Need help, URGENT maths question: (7 Viewers)

InteGrand · Mar 19, 2016

1008 said:
Thanks so much InteGrand. Just another question, for one sided limits, when we're asked to find lim(x->a+) f(x) or lim(x-->a-) f(x), are we meant to just calculate f(a) in both cases if it is defined, or are we meant to find f(a) and f(-a) respectively?

$\noindent You need to find each limit separately (using the behaviour of the function on the relevant side of $a$). Even if $f(a)$ is defined, there is no need for the limits to equal $f(a)$. And $f(-a)$ is unrelated to these limits.$

1008 · Mar 25, 2016

Ok thanks! I'll post some more questions soon, BTW got 100% in the online calculus test thanks to you! But there is another algebra + calculus test coming up in two weeks...

1008 · Mar 29, 2016

Hey Everyone:

Here are a few additional questions I need help with, if anyone of you guys could help me that would be great:

1) A rower is rowing across a river. His rowing speed is 2km per hour and there is a current flowing in the river at 1km per hour. Find the direction that the rower must row to go directly across the river. If the river is 300m wide, how long will it take him to cross the river?

2) There are three ropes attached to a block of wood and a man is pulling on each rope. The first man is pulling with a force of 200N due north, the second is pulling with a force of 250N due east and the third is pulling with a force of 150N southwest. What is the net force on the block?

3) Town B is 18km N 18 degrees west from Town A. Town C is 25km N 36 degrees E from B. Town D is 20 km S 72 degrees E from C. Town F is 15km South 25 degrees west from D. What are the distance and bearing of Town F from Town A?

4) Find the parametric vector forms for the plane: 5x^(2) - 6x^(3) = 5 where x =((x(1)),(x(2)),(x(3))) in R3

davidgoes4wce · Mar 29, 2016

1008 said:
Hey Everyone:

Here are a few additional questions I need help with, if anyone of you guys could help me that would be great:

1) A rower is rowing across a river. His rowing speed is 2km per hour and there is a current flowing in the river at 1km per hour. Find the direction that the rower must row to go directly across the river. If the river is 300m wide, how long will it take him to cross the river?

2) There are three ropes attached to a block of wood and a man is pulling on each rope. The first man is pulling with a force of 200N due north, the second is pulling with a force of 250N due east and the third is pulling with a force of 150N southwest. What is the net force on the block?

3) Town B is 18km N 18 degrees west from Town A. Town C is 25km N 36 degrees E from B. Town D is 20 km S 72 degrees E from C. Town F is 15km South 25 degrees west from D. What are the distance and bearing of Town F from Town A?

4) Find the parametric vector forms for the plane: 5x^(2) - 6x^(3) = 5 where x =((x(1)),(x(2)),(x(3))) in R3

Do you happen to have answers for them? Just want to compare my answers against the correct ones.

1008 · Mar 29, 2016

davidgoes4wce said:
Do you happen to have answers for them? Just want to compare my answers against the correct ones.

Yep sure, here they are:
1) The rower must row 30 degrees upstream
2) 172 N 57 degrees E
3) Approximately 28 km N 51 degrees 9 minutes E from A
4) x = (0,1,0) + a(1,0,0) + b(0,6/5,1) for a,b in R

I think the answer in 4 could be different and still correct...

InteGrand · Mar 29, 2016

1008 said:
Hey Everyone:

Here are a few additional questions I need help with, if anyone of you guys could help me that would be great:

1) A rower is rowing across a river. His rowing speed is 2km per hour and there is a current flowing in the river at 1km per hour. Find the direction that the rower must row to go directly across the river. If the river is 300m wide, how long will it take him to cross the river?

2) There are three ropes attached to a block of wood and a man is pulling on each rope. The first man is pulling with a force of 200N due north, the second is pulling with a force of 250N due east and the third is pulling with a force of 150N southwest. What is the net force on the block?

3) Town B is 18km N 18 degrees west from Town A. Town C is 25km N 36 degrees E from B. Town D is 20 km S 72 degrees E from C. Town F is 15km South 25 degrees west from D. What are the distance and bearing of Town F from Town A?

4) Find the parametric vector forms for the plane: 5x^(2) - 6x^(3) = 5 where x =((x(1)),(x(2)),(x(3))) in R3

$\noindent 1) Say the river has its banks parallel to the $x$-axis in the Cartesian plane. Say the current is $1$ km/h to the right, so we can draw the current vector $\vec{c}$ as a vector from the origin pointing right 1 unit. Say we want to go from the lower bank to the top. Our velocity vector has a magnitude of 2 (units being km/h). Draw the velocity vector $\vec{v}$ as 2 units in length and making some angle $\theta$ with the $x$-axis (and hence with the current vector). This angle $\theta$ is the direction we want to find. The triangle formed by $\vec{v}$, $\vec{c}$, and the vector connecting the heads of these two vectors is the famous ``30, 60, 90'' right-angled triangle. So the angle $\theta$ is just $60^\circ$. So the direction we need to row in is $60^\circ$ from the bank$.$

$\noindent To find the time taken to traverse the river, observe that the component of the velocity vector in the direction of interest (the $y$-direction, i.e. direction towards the opposite bank) is of length $\sqrt{3}$, using Pythagoras. Hence we are rowing at a rate of $\sqrt{3}$ km/h in that direction (we are also going along the current direction, but as this is perpendicular to the direction of interest, it is irrelevant to the time required). Since time is distance divided by speed, the time to reach the opposite bank is $\frac{300 \text{ m}}{\sqrt{3}\text{ km/h}}=\frac{0.3 \text{ km}}{\sqrt{3} \text{ km/h}}=\frac{\sqrt{3}}{10}\text{ h}$, which is $0.173205\ldots$ hours, or about $10.4$ minutes.$

$\noindent 2) This can be done by simple vector addition. Consider the forces in the $x-y$ plane. Let North be in the positive $y$-direction. The force from the first man is the vector $\vec{F}_1 = 200 (0,1)$ (units of magnitude being newtons for all vectors here). The force from the second man is the vector $\vec{F}_2 = 250 (1,0)$. Now, the magnitude of the force $\vec{F}_3$ from the third man is given to be $\left \|\vec{F}_3 \right \|=150$. The direction is southwest, which is in the direction of this vector: $\vec{v}_\text{SW}:= (-1,-1)$. Normalising, the unit vector in the southwesterly direction is $\hat{u}_\text{SW}:= \frac{1}{\sqrt{2}}(-1,-1)$. Since we have the magnitude of $\vec{F}_3$ and a unit vector in its direction, we have that $\vec{F}_3=\left \|\vec{F}_3 \right \|\hat{u}_\text{SW}=\frac{150}{\sqrt{2}}(-1,-1)$.$

$\noindent Now, the net force vector is just the sum of the three force vectors, by definition. Hence $\vec{F}_\text{net}=\sum \limits _{i=1}^3 \vec{F}_i =200 (0,1) + 250 (1,0) + \frac{150}{\sqrt{2}}(-1,-1) = (0,200) + (250,0) + \left(-75\sqrt{2},-75\sqrt{2}\right)=\left(250-75\sqrt{2},200-75\sqrt{2}\right)$.$

$\noindent 3) We seek the vector $\overrightarrow{AF}$. Consider vectors in the $\mathbb{R}^2$ plane with the positive $y$ direction being North. Using the $\bold{i}$ and $\bold{j}$ vector notation, we deduce from the given data that:$

$\bullet \overrightarrow{AB} = 18\cos(108 ^\circ)\bold{i}+18\sin(108^\circ)\bold{j}$

$\bullet \overrightarrow{BC} = 25\cos(54 ^\circ)\bold{i}+25\sin(54^\circ)\bold{j}$

$\bullet \overrightarrow{CD} = 20\cos(-18 ^\circ)\bold{i}+20\sin(-18^\circ)\bold{j}$

$\bullet \overrightarrow{DF} = 15\cos(245 ^\circ)\bold{i}+15\sin(245^\circ)\bold{j}$

$Now,$

$\overrightarrow{AF} =\overrightarrow{AB}+ \overrightarrow{BC}+ \overrightarrow{CD}+\overrightarrow{DF} =(18\cos(108^\circ)+25\cos(54^\circ)+20\cos(18 ^\circ)+15\cos(245^\circ))\bold{i}+(18\sin(108 ^\circ)+25\sin(54^\circ)-20\sin(18^\circ)+15\sin(245^\circ))\bold{j}$

$\noindent You can find the length and direction of this vector now, since the components are known.$

$\noindent 4) The plane is $5x_2- 6x_3 = 5$. Since $x_1$ doesn't appear, we can set it equal to a free parameter, so let $x_1 = \xi$, $\xi \in \mathbb{R}$. We can also set $x_2 = \eta$, $\eta \in \mathbb{R}$. Now from the Cartesian equation, we have $x_3 = \frac{5}{6}x_2 - \frac{5}{6} = \frac{5}{6}\eta - \frac{5}{6}$. Hence a parametric equation for the plane is:$

$\begin{align*}\vec{x} &= \left(x_1,x_2,x_3\right) \\ &= \left(\xi, \eta, \frac{5}{6}\eta - \frac{5}{6}\right) \\ &\Rightarrow \boxed{\vec{x} = \left(0,0,- \frac{5}{6}\right) + \xi \left(1,0,0 \right) +\eta \left(0,1,\frac{5}{6}\right), \xi,\eta \in \mathbb{R}}.\end{align*}$

InteGrand · Mar 29, 2016

davidgoes4wce · Mar 29, 2016

For Question 2 I drew out a diagram and worked out the net force in the vertical and horizontal directions. I then applied the Pythagoras theorem to work out the net total Force and direction.

davidgoes4wce · Mar 29, 2016

With Question 1, I split up the velocities into horizontal components (current) and the applied force of the rower (diagonal) to work out the resultant velocity.

In the question they also mention the vertical distance 'directly' to the river , by using the formula speed=distance/time to work out the time required to cross the river

$time=\frac{distance}{speed}= \frac{0.300 \ km}{\sqrt{3} \frac{km}{hr}}= \frac{0.300 hr}{\sqrt{3}}$

$$ The angle is obtained by equating tan $ \theta= \frac{1}{\sqrt{3}}, $ which in this case is $ \theta=30^\circ$

1008 · Mar 29, 2016

Wow thanks guys! Really appreciate it.

1008 · Mar 29, 2016

Got another question though:

5) You are an auditor for a company whose four executives make regular business trips on four routes and you suspect that at least one of the executives has been overstating her expenses. You don't know how much it costs to travel each route, but you know that it is the same for all the executives. You know the number of trips each executive made on each route in a certain period and you know the total expenses claimed by each executive for this period. If the numbers of trips are as shown in the table below, do you have sufficient information to be sure that someone is cheating? State your reasoning clearly

InteGrand · Mar 29, 2016

1008 said:
Got another question though:

5) You are an auditor for a company whose four executives make regular business trips on four routes and you suspect that at least one of the executives has been overstating her expenses. You don't know how much it costs to travel each route, but you know that it is the same for all the executives. You know the number of trips each executive made on each route in a certain period and you know the total expenses claimed by each executive for this period. If the numbers of trips are as shown in the table below, do you have sufficient information to be sure that someone is cheating? State your reasoning clearly

$\noindent Let $x_i$ be the cost for route $i$ ($i=1,2,3,4$). Let $b_A,b_B,b_C,b_D$ be the amounts claimed in total by Executive $A,B,C,D$ respectively. Then we must have the following matrix equation: $A\vec{x} = \vec{b}$, where $A$ is the matrix (table) given to us, $\vec{x} = \left(x_1,x_2,x_3,x_4\right)$, and $\vec{b} = \left(b_A,b_B,b_C,b_D\right)$.$

$\noindent It can be shown using Gaussian Elimination that $A$ row reduces to the identity matrix. Hence no matter what $\vec{b}$ is, there indeed exist solutions for the costs $x_i$; if it turned out there didn't exist solutions, we could be quite there was someone cheating, because of course we know there need to exist solutions, since the routes obviously do have costs. If there were no solutions, it would be because someone was reporting a false value for their total cost $b$. However, even if solutions did exist, there could be cheating going on. In practice, the costs per trip should be positive and integers. If the solutions came out negative, it would be likely cheating was occurring, or some $b$ value was mis-presented etc. Did they provide us with the $b$ values? It says we know them.$

$\noindent (In this question, `cheating' simply refers to there being a mistake in the $b$ values. We can find out whether there is a mistake, but obviously can't know just from this data what the mistake's cause was, i.e. if the mistake was caused by actual lying, or simply things like transcriptions errors when recording the $b$ values etc.)$

1008 · Mar 29, 2016

InteGrand said:
$\noindent Let $x_i$ be the cost for route $i$ ($i=1,2,3,4$). Let $b_A,b_B,b_C,b_D$ be the amounts claimed in total by Executive $A,B,C,D$ respectively. Then we must have the following matrix equation: $A\vec{x} = \vec{b}$, where $A$ is the matrix (table) given to us, $\vec{x} = \left(x_1,x_2,x_3,x_4\right)$, and $\vec{b} = \left(b_A,b_B,b_C,b_D\right)$.$

$\noindent It can be shown using Gaussian Elimination that $A$ row reduces to the identity matrix. Hence no matter what $\vec{b}$ is, there indeed exist solutions for the costs $x_i$; if it turned out there didn't exist solutions, we could be quite there was someone cheating, because of course we know there need to exist solutions, since the routes obviously do have costs. If there were no solutions, it would be because someone was reporting a false value for their total cost $b$. However, even if solutions did exist, there could be cheating going on. In practice, the costs per trip should be positive and integers. If the solutions came out negative, it would be likely cheating was occurring, or some $b$ value was mis-presented etc. Did they provide us with the $b$ values? It says we know them.$

$\noindent (In this question, `cheating' simply refers to there being a mistake in the $b$ values. We can find out whether there is a mistake, but obviously can't know just from this data what the mistake's cause was, i.e. if the mistake was caused by actual lying, or simply things like transcriptions errors when recording the $b$ values etc.)$

No, I'm afraid they don't give anything else with the question. I suppose they just want us to deduce if we would be able to find who was overstating their expenses supposing we knew the b values.

BTW how do you use LaTex in the reply form?

InteGrand · Mar 29, 2016

1008 said:
No, I'm afraid they don't give anything else with the question. I suppose they just want us to deduce if we would be able to find who was overstating their expenses supposing we knew the b values.

BTW how do you use LaTex in the reply form?

To use LaTeX on this site, we enclose LaTeX code in tex tags. There's a Short Guide to LaTeX on this site here: http://community.boredofstudies.org/14/mathematics-extension-2/234259/short-guide-latex.html

e.g. To get $x^2$ , write [ tex] x^2 [ /tex] (without the spaces I put in the tex tags; I put the spaces in so that you can see the tags, as otherwise it'd get rendered as LaTeX).

jathu123 · Mar 29, 2016

oh and if you're having trouble typing the latex code, use this website to generate it easily : https://www.codecogs.com/latex/eqneditor.php if you haven't already

1008 · Mar 29, 2016

Thanks! Also @InteGrand, Wouldn't this be correct, I just formed this augmented matrix:
$\begin{bmatrix}\left.\begin{matrix} 0& 1 & 1 & 2 & \\ 1& 2& 0& 1& \\ 3& 4&0 &1 & \\ 2&1 &3 &3 &\end{matrix}\right|\begin{matrix}A& \\ B& \\ C& \\ D& \end{matrix}\end{bmatrix}$
Reducing it to row echelon form:

$\begin{bmatrix}\left.\begin{matrix} 1& 2 & 0 & 1 & \\ 0& 1& 1& 2& \\ 0& 0& 2 &2 & \\ 0&0 &0 &1 &\end{matrix}\right|\begin{matrix}B& \\ A& \\ C-3B+2A& \\ D+7B-3A-3C& \end{matrix}\end{bmatrix}$

But I didn't know what to do next.

As only the b column is non-leading there is a unique solution, so the possibility of having no solution is ruled out. However, the answers at the back say "Perhaps if the costs are negative or very large then you can be sure that someone is cheating". I get the bit about costs being negative thanks to your explanation, but what of the costs being very large?

1008 · Mar 29, 2016

Oh, excuse my LaTex code, I'm still learning!

Drongoski · Mar 29, 2016

For (1):

Simply solve vector triangle (right -angled) with hypotenuse = 2 and shortest side 1. So resultant speed (2nd shortest side) = sqrt(3). The 2 other angles in said triangle are 30 deg and 60 deg resp. Hence must row 30 deg up river.

So time taken should be 0.3/sqrt(3) = 0.1732 . . . hr = 10.3923 . . . minutes.

ps
ha ha. Didn't see all the earlier-posted solutions, as I completely missed Page 2.

InteGrand · Mar 29, 2016

1008 said:
Thanks! Also @InteGrand, Wouldn't this be correct, I just formed this augmented matrix:
$\begin{bmatrix}\left.\begin{matrix} 0& 1 & 1 & 2 & \\ 1& 2& 0& 1& \\ 3& 4&0 &1 & \\ 2&1 &3 &3 &\end{matrix}\right|\begin{matrix}A& \\ B& \\ C& \\ D& \end{matrix}\end{bmatrix}$
Reducing it to row echelon form:

$\begin{bmatrix}\left.\begin{matrix} 1& 2 & 0 & 1 & \\ 0& 1& 1& 2& \\ 0& 0& 2 &2 & \\ 0&0 &0 &1 &\end{matrix}\right|\begin{matrix}B& \\ A& \\ C-3B+2A& \\ D+7B-3A-3C& \end{matrix}\end{bmatrix}$

But I didn't know what to do next.

As only the b column is non-leading there is a unique solution, so the possibility of having no solution is ruled out. However, the answers at the back say "Perhaps if the costs are negative or very large then you can be sure that someone is cheating". I get the bit about costs being negative thanks to your explanation, but what of the costs being very large?

Well if the costs (solutions for x1, x2, x3, x4 from my post) came out to be very large (say 1,000,000 or something), that'd be fishy, because clearly a trip wouldn't cost that much. Basically, if the costs came out to be anything "unreasonable", something must be up.

1008 · Mar 29, 2016

InteGrand said:
Well if the costs (solutions for x1, x2, x3, x4 from my post) came out to be very large (say 1,000,000 or something), that'd be fishy, because clearly a trip wouldn't cost that much. Basically, if the costs came out to be anything "unreasonable", something must be up.

Alright, thanks!

Need help, URGENT maths question: (7 Viewers)

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