Differentiation :Help please! :> (1 Viewer)

[HKHSCstudent]

Please help me , I got stuck in the following questions!

1.
16 Koala bears were released from capavity into a wildlife park. The population P of koalas in the park t years later is given by

P=-t^3+6t^2+16,for 0<t<5
*^=power

a) after how many years was the popualtion a maximum?
b) What was the maximum population?
c)Sketch the curcve P=-t^3+6t^2+16 for 0<5<5
d) When was the population increasing most rapidly?

2. an open rectanglar box has four sides and a base, but no lid. The figure below shows the box which has dimension of 3xcm, 2xcm and a height of y cm.

(a) Write down the formulae for the outer surface area Acm^2 of the box and the volume Vcm^3 contained by the box.
(b) It is known that A+240. Elimnate y to obtain a formula V(x) for the volume as a function of x.
(c) show that x<2( (root of)10 )
(d) Find the value of x for which V is a maximum and verify the maximum value of V is 64 (root 30).

Can' t work out part (c) and (d) in question 2 .





3. a piece of wire of length 5 m is bent to form the hypotenuse and side of a right-angled triangle ABC, as shown in the diagram. Let the length of the side AB be x metres.
a) What is the length of the hypotenuse AC in terms of x?
b) show that the area of the triangle ABC is 1/2 x (root of (25-10x) ) square metres.
C) What is the maximum possible area of the triangle?

4. A cylinder of radius r cm and height h cm is inscribed in a cone with base radius 3 cm and height 10cm, as in the diagram.
a) Show that the volume V of the cylinder is given by

V=(10(22/7)r^2)(3-r) ) / 3

b) Hence find the values of r and h for the cylinder which has maximum volume.

c) What is the maximum volume?


Thanks , I got struck for it a long time!!!!

 

[loser101]

16 Koala bears were released from capavity into a wildlife park. The population P of koalas in the park t years later is given by

P=-t^3+6t^2+16,for 0<5
*^=power

a) after how many years was the popualtion a maximum?

P=-t^3+6t^2+16

Differentiate

P'=-3t^2+12t


Factoring P'=-3t(t-4)

Max when P'=0; t=4 ; 0 < t < 5

i.e 4 years when t is max

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max population => put t=4 into P=-t^3+6t^2+16 = 48


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increasing most rapidly => this means the rate of change of the tangent.

i.e take the 2nd derivative

P'' = -6t+12

Max when P''=0, => t=2

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[ssglain]

All of these questions rely on the same principle: find an expression for one variable in terms of another (this is often given, if not then it requires some simple algebraic manipulation of given conditions to obtain) and then using differentiation to find the rate of change of one variable with respect to another and finally finding the stationary point of the required nature (max or min).

Here's Q1. I'll do the others shortly.

 

[HKHSCstudent]

回覆: Differentiation :Help please! :>

Diagrams for the questions 2.

Question 4

 

[HKHSCstudent]

回覆: Differentiation :Help please! :>


Question 4 Diagram


Question 3 diagram

 

[ssglain]

Sorry.. I totally forgot about this.

 

[Steth0scope]

Your writing is so neat ssglain lol.

 

[ssglain]

Your writing is so neat ssglain lol.

Lol. Thanks.

It does make me wonder how messy your writing must be.