VCE Methods Question Thread!

[luffy]

Hey guys,

Can anyone help me find the x-intercepts to his equation. I'm trouble solving this. Thank you. 

Graph it. The absolute value of the log graph has been taken, so the minimum y-value is 0. However, it has then been shifted up 1. Hence, the minimum y-value is 1. (i.e. range of [1, infinity). )

Therefore, the graph has no x-intercepts.

[Phy124]

Hey guys,

Can anyone help me find the x-intercepts to his equation. I'm trouble solving this. Thank you. 

majimakinsa, the stated graph won't have any x-intercepts.

1. Think about the logarithmic function in it's normal form, but translated 3 units left. It has an asymptote at x= -3 and an intercept at x = -2 (increasing from (-3,inf))

2. Think about the absolute function of this, it also has asymptote x = -3 and an intercept at x = -2(but will be decreasing from (-3,2) and increasing (2,inf) i.e. can only produce y-values of 0 or greater)

3. If the graph in part 2 is translated 1 unit up, instead of producing y-values of 0 or greater, it will produce values of 1 or greater and hence won't have an x-intercept.



Alternatively, try solving it this way:





We know at this point we can't solve the equation because the left hand side will always be positive whilst the right hand side is negative.

edit: beaten by luffy

[Insa]

Ohhhh! I see now. Thanks luffy and ~My♥Little♥Pony!

[dinosaur93]

Hey guys,

Can anyone help me find the x-intercepts to his equation. I'm trouble solving this. Thank you. 

majimakinsa, the stated graph won't have any x-intercepts.

1. Think about the logarithmic function in it's normal form, but translated 3 units left. It has an asymptote at x= -3 and an intercept at x = -2 (increasing from (-3,inf))

2. Think about the absolute function of this, it also has asymptote x = -3 and an intercept at x = -2(but will be decreasing from (-3,2) and increasing (2,inf) i.e. can only produce y-values of 0 or greater)

3. If the graph in part 2 is translated 1 unit up, instead of producing y-values of 0 or greater, it will produce values of 1 or greater and hence won't have an x-intercept.

(Image removed from quote.)

Alternatively, try solving it this way:





We know at this point we can't solve the equation because the left hand side will always be positive whilst the right hand side is negative.

edit: beaten by luffy

Which software did you use to generate these graphs?

[Phy124]

Which software did you use to generate these graphs?

https://www.desmos.com/calculator/c

[Hutchoo]

Just wondering, do 'sign graphs' (NOT sine graphs btw) ever pop up in exams and such?
I'm doing some methods now, and the concept popped up. It's page 153 (Exercise 4e) for Essentials if anyone was wondering.

[pi]

The sign (or signum) function will never pop up in a question, but you may have to take the derivative of an absolute function which will ultimately mean that you have to sketch the associated sign function

Here's a trick to get from the calc screen to the VCAA approved notation (you can't write a sign function in the exam btw):

   <-- On CAS screen if you just derive the absolute function


^What VCAA likes


edit: random line in the latex removed

[Hutchoo]

Sweet.
Yeah the questions asking to draw the sign function were for cubic/quadratic equations. Glad they won't pop up anywhere funky.

[dinosaur93]

Could someone help me verify whether or not Im in the right track?

Could anyone also help me the questions that Ive missed out because Im rather confuse of how to approach the questions, thanks heaps!~


MODELLING TRIGONOMETRIC GRAPHS

QUESTION 1
A top secret satellite is launched into orbit from a remote island not on the equator. When the satellite reaches orbit, It follows a sinusoisal pattern that takes it north and south of the equator. Twelve minutes after it is launched it reaches the farthest point of the equator. The distance north or south of the equator can be represented by the function

where d(t) is the distance of the satellite north of the equator t mins. after being launched.

a.) Show that the launch site is 2369km. Is it far north or far south of the equator?


b.) Is the satellite north of south of the equator after 20 mins.? What is this distance to the nearest kilometres?



c.) When, to the nearest tenth of a minute, will the satellite first be 2500km south of the equator?

2500km south of the equator = below the equator by 2500km





QUESTION 2
The height of a tidal wave approaching the face of the cliff on an island is represented by the equation

where h(t) is the height in metres, of the wave above the normal sea level t mins after the wave strikes the cliff.


a.) What are the maximum and minimum heights of the wave relative to normal sea level?

Maxmimum = 7.5 metres
Minimum = -7.5 metres


b.) What is the period of the function?



c.)How high will the wave be, relative to the normal sea level one minute after striking the cliff?


d.) Normal sea level is 6 metres at the base of the cliff.

i.) For what values of 'h' would the sea bed be exposed?

ii.) How long, to the nearest tenth of a minutes, after the wave strikes the cliff does it take for the sea bed to be exposed

iii.) For how long, to the nearest tenth of a minute, is the sea bed exposed?


QUESTION 3
A city water authority determined that, under normal conditions, the approximate amount of water, W(t), in millions of litres, stored in a reservior t months after May 1, 2003, is given by the formula

a.) sketch the graph of this function over the next 3 years

b.) The authority decided to carry out simulation to determine if they had enought water to cope with a serious fire.

"If on Novermber 1, 2004, there is a serious fire which uses 300, 000 litres of water to bring under control, will the reservoir run dry if water rationing is not imposed?"

i.) Explain how to use the graph in a.) to solve the problem.


ii.) Will the reservoir run dry if water rationing is not imposed? If so, what month will this occur?

[dinosaur93]

MORE TRIG MODELLING

A pain spot X lies on the outer rim of the wheel of a paddle steamer. The wheel has a radius of 3 metres  and it rotates clockwise at a constant rate.

h metres is the distance of X, above the bottom of the boat, t seconds after the wheel begins to rotate.

Point X is seen entering the water every 4 seconds.

At time t=0, X is at its highest position at the top of the wheel

This situation is modelled by

Find a, b, and c

a = 3

b =

c = 1

[InsaneMcFries]

Although it sounds like an easy question, I cannot for the life of me figure it out. :S

"A window is in the shape of an equilateral triangle on top of a rectangle so that the side length of the triangle is equal to the width of the rectangle. The frame of the window is 360 cm in length (i.e. the window's perimeter is 360cm). Find the exact length of the triangle if the area of the window is the maximum possible."

The answer the book gives is

I tried to solve it by solving 360=3x+2y, getting y=180-3x/2, deriving area=x(180-3x/2), letting it =0, then getting x=60, but that just ended up in me zeroing out, I'm not sure how else to attack it.

Obviously the main flaw in my working is that I only found the area of the rectangle, but I used CAS for the area of both and I just got 360 again. :S

[brightsky]

you forgot the equilateral triangle atop the rectangle in your area calculation.

[Phy124]

Resolve the following rational expressions into partial fractions.
 (3x^2+2x+5) / (x^2+2)(x+1)
 AND
 (x^2+2x-13) / (2x^3+6x^2+2x+6)
 
show workingg please.. thanks!

I don't think partial fractions are a part of the methods syllabus?

[ashoni]

oh my bad! posted on wrong thread... LOL

[kensan]

I just need some clarification, so,



Have I simplified this correctly and fully? thanks