Help with several past Trial Qs (1 Viewer)

[5q7]

I'm practicing past trial exams and there are some questions in the JRAHS Trial from 1994 that I am having trouble with.

Question 3b.
How do I integrate (1 + tan^2 x)dx between pi/4 and 0?

Question 9a.
The gradient function of a curve is given by y'= 2 / (x - 1), if the curve passes through (4,0) find the equation of the curve

Question 9c.
A tank containing 18000 litres of water is to be drained. After t minutes, the rate at which the water is decreasing is given by: dV/dt = -40(30 - t)
i) Derive a formula for the volume of water remaining after t minutes
ii) How much water will be left after 10 mins?
iii) How long will it take for the tank to empty?

 

[Forbidden.]

Q3 b) is a classic ...

By Pythagorean Identity
sin²x + cos²x = 1
tan²x + 1 = sec²x

So
1 + tan²x = 1 + ( sec²x - 1 ) = sec²x


Q9 a)

Integrate then substitute (4,0) into the x and y values respectively into the primitive function AND to solve for the value of the constant and declare that your final answer.

 

[watatank]

Question 9a.
The gradient function of a curve is given by y'= 2 / (x - 1), if the curve passes through (4,0) find the equation of the curve

y'= 2 / (x - 1)

y = 2 * integral [ 1/ (x - 1) ] + c

= 2 * [ ln(x - 1) ] + c

when x =4, y = 0

2 * [ ln(4 - 1) ] + c = 0

2*ln3 + c = 0

.'. c = -2ln3

y = 2[ ln(x - 1) ] - 2ln3

 

[watatank]

Question 9c.
A tank containing 18000 litres of water is to be drained. After t minutes, the rate at which the water is decreasing is given by: dV/dt = -40(30 - t)
i) Derive a formula for the volume of water remaining after t minutes
ii) How much water will be left after 10 mins?
iii) How long will it take for the tank to empty?

i) integrate dV/dt, find the constant by letting V = 18000 and t = 0

ii) using i) sub 10 for t

iii) using i) sub 0 for V

 

[5q7]

Thanks everyone for your help

 

[kloudsurfer]

Q3 b) is a classic ...

By Pythagorean Identity
sin²x + cos²x = 1
tan²x + 1 = sec²x

So
1 + tan²x = 1 + ( sec²x - 1 ) = sec²x


Q9 a)

Integrate then substitute (4,0) into the x and y values respectively into the primitive function AND to solve for the value of the constant and declare that your final answer.

Random q...are pythagorean identities part of 2 unit? Cos I learnt it last yr when i was doing 3 unit and I always assumed it was 3 unit stuff?

 

[watatank]

yes they are.

 

[kloudsurfer]

yes they are.

Damn. Now I have 2 remember them.

Thanks

 

[jb_nc]

you only need sin^2 + cos^2 = 1 all the rest can be derived from that

 

[watatank]

you only need sin^2 + cos^2 = 1 all the rest can be derived from that

yeah

sin^2 + cos^2 = 1

if you divide all by sin^2

1 + cot^2 = cosec^2

if you divide all by cos^2

tan^2 + 1 = sec^2

 

[kloudsurfer]

yeah

sin^2 + cos^2 = 1

if you divide all by sin^2

1 + cot^2 = cosec^2

if you divide all by cos^2

tan^2 + 1 = sec^2

Oh wow. I never caught onto that when i did it the first time around.

Thanks.

 

[jannny]

I'm confused, when do u use the V=(V0) e^kt equation.

coz in question 9c, could you use the dv/dt as k and V0 as 18000?

or we should only use this equation if it said "is proportional to blah blah" or pertaining to population or half life?

thanks