2u Mathematics Marathon v1.0 (1 Viewer)

[skyrockets1530]

cant resist the simpsons...

Spoiler

sub x=2, 3, 4 into ln(x-1) to get the following...
1/3 { 0 + (4x0.6931) + 1.0986 }
~1.2904

someone else can post a question.. im off to bed.. for real this time..

Ok, i'll post a question

Serena invests 8000 at the begginning of each year into her supperannuation account which pays 10% pa compounded half-yearly, after 10 years, serena reduces her investment to 5000$, find the amount in the supper account after 20 years.

 

[skepticality]

Question 85

There are two points on a wall of a building, A and B. The two points, A and B are 200meters apart. A pole has the perpendicular distance to the building 50m and point A is placed such that the line drawn from A to the pole is perpendicular to the wall. A person standing has the perpendicular distance to the building 80m and point B is placed such that the line drawn from B to the person is perpendicular to the wall. The two perpendicular lines drawn to the wall are 200m apart. The person is sqrt[40900] meters from the pole. The person is asked to reach the pole as fast as he can, ie with minimal distance. However he must touch the wall first. Find the point on the wall the person touches, by expressing it as the distance from B.

theres actually a dumbass(proper) way and a cheapass way

 

[sunjet]

q83 off the top of my head, check it

Spoiler

1/3 . ln48

 

[skyrockets1530]

Alright then DeanM, here goes:
Question 84
Find the equation of the tangent to the curve y = e^xln(x) at x = 2.

Spoiler

y = e^xln(x)
y' = e^xlnx + e^x/x
m = e²ln2 + e²/2
y-(e²ln2) = (e²ln2 + e²/2)(x-2)
y = (e²ln2 + e²/2)(x-2) - e²ln2
xe²ln2 + xe²/2 - 2e²ln2 - e² - y - e²ln2 = 0
thats too long

already posted a question

 

[yabby]

Alright then DeanM, here goes:
Question 84
Find the equation of the tangent to the curve y = e^xln(x) at x = 2.

Spoiler

y=e^xlnx
using product rule
dy/dx= e^x.1/x + e^x.lnx
sub x=2
dy/dx= e^2.1/2 +e^2.ln2

When x=2
y= e^2ln2

Sub into formula y-y1=m(x-x1)
y - (e^2.ln2) = (1/2.e^2+e^2.ln2)(x-2)
y= (1/2.e^2+e^2.ln2)(x-2) + (e^2.ln2)
Put that in the calc, and there u go

Umm there was 2 other questions..so i wont add one to make it even more confusing

 

[icycloud]

thats too long

Sorry about that, you don't have to fully expand the equation.

 

[yabby]

lol sky,doh! u did it way fsater then me..anyways how do u do those square signs..?

 

[skyrockets1530]

Sorry about that, you don't have to fully expand the equation.

haha, thats ok, were just taught to always put it in general form

 

[skyrockets1530]

lol sky,doh! u did it way fsater then me..anyways how do u do those square signs..?

haha, for those ² things just do, sup in brackets and for the ones below, like ₂ just do sub in brackets, and close the tags with the /, just click 'reply with quote' and you'll see em

 

[skepticality]

Question 85

There are two points on a wall of a building, A and B. The two points, A and B are 200meters apart. A pole has the perpendicular distance to the building 50m and point A is placed such that the line drawn from A to the pole is perpendicular to the wall. A person standing has the perpendicular distance to the building 80m and point B is placed such that the line drawn from B to the person is perpendicular to the wall. The two perpendicular lines drawn to the wall are 200m apart. The person is sqrt[40900] meters from the pole. The person is asked to reach the pole as fast as he can, ie with minimal distance. However he must touch the wall first. Find the point on the wall the person touches, by expressing it as the distance from B.

theres actually a dumbass(proper) way and a cheapass way

Answer is..

Spoiler

123

for those interested

 

[skyrockets1530]

I'll answer my own question...
Serena invests 8000 at the begginning of each year into her supperannuation account which pays 10% pa compounded half-yearly, after 10 years, serena reduces her investment to 5000$, find the amount in the supper account after 20 years.

Spoiler

First do the first ten years
deposit is made once a year and interest is paid twice yearly
A₁ = 8000(1.05)
A₂ = [8000(1.05)]1.05
= 8000(1.05²)
A₃ = 8000(1.05³) + 8000(1.05)
A₄ = 8000(1.05⁴) + 8000(1.05²)
A₄ = 8000(1.05⁴) + 8000(1.05²)
A_n = 8000(1.05ⁿ + 1.05^n-2 + 1.05^n-4 + ....)
A₂₀ = 8000(1.05²⁰ + 1.05¹⁸ + 1.05¹⁶ + .... + 1.05²)
A₂₀ = 8000[(1.05²(1.1025¹⁰ - 1)) / .1025]
NOTE: the rate is 1.1025 as the series goes 1.05² + 1.05⁴...
A₂₀ = 142,264.2513
Now for the second 10 years
A₂₁ = [142,264.2513 + 5000]1.05
A₂₂ = 147,264.2513(1.05²)
A₂₃ = 147,264.2513(1.05³) + 5000(1.05)
A₂₄ = 147,264.2513(1.05⁴) + 5000(1.05²)
...
A₄₀ = 147,264.2513(1.05²⁰) + 5000[(1.05²(1.1025⁹ - 1)) / .1025]
Note the second series of interest only has 9 terms
A₄₀ = $466,384.57 total value of super account
That should be all correct, please let me know if theres an error

Next question:
given that d/dx = e^x2 = 2xe^x2 , evaluate the integral of xe^x2 between 1 and 0

 

[rsingh]

Question: Find the integral of 1/x between 1 and 0.

Thanks.

 

[skyrockets1530]

Question: Find the integral of 1/x between 1 and 0.

Thanks.

[lnx]1,0
ln1 - ln0
=0

 

[skepticality]

Next question:
given that d/dx = e^x2 = 2xe^x2 , evaluate the integral of xe^x2 between 1 and 0

Spoiler

d/dx = e^x2 = 2xe^x2
therefore 1/2 [integral] boundary 0..1 2xe^x2 = [e^x2]¹₀
=2[e^x2]¹₀
=2(e-1)

Next Question (do it guys..)

There are two points on a wall of a building, A and B. The two points, A and B are 200meters apart. A pole has the perpendicular distance to the building 50m and point A is placed such that the line drawn from A to the pole is perpendicular to the wall. A person standing has the perpendicular distance to the building 80m and point B is placed such that the line drawn from B to the person is perpendicular to the wall. The two perpendicular lines drawn to the wall are 200m apart. The person is sqrt[40900] meters from the pole. The person is asked to reach the pole as fast as he can, ie with minimal distance. However he must touch the wall first. Find the point on the wall the person touches, by expressing it as the distance from B.

theres actually a dumbass(proper) way and a cheapass way

 

[rama_v]

Spoiler

d/dx (e^x2) = 2x e^x2
.: ∫xe^x2 dx
= (1/2) e^x2
= (1/2)(e¹ - e⁰)
= (1/2)(e - 1)

Next question:
A car, of initial value $15000 depreciates at a rate of 3% p.a.
Calculate the car's worth after five years.

 

[Twiggyy]

question:
A car, of initial value $15000 depreciates at a rate of 3% p.a.
Calculate the car's worth after five years.

15000(1-0.97)^5 = $12881


next q -

What values of k are always positive?
x^2 - 2kx + 6k

 

[icycloud]

What values of k are always positive?
x^2 - 2kx + 6k

Spoiler

Positive definite: a > 0 and Discriminant < 0

a = 1 > 0

Discriminant = (-2k)^2 - 4(6k)
= 4k^2 - 24k
< 0

4k^2 - 24k < 0
k^2-6k < 0
k(k-6) < 0

Therefore, 0 < k < 6 #

There's already an unanswered question above, post #174.

 

[sunjet]

15000(0.97)^5

 

[lie_stella]

Next Question

Solve ln(7x-12) = ln x

 

[Kutay]

Next Question

Solve ln(7x-12) = ln x

Spoiler

well because they got the same base they cancel out ---> log laws there for it becomes 7x - 12 = x and x = 2

Hope thats right, correct me if i am wrong

Next Question:

The point P(x,y) lies on the line y = 2x - 9 and is equidistant from A(0,1) and B(2,3) . Find the Equation of the Locus P

Answer:

Spoiler

ROOT {(x-0)^{2 + (y-1)2} = ROOT {(x-2)2 + (y-3)2}
Then jus square both sides and u you in up getting 4y +4x -12 = 0 by putting to one side}