2u Mathematics Marathon v1.0 (1 Viewer)

[skyrockets1530]

Question 63
Integrate 3/Sqrt{e^x} dx

Spoiler

Integral 3/Sqrt{e^x} dx
= 3/e^x/2
= 3e^-x/2
now integrate
= 3e^-x/2 / -.5
= -6e^-x/2 + C

Find the equation of the normal to the curve y=tan²x at x=4

 

[skyrockets1530]

Ahh. someone beat me to it, and i appear to be wrong, meh

 

[icycloud]

Ahh. someone beat me to it, and i appear to be wrong, meh

Ahh your answer and mine are equivalent, except I did it a really long way >_> Can't believe I didn't see the simplification there lol... I've been doing too many harder integration problems.

 

[skyrockets1530]

Question 64
A glass has a shape obtained by rotating part of the parabola x = y^2 / 30 about the y-axis. The glass is 10 cm deep. Find the volume of liquid which the glass will hold.

Spoiler

( x = y^2 / 30 )²
x² = y⁴ / 900
pi∫ between 10 and 0 of y⁴ / 900
= pi(1/900)(y⁵/5)
= pi[y⁵/4500]10,0
= pi[100000/4500 - 0/4500]
= 22.22pi cm³
that should be right

the population of a city is increasing according to the formula P = Ae^kt
where p is the population, and t is the time in years
the population in 2005 is 60000, while the population is projected to be 87000 in the year 2045 - find hte rate at which the population is growing, and the population in the year 1990

 

[skyrockets1530]

Ahh your answer and mine are equivalent, except I did it a really long way >_> Can't believe I didn't see the simplification there lol... I've been doing too many harder integration problems.

ahh, cool- i always simplify first, cause its the only thing i ever think of doing, atleast you'll be good for the harder ones

 

[DeanM]

the population of a city is increasing according to the formula P = Ae^kt
where p is the population, and t is the time in years
the population in 2005 is 60000, while the population is projected to be 87000 in the year 2045 - find hte rate at which the population is growing, and the population in the year 1990

Spoiler

in 2005 pop = 60 000, in 2045 pop = 87 000
so 87 000 = 60 000e^40k
divide both sides by 60 000
therefore 1.45 = e^40k
40k = ln1.45
divide by 40
k = 0.00929 ( to 5 decimal places )

P= 60 000e^-15x0.00929
=52 195 people in 1990

Next question: solve 2sin^2x - 3sinx - 2 = 0 x is between 0 and 2pi

 

[icycloud]

Next question: solve 2sin^2x - 3sinx - 2 = 0 x is between 0 and 2pi

Spoiler

2sin^2 (x) - 3sin(x) - 2 = 0

Let u = sin(x)

Equation becomes:

2u^2 - 3u - 2 = 0
(2u-4)(2u+1) / 2 = 0
(u-2)(2u+1) = 0

Therefore, u = -1/2, 2
sin(x) = -1/2, 2

sin(x) = 2 yields no solutions
sin(x) = -1/2,
x = arcsin (-1/2)
= 7pi/6, 11pi/6 (0<=x<=2pi) #

Question 66:
Find the volume of solid generated when the curve y = ln(x) is rotated about the y-axis between y=0 and y=1.

 

[rama_v]

Solution to Question 66

Spoiler

y = ln(x) is rotated about the y-axis between y=0 and y=1.

V = pi [Int from 0 to 1] x² dy
y = ln (x)
x = e^y
x² = e^2y

V = pi [Int from 0 to 1] e^2y dy
= pi [1/2 e^2y ]¹₀
= pi/2 (e² - e⁰)
= pi/2 (e² -1) u³

Question 67
Express log₃8 / log₃2 as an integer

 

[DeanM]

Question 67
Express log₃8 / log₃2 as an integer

Spoiler

ive got afew different answers.. but i think this one is correct
change log₃8 into log₃2^3

so then we put the power out the front so;
3log₃2 / log₃2
the logs cancel and where left with 3 ???

 

[rama_v]

Thats correct. You can also use the change of base formula, u get the same answer

log₃8 / log₃2 = log₂8
= log₂2³
= 3log₂2
= 3

 

[DeanM]

ahh yea... thanks !
okay question 68
lets keep doing logs shall we...
nice and eay one, just to boost our confidence for monday...
simplify 4log3 - log27

 

[Riviet]

simplify 4log3 - log27

Spoiler

4log3-log27=4log3-log3³
=4log3-3log3
=log3

Next one, Integrate (x+3)/(x²+3x)

 

[icycloud]

Next one, Integrate (x+3)/(x^2+3x)

Spoiler

∫(x+3)/(x^2+3x) dx
= ∫(x+3)/x(x+3) dx
= ∫dx/x (x not equal to -3 or 0)
= ln(x) + C (x>0) #

Question 70
Differentiate y = xln(x)
Hence or otherwise, find ∫Sqrt(13) ln(x) dx.

 

[Nerd Queen]

Question 70
Differentiate y = xln(x)
Hence or otherwise, find ∫Sqrt(13) ln(x) dx.

Ok this is probably WAY rong but ... could sum 1 check it?

Spoiler

y=xln(x)
y' = x(1/x) + ln(x)*1
= 1 +ln(x)

∫ sqrt(13)ln(x)
= (1/x)*sqrt(13)
= (sqrt[13]/x)

sorry thats probably wrong i have a couple o questions though, how do u drop numbers so that they go to the base of, like if you have log to the base of 5 how do u drop the little 5 to read like that? and how do u do the integrate sighn thing?

anyway: if i must - the next question can be:

A man earns $41500 in 1994 and invests 15% of his earnings in an account earning 10% interest per annum, compounded annualy. How much does he earn oh his investment at the end of 5 years?



K sorry just read over my answer: makes no sense at all: IGNORE ME

 

[DeanM]

Ok this is probably WAY rong but ... could sum 1 check it?



sorry thats probably wrong i have a couple o questions though, how do u drop numbers so that they go to the base of, like if you have log to the base of 5 how do u drop the little 5 to read like that? and how do u do the integrate sighn thing?

anyway: if i must - the next question can be:

A man earns $41500 in 1994 and invests 15% of his earnings in an account earning 10% interest per annum, compounded annualy. How much does he earn oh his investment at the end of 5 years?



K sorry just read over my answer: makes no sense at all: IGNORE ME

lol yea... do you mean 15% of $41500, so $6625 ???

 

[Nerd Queen]

lol yea... do you mean 15% of $41500, so $6625 ???

yep. Theres a lesson 2 b learnt with that question! while i was doing it i didnt READ IT PROPERLY and so i did it as 41500 and 15% interest!!!! i dont think i need to say that I WAS WRONG!

 

[DeanM]

Question 71
A harder probability question :
The probability that a particular type of scrub will flower in the first year after plating is 1/4.
How many of these scrubs need to planted in order to be more than 90% certain of having atleast one flowering in the first yr?

 

[word.]

typing log[sub]5[/sub]x will output log₅x.
and use [sup] for superscripts

Answer to 70:

Spoiler

d/dx xlnx
= x*1/x + lnx
= 1 + lnx

so, ∫Sqrt(13)*ln(x)dx = sqrt(13)(xlnx - ∫1dx)
= Sqrt(13)(xlnx - x) + C

Answer to Nerd's question:

Spoiler

15% of $41500 = $6225
after 5 years his investment is worth 6225 * 1.1⁵ = $10025.42

Answer to 71:

Spoiler

P(flower) = 1/4
P(n shrubs not flowering) = 0.75ⁿ
0.1 = 0.75ⁿ
n = ln0.1/ln0.75 = 8.003
soo, technically the answer should be 9 shrubs.

Question 72
Find ∫[(4x⁵ + 1)/4x⁴]dx.

 

[icycloud]

Question 72
Find ∫[(4x^5 + 1)/(4x^4)]dx.

Spoiler

∫[(4x^5 + 1)/(4x^4)]dx
= ∫((4x^5)/(4x^4))dx + ∫1/(4x^4) dx
= ∫x dx + 1/4 ∫ x^(-4) dx
= (x^2) / 2 + 1/4 * (x^(-3) / -3) + C
= x²/2 - 1/(12x³) + C #

Question 73
What is the probability of rolling three odd numbers in a row on a fair eleven-sided dice marked with the numbers 1 to 11?

 

[Riviet]

Q73 answer:

Spoiler

P(odd)=6/11
.: p(odd, odd, odd)=(6/11)³
=216/1331

Q74: A army base has 2 defence guns. The first one has a success rate of 0.8 of destroying incoming enemy aircraft and the second has a success rate of 0.9.
a) Find the probability of an enemy aircraft passing through both guns without being shot down.
b) Find the probability of either the first or second gun shooting down the first 100 enemy aircraft that attack the army base.