Assume a function y = f(x) , differentiable everywhere. Now we have for some Δx

Δy = f(x + Δx) - f(x)

The differential of x, is defined as “dx”, can be any real number, and dx = Δx

The differential of y, is defined by “dy” and

dy = f’(x) dx

Clearly,

Δy ≈ dy, depending on the magnitude of Δx.

In calculus an expression like “dx” usually denotes something infinitesimally small.

Why is it necessary to have dy and dx used as real...

Question about the differential in Calculus]]>

$\tiny{\textbf{6.8.7}}$ Kiaser HS

Population Growth The population of a southern city follows the exponential law

(a) If N is the population of the city and t is the time in years, express N as a function of t.$N(t)=N_0e^{kt}$

(b) If the population doubled in size over an 18-month period and the current population is 10,000, what will

the population be 2 years from now?

$\begin{array}{rl}

2&=e^{k(1.5)} \\

\ln 2&=k...

6.8.7 Population Growth]]>

a) Find the magnitude of the avg resistance force acting on the golf ball.

I got the and resistance =0.2N

The ball travels a further 105.8m along a curved path to land on the green. The green is 4m lower than the tee. The average resistance remains unchanged

b) find the...

Work energy principle and power]]>

Electrical Resistance of a Wire

The electrical resistance of a wire varies directly with the length of the wire and inversely with the square of the diameter of the wire.

If a wire 432 feet long and 4 mm in diameter has a resistance of 1.24 $\Omega$

find the length of a wire of the same material whose resistance is 1.44 $\Omega$ and whose diameter is 3 mm

y varies inversely with x $\quad y=\dfrac{k}{x}$

y varies directly with x $\quad y=kx$

OK not real sure how to set this...

2.5.1 varies directly with the length of the wire and inversely with the square of the diameter of the wire.]]>

Let $ ( , ):V \times V \rightarrow \mathbb{R} $ be a real-valued non-degenerate inner product on the real vector space $V$.

Given, for all $v \in V$ we have $(v,v) \geq 0$

Now prove that if $(x,x)=0$ then $x=0$ for $x \in V$, that is, prove that the inner product is Euclidean.

I think it is easy, but I cannot find it. Thank you.]]>

However, this problem appears in my problems book quite early right after the definition of $\epsilon$-language definition of limit of a sequence, the reader is supposed not to know anything about continuity.

My question is: Is there any proof for this result in $\epsilon-delta$ language that is more elementary?

Please help me.

Thanks.]]>

a)Prove that a sequence $\{a_n\}$ converges on $X$ if and only if the sequence $\{p_{\alpha}(a_n)\}$ converges on $X_{\alpha}$ for all $\alpha \in I$.

b) Let $I$ the set of all sequences $\alpha:\mathbb{Z}_{\geq 1}\rightarrow \{-1,1\}$. Let the sequense $a_n=\prod_{\alpha...

Prove that the sequence does not have a convergent subsequence]]>

PS: I JUST NEED THE ANSWER AND SOLUTIONS]]>

- Fund 1, which pays a rate of 8.6% per year;
- Fund 2, which pays a CPI rate (inflation rate) + 3.75% per year.

a) Liam should invest in fund 1 and...

Basic problem about investment (basic percentages, and econ/finance).]]>

Let $H:=\{(x,y)\in \mathbb{R}^2\;:\; y>0\}$ and $R=\{(x,y)\in \mathbb{R}^2\;:\; y=0\}$. Notice that $\tau$ he topology of $H$ induced by $ \mathbb{R}^2$. Let the set $X(S):=H\cup S$, where $S\subset R$.

Define the topology $\tau^{\star}$ over the set $X(S)$ as the generated topology $\tau$ and the set $\mathcal B$, where $\mathcal{B}$ is formed by all the sets of form$\{x\}\cup B$, where $x\in S$ and $B\subset H$ is an open ball tangent to$R$...

Prove that it is first countable]]>

payout= 1:1

Using 4 rounds the strategy wins 1 of 4 rounds 90% of the time, the other 10% it loses all 4 rounds

Rules: if any round is won that game is over, else keep betting till all 4 rounds over

The question is how to I change the betting method to turn positive results

Key= g=game, r=round, b=betting amount, pl=current running Profit/lose

Ex1: 10 games of 4 rounds

g1: r1: b=1, lost, pl=-1 | r2: b=2, lost, pl=-3 | r3...

Betting Strategy]]>

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Consider that $\{a,\,b,\,c,\,d\}\in \mathbb{R} $ and that $a^2+b^2=c^2+d^2=1$ and $ac+bd=0$, evaluate $ab+cd$.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!]]>

Find $x \quad |2x-5|<9$

divide into 2 solution sets

$\begin{array}{rl|rl}

(2x-5)&=9&-(2x-5)&=9 \\

2x&=14&-2x&=4 \\

x&=7&x&=-2

\end{array}$

x is $-2<x<7$

kinda maybe...]]>

a) Find the speed of the ball just before it hits the water

I got the ans 22.5m/ s

The water immediately absorbs 8J of energy from the ball. The ball then sinks vertically downwards to reach the bottom of the pond. The resistance acting on the ball...

Work energy principle and power]]>

An object is propelled vertically upward with an initial velocity of 20 meters per second.

The distance s (in meters) of the object from the ground after t seconds is

$s=-4.9t^2+20t$

(a) When will the object be 15 meters above the ground?

$15=-4.9t^2+20 \implies -4.9t^2 =-5$

ok there is no term b so decided not to use quadratic formula

so far....

$49t^2=50$

(b) When will it strike the ground?

(c) Will the object reach a height of 100 meters]]>

payout= 1:1

Using 4 rounds the strategy wins 1 of 4 rounds 90% of the time, the other 10% it loses all 4 rounds

Rules: if any round is won that game is over, else keep betting till all 4 rounds over

The question is how to I change the betting method to turn positive results

Key= g=game, r=round, b=betting amount, pl=current running Profit/lose

Ex1: 10 games of 4 rounds

g1: r1: b=1, lost, pl=-1 | r2: b=2, lost, pl=-3 |...

Gambling Strategy Help]]>

My attempt to answer this question: Let the actual velocity of wind is $\vec{v}=x\hat{i} + y\hat{j}$ where $\hat{i}$ and $\hat{j} $ represents velocities of 1KM per hour towards east and north respectively. As the person is going northeast with a velocity of 6KM/hr, his actual velocity is $ 3\sqrt{2} \hat{i} +3\sqrt{2}\hat{j}$

Then the velocity of wind relative to person is $x\hat{i} -y\hat{j}- 3\sqrt{2}\hat{i} -3\sqrt{2}\hat{j}$...

proof of known velocity of wind]]>

I don't understand this. Pls help]]>

i)How many students will be aware of the rumour after 7 days.

ii)How long will it take for 850 students to hear the rumour]]>

How to answer this question? I am working on this question. Any math help, hint or even correct answer will be accepted.]]>

How to answer this question?]]>

Find all complex numbers x which satisfy the given condition

$\begin{array}{rl}

1+x&=\sqrt{10+2x} \\

(1+x)^2&=10+2x\\

1+2x+x^2&=10+2x\\

x^2-9&=0\\

(x-3)(x+3)&=0

\end{array}$

ok looks these are not

a) Find the speed of the ball when it is 1m above the ground

Increase in GPE= loss of KE

mgh=1/2mu^2-1/2mv2

10×(1.45-1)= 1/2×15^2-v^2/2

V= 14.7 m/ s

But text book ans is 14.3m/s

Show that the ball was kicked at an angle of 21.0 degree

10× 1.45 sin theta= 1/2× 15^2- 1/2× 14.7^2

I don't get the ans.]]>

I don't understand how to solve this. Pls help]]>

How to answer this question? Any math help, hint, or even correct answer will be accepted.]]>

a) Find the child's loss of GPE

I got the ans 800J

there is a constant resistance of 112N throughout

b) find the distance the child has travelled when she comes to rest.

Using work energy principle

Increase in KE =0J

Increase in GPE= -800J

Work done against resistance = -112SJ

So I get S= 7.14m

The slide is inclined...

Work energy principle and power]]>

a) Assuming that there is no resistance, find her speed when she reaches the bottom of the slide

I got the ans 10.2m/ s

b) the girl's actual final speed is 8m/s because there is resistance of average value of 40N. Find the length of the water slide.

I get the ans 37.5

The ans in textbook is 25m

Pls help]]>

If $\rho =\frac{d\psi}{ds}$, then the term 2 should be $\upsilon^2 \frac{d\hat{T}}{d\psi}\rho$, but instead, it was written $\frac{\upsilon^2}{\rho}\frac{d\hat{T}}{d\psi}$

How is that computed? How to compute radius of curvature($\kappa$) if $\frac{d\hat{T}}{ds}= \kappa\hat{N}$

[ATTACH type="full"...

Tangential and normal acceleration of a particle moving in a plane curve in the cartesian coordinates]]>

Iam getting the ans 1.97m/s.

The text book ans is 1.39 m/s

Pls help]]>

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Let $a,\,b,\,c$ and $d$ be the roots to the polynomial $f(x)=x^4-3x^3+2x^2+5x-4$ . Evaluate $\left(a+1+\dfrac{1}{a}\right)\left(b+1+\dfrac{1}{b}\right)\left(c+1+\dfrac{1}{c}\right)\left(d+1+\dfrac{1}{d}\right)$.

-----

Remember to read the POTW submission guidelines to find out how to submit your...

Problem Of The Week #476 July 20th 2021]]>

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Evaluate $\dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{c+a}$, given that $\dfrac{(a-b)(b-c)(c-a)}{(a+b)(b+c)(c+a)}=\dfrac{1}{11}$.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!]]>

I found the maximum, minimum for this, but how can I find the period of from this table.

As I already know formula to find period is =2pi/k but how can I find K-value from this chart??]]>

(Can someone help me for this)]]>

Describe the transformations that are applied to y= -4cos[2(x-30°)] +5 (State any shifts, stretches, compressions, or reflections).]]>

For example: rot90 ([1, 2; 3, 4], -1) ≡ rot90 ([1, 2; 3, 4], 3) ≡ rot90 ([1, 2; 3, 4], 7)

What is the meaning of 'rot90;?

What is the meaning of this example?

How to write equivalence relation in octave?

How does all of the above expressions have equivalence relation?]]>

Let $\mathcal{B}_\mathbb{R}$ the $\sigma-algebra$ Borel in $\mathbb{R}$ and let $\mu : \mathcal{B}_\mathbb{R} \rightarrow{} \mathbb{R}_{+}$ a finite measure. For each $x \in \mathbb{R}$ define

$$f_{\mu} := \mu((- \infty,x]) $$

Prove that:

a) $f_{\mu}$ is a monotonic non-decreasing function

b) $\mu((a,b]) = f_{\mu}(b)- f_{\mu}(a)$ for all $a,b \in \mathbb{R}$

The definition ($\sigma-algebra$ borel , is this:

Definition $\sigma-algebra$...

Prove that the function is monotonic and not decreasing]]>

Let $p \in \mathbb{Z}$ be a prime number. Consider $R = \{m/n \in \mathbb{Q}: p$ does not divide $n \}$

How can I prove that $R $ is a sub-ring of $\mathbb{Q}$? (only the obvious parts) and find the group of units of $R, R^{\times}$

I have no idea.How can I solve ?

Thanks]]>

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Four real numbers $p,\,q,\,r,\,s$ satisfy the equations $p+q+r+s=9$ and $p^2+q^2+r^2+s^2=21$. Prove that there exists a permutation $(a,\,b,\,c,\,d)$ of $(p,\,q,\,r,\,s)$ such that $ab-cd\ge 2$.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!]]>

- First, he draws $6$ segments from the origin to the boundary of the circle, which splits the upper (positive $y$) semicircle into $7$ equal pieces.

- Next, starting from each point where a segment hit the circle, he draws an altitude to the $x$-axis.

- Finally, starting from each point where an altitude hit the $x$-axis, he draws a segment directly away from the bottommost point of the circle $(0,-1)$, stopping...

18 segments]]>

$\begin{array}{rl}

(x+4)^2+(y+11)^2&=169 \\

(x-9)^2+(y+5)^2&=100 \\

(x-4)^2+(y-5)^2&=25

\end{array}$

ok i solved this by a lot of steps and got (1,1) as the intersection of all 3 circles

these has got to be other options to this.

basically I expanded the equations then set them equal to each other but what a mess

suggestions???

I was thinking about a matrix but not sure how to set it up...

2.4.10 3 circles one intersection]]>

The reciprocal of half a number increased by half the recipical of the number is $\dfrac{1}{2}$

$\begin{array}{rl}

n= & \textit{the number} \\ \\

\dfrac{n}{2}= &\textit{half the number}\\ \\

\dfrac{2}{n} = &\textit{the reciprocal of half the number}\\ \\

\dfrac{1}{2n}= & \textit{half the reciprocal of the number}\\ \\

\dfrac{2}{n}+\dfrac{1}{2n} &=\dfrac{1}{2}\\ \\

&\textit{Multiply...

3.1.2 The reciprocal of half a number increased by half the recipical of the number is]]>

The box is travelling at 2 m/s when it reaches the bottom of the ramp.

Find the length of the ramp

Find the loss in the potential energy of the box.

I don't understand how to calculate. Pls help.]]>

1. The midpoint of $AB$ is $E$.

2. The points $A,\,G$ and $F$ are on the same line.

3. There is a point $C$ at which $BG$ and $EF$ intersect.

4. $CE=1$ and $AC=AE=FG$.

Prove that if $AG=x$, then $AB=x^3$.]]>

Iam getting the ans 24.8J

PE = mgh= 1.2× 12 sin 35 ×3

The ans in the textbook is 20.6J]]>

a) show that the speed of ball A after the impact is 3/10 m/s

b) Find the speed of ball B after the impact.

I don't understand how to calculate this.

Pls help]]>

Find x for $f(x)=0 \quad 5+i\quad 5-i\quad $

$\begin{array}{rl}

\textsf{factored} &f(x)=(x-1)[x-(5+i))(x-(5-i)]\\

\textsf{foil} &x^2-x(5+i)-x(5-i)+(5-i)^2\\

\textsf{expand} &x^2-5x-xi-5x+xi+25-2i+i^2 \\

\textsf{simplify} &x^2-10x+26\\

\textsf{observation } &(x-1)=0,\quad x=1\\

\textsf{quadratic formula} &=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\\...

5.t.11 find x for the imaginary factors]]>

Find (A)mplitude, (P)eriod, PS, VS. graph 2 periods

$y=3\cos(\pi x-2)+5$

by observation we have A=3 and VS=5

ok assume $\omega=\pi$

so if period is $T=\dfrac{2\pi}{\omega}$ then $T=\dfrac{2\pi}{\pi}=2$]]>

Let $1\leq n\in \mathbb{N}$ and for $x=\begin{pmatrix}x_1\\ x_2\\ \vdots \\ x_n\end{pmatrix}, \ x=\begin{pmatrix}x_1\\ x_2\\ \vdots \\ x_n\end{pmatrix}\in \mathbb{R}^n$ and let $x\cdot y=\sum_{i=1}^nx_iy_i$ the dot product of $x$ and $y$.

Let $S=\{v\in \mathbb{R}^n\mid v\cdot v=1\}$ and for $v\in S$ let $\sigma_v$ be a map defined by $\sigma_v:\mathbb{R}^n\rightarrow \mathbb{R}^n, \ x\mapsto x-2(x\cdot v)v$.

I have shown that it holds for $v\in S$ and $x,y\in \mathbb{R}^n$...

Diagonalizable transformation - Existence of basis]]>

I'm having a problem solving this, My approach is solving $x_1$ as a variable and rest as constants first and then going on further. But it is getting too lengthy. Is there any short method?]]>

Find the rectangular equation of the curve $r=\sin\left(\theta+\dfrac{\pi}{4}\right)$

$r=\sin \theta{\cos \dfrac{\pi}{4}

+{\cos \theta{\sin \dfrac{\pi}{4}}}}

=\sin \theta\left(\dfrac{\sqrt{2}}{2}\right)+\cos \theta\left(\dfrac{\sqrt{2}}{2}\right)

=\left(\dfrac{\sqrt{2}}{2}\right) (\sin \theta+\cos\theta)$

well so far anyway

Desmos plotted a circle]]>

I know only that $...=2^{1+{1\over2}+{1\over10}+{1\over80}+{1\over880}+\ldots}$]]>

We consider the $4\times 4$ matrix $$A=\begin{pmatrix}0 & 1 & 1 & 0\\ a & 0 & 0 & 1\\ 0 & 0 & b & 0 \\ 0 & 0 & 0 & c\end{pmatrix}$$

(a) For $a=1, \ b=2, \ c=3$ check if $A$ is diagonalizable and find a basis of $\mathbb{R}^4$ where the elements are eigenvectors of $A$.

(b) Show that if $a>0$ and $b^2\neq a\neq c^2$ then $A$ is diagonalizable.

(c) Show that if $a\leq 0$ then $A$ is not diagonalizable.

For (a) I have done the following :

The eigenvalues of $A$ are...

Diagonalizable matrix A]]>

terms of x_1, y_1, x_m, and y_m.

54. Use the result of Exercise 53 to find the endpoint (x_2, y_2) of each line segment with the given endpoint (x_1, y_1) and midpoint (x_m, y_m).

(a) (x_1, y_1) = (1, −2)

(x_m, y_m) = (4, −1)

(b) (x_1, y_1) = (−5, 11)

(x_m, y_m) = (2, 4)

I need help with 53 and 54.]]>

Answer: \[ 2/3 sqrt3 \]

Thanks]]>

Find amplitude, period, PS, VS. then graph.

$y=\cos\left(x+\dfrac{\pi}{2}\right)$

For the graphs of $y=A\sin(\omega x - \phi)$ or $y=A\cos(\omega x - \phi),\omega>0$

Amplitude $=|A|$

Period $T=\dfrac{2\pi}{\omega}=\dfrac{2\pi}{2}=\pi$

PS $=\dfrac{\phi}{\omega}=\dfrac{\pi}{4}$

well so far

I don't know what the greek letter is for VS or Vertical Shift? which is usually D]]>

]]>

How much time would it take for Alan and Brian to paint it together?

If Alan had to paint it on his own, it would take him one hour more than the time...

[ASK] Paint Problem]]>

Write the slope-intercept equation of the line that is parallel to -9x-7y=4 and has the same y-intercept as the graph of -5x+11y=-22.]]>

Let me see.

Renaldo's commission is 0.105 of his total sales.

I understand this to be 0.105x, where x = his total sales.

Plus a salary of 2,500 dollars.

So far, I have 0.105x + 2,500.

In a certain month, Renaldo earns 3,025 dollars.

I say the equation needed is as follows:

0.105x + 2,500 = 3,025

You say?]]>

Here goes the question...and if this question intrigues you, please feel free to try it and in case you have solved it, please share it with us!

In the isosceles triangle $ABC$, the angle at the base $BC$ is equal to...

Find angle EDC]]>

find x in degrees $\quad 3\sin^2 x -\sin x-1=0$

rewrite as $3u^2-u-1=0$

quadradic eq

$u=\dfrac{1\pm \sqrt{13}}{6}$

ok this is ?? are we going to have decimal degrees?]]>

Simplify the expression

$\dfrac{{\cos 2x\ }}{{\cos x-{\sin x\ }\ }}

=\dfrac{{{\cos}^2 x-{{\sin}^2 x\ }\ }}{{\cos x\ }-{\sin x\ }}

=\dfrac{({\cos x}-{\sin x})({\cos x}+{\sin x\ })}{{\cos x}-{\sin x}}

=\cos x +\sin x$

ok spent an hour just to get this and still not sure

suggestions????]]>

Find domain asymptotes.

$g(x)=\dfrac{2x^2-14x+24}{x^2+6x-40}$

$\begin{array}{rll}

\textsf{factor}&=\dfrac{2(x-3)(x-4)}{(x-4)(x+10)}

=\dfrac{2(x-3)\cancel{(x-4)}}{\cancel{(x-4)}(x+10)}

=\dfrac{2(x-3)}{x+10}\\

\textsf{Domain} & -\infty<-10<\infty\\

HA \quad y&=3 \\

VA \quad x&=-10

\end{array}$

I think there is an oblique asymptote but ??

Also the OP has a hole at $x=4$ but they didn't ask for it ???

btw how come...

s5.t.5 Find domain and asymptotes.]]>

Find x

$5^{x^2+8}=125^{2x}$

$\begin{array}{rlll}

\textsf{common base}&125^{2x}=(5^3)^{2x}=5^{6x}\\

\textsf{then } &x^2+8=6x\implies x^2-6x+8=0 \\

\textsf{factor}&(x-2)(x-4)=0\\

\textsf{get zeros}&x=2, \quad x=4\\

\end{array}$

should be ok

suggestions......]]>

1. What is the growth rate pf the given problem?

2. Is the number of bacterial population important to the problem?

3. At what time should

a) Find the downward momentum of each Ball just before they meet.

The Ball coalesce and the combined object falls to the ground.

b) show that the combined object reaches the ground 2.68...

Momentum]]>

Express y as a function of x. $\quad C>0$

$3\ln{y}=\dfrac{1}{2}\ln{(2x+1)}-\dfrac{1}{3}\ln{(x+4)}+\ln{C}$

rewirte as

$\ln{y^3}=\ln{(2x+1)^{(1/2)}}-\ln{(x+4)^{(1/3)}}+\ln{C}$

then e thru and isolate y

i think

looks like it will be ugly]]>

$x^3-5x^2+3x+1;\quad x+1$

\item \textit{apply synthetic division}

\item$\begin{array}{c|rrrrr}

1 &1 &-5 &3 &1\\

& &1 &-4 &-1\\

\hline &1 &-4 &-1 &0

\end{array}$

$(x-1)$ so $x^2-4x-1$

$\begin{array}{rl}

x &=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\\

\textsf{a,b,c} &=\dfrac{-(-4)\pm\sqrt{(-4)^2-4(1)(-1)}}{2(-1)}

=\dfrac{4\pm\sqrt{20}}{-2}...

5.12 synthetic division]]>

$\log_9{(x+1)}+3\log_3{x}=14$

ok not sure as to best approach to this

assume change the base 9?]]>

We have the points $Z=(-1,1)$, $A=(-1,6)$ and $B=(3,4)$.

Let $\delta$ be the rotation with center $Z$ and $\delta (A)=B$.

Let $C$ be the point on the circumcircle of the triangle $ABZ$ such that the segment $\overline{CZ}$ goes through the center of circumcircle.

Let $\gamma$ be a rotation with center $C$ and $\gamma (B)=A$.

Show that $\gamma\circ\delta$ is a point reflection at $A$.

To show that do we have to show that there is exactly one fixed point? Or do we...

Show that it is a point reflection at A]]>

Solve $2+5\ln{x}=21$

\$\begin{array}{rlll}

\textsf{isolate} &\ln{x} &=\dfrac{19}{5} &(1)\\

\textsf{then} & &= &(2)\\

\textsf{then} & &= &(3)\\

\textsf{hence} & &= &(4)

\end{array}$

ok for (2) I presume e thru then calculate for x

just strange to see a fraction as an e exponent

tryin array on these no sure if its better ]]>

$\begin{array}{rll}

\textsf{rewrite} &(x^2+6x )+(y^2+8y)=-9\\

\textsf{complete square} &(x^2+6x+9)+(y^2+8y+16)=-9+9+16\\

\textsf{simplify equation} &(x+3)^2+(y+4)^2=16=4^2\\

\textsf{observation} &C(-3,-4), \quad R=4

\end{array}$

hopefully ok

is there another way to do this other than complete the square

if you are inclined to do so I would be interested in a tikz code would be fore this ]]>

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Find the largest natural number $n$ for which $3^{2016}-1$ is divisible by $2^n$.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!]]>

$sinx + cosx + sin2x + cos2x + sin3x = -1 $

Please help anyone wihout using desmos]]>

actually I didn't find this a slam dunk question....]]>

1. If the temperature of the turkey is 150 F after half an hour, what is the temperature after 45 minutes?

2. When will the turkey have cooled to 100 F?]]>

Solve for x give exact for\\

$\log{(x-10)}-\log{(x-6)}=\log{2}$

$\begin{array}{rrll}

\textsf{subtraction rule} &\log\left(\dfrac{x-10}{x-6}\right)&=\log{2} \\

\textsf{drop logs} &\dfrac{x-10}{x-6}&=2 \\

&x-10&=2(x-6)=2x-12\\

\textsf{isolate x} &2&=x

\end{array}$

hopefully ok but???

quess we could just put all the logs on r side and set em to zero]]>

A pinball moving in a plane with velocity

I don't understand how to answer this question? Any math help, hint or even correct answer will be accepted?]]>