were the crystals dissolving after 5minutes? Give your answer to the nearest gram/minute.
«® i)Proof ii)1gram/min »
27)! ¥2U93-10c¥
Boxes in the shape of rectangular prisms are to be constructed from special materials. The width (xmetres) of the base is to be half the length of the base and each box is to hold a volume of 4cubic metres.
Material that is used to build the base and top costs $15per m2. A cheaper material at $10perm2 is used for the four sides.
i. Show that the total cost ($C) of building each box is given by: .
ii. What is the width of the base of the cheapest box that can be constructed?
«® i)Proof ii)x=1m »
28)! ¥2U94-3d¥
When heat was applied to a metallic disc for tseconds, its area, Acm2, increased at a rate given by: .
i. At what rate was the area of the metallic disc increasing at the end of the third second?
ii. Before heat was applied, the area of a metallic disc was 10cm2. Heat was applied to this metallic disc for 3seconds. What was the area of this metallic disc at the end of three seconds?†
«® i)4 cm2/s ii)13 cm2 »
29)! ¥2U94-8c¥
The diagram shows the displacement, xmetres, of a particle from the origin Oafter tseconds for 0£t£2p.