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If $4000 is borrowed at a rate of 5.75% interest per year, compounded quarterly, find the amount due at the end of the given number of years. (Round your answers to the nearest cent.)

Let N be the number of times per year the interest is compounded and let T be in years.

$$FV=PV\left(1+\dfrac {rate}{N}\right)^{T \cdot N}$$

$$FV=4000\left(1+\dfrac{0.0575}{4}\right)^{4T}$$

$${\mathtt{4\,000}}{\mathtt{\,\times\,}}{\left({\mathtt{1}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{0.057\: \!5}}}{{\mathtt{4}}}}\right)}^{\left({\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{5}}\right)} = {\mathtt{5\,321.459\: \!175\: \!499\: \!329\: \!081\: \!7}}$$

$${\mathtt{4\,000}}{\mathtt{\,\times\,}}{\left({\mathtt{1}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{0.057\: \!5}}}{{\mathtt{4}}}}\right)}^{\left({\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{7}}\right)} = {\mathtt{5\,965.117\: \!902\: \!958\: \!065\: \!897\: \!6}}$$

$${\mathtt{4\,000}}{\mathtt{\,\times\,}}{\left({\mathtt{1}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{0.057\: \!5}}}{{\mathtt{4}}}}\right)}^{\left({\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{9}}\right)} = {\mathtt{6\,686.630\: \!569\: \!302\: \!827\: \!466\: \!9}}$$

so 

5yrs -> $5321.46

7yrs -> $5965.12

9yrs -> $6686.63

you'll need to use tan inverse to find theta.

If you go the calculator on the home page of this website press"2nd" then "atan" to bring up arctan which is equivalent to tangent inverse.  Then type in 1.47/1.59 and hit equals and your result will pop right up!

as a first step lets make this look a little more friendly by multiplying both sides of the equation by 100

5x+4=30-20x

Now add 20x to both sides and subtract 4

25x=26

x=26/25

More than likely you mean cosine inverse of (6.4/9.6).  Since cosine and cosine inverse are functions they have no value and cannot be multiplied by things. To compute cos^-1(6.4/9.6) you can go to the calculator on this site and click the "2nd" button, then "acos" which stands for arc cosine with is the same thing as cosine inverse, then type in your fraction in the parentheses and your answer will pop right out!

Inverse trig functions use ratios or function values as their inputs and produce angle measures as their outputs versus a regular trig function that does the exact opposite.  When I teach this in geometry i use the phrase "inverses find angles" as a way to remember which is which.

As an example, if you wanted to know what measure on the unit circle would produce a y value of 1 you would use sin^-1(1)=pi/2  

Never mind; what's a factor of 100 among friends!

The mass at time 0 is found by replacing t by zero in 10e^-0.013t

Since e⁰ = 1 this means the mass at time zero is just 10 kg.

Replace t by 42 to find the mass after 42 days (assuming that the decay constant has units of days^-1)

$${\mathtt{massafter42days}} = {\mathtt{10}}{\mathtt{\,\times\,}}{{\mathtt{e}}}^{{\mathtt{\,-\,}}\left({\mathtt{0.013}}{\mathtt{\,\times\,}}{\mathtt{42}}\right)} \Rightarrow {\mathtt{massafter42days}} = {\mathtt{5.792\: \!622\: \!313\: \!807\: \!82}}$$

So approximately 5.79kg remain after 42 days.

Glad I saw your post, Alan.  In my inital setup i missed a decimal point accompanied with 39 so my answers were off by a factor of 100!

Here's another inequality I want to turn into equalities.

Separately find the extreme values.

1. -8 = 4 - 2x₁

Add 8 to both sides:  0 = 12 - 2x₁

Add 2x₁ to both sides 2x₁ = 12

Divide by 2:  x₁ = 6 

2. 4 - 2x₂ = -5

Add 5 to both sides: 9 - 2x₂ = 0

Add 2x₂ to both sides: 9 = 2x₂

Divide by 2 to get x₂ = 4.5

The extreme values are 4.5 and 6. The only point to beware of here is that x can equal the second (lower) limit, but is strictly less than the first (upper) limit. So  4.5 ≤ x < 6 

I don't like inequalities!  I try to turn them into equalities.  We can do this here by separately finding the two extremes:

1. 0.39m₁ + 2600 = 6890

Subtract 2600 from both sides:  0.39m₁ = 4290

Divide both sides by 0.39:  m₁ = 4290/0.39 = 11000

2. 0.39m₂ + 2600 = 8840

Subtract 2600:  0.39m₂ = 6240

Divide by 0.39: m₂ = 6240/0.39 = 16000

Hence the range is between 11000 and 16000 miles.

Focus on the meaning of"percent" as being "out of 100 so 3.5% really means 3.5/100=.035 

we want this much of 50 which means .035(50)=1.75

In this case we cannot subtract 39. In algebra we always need to look for inverses (opposites) to cancel things out.  When we look at the original setup the 39 is connected to the m by multiplication and then 2600 is added.  We have two options: a, cancel out the multiplication with division or b, cancel out addition with subtraction.  We should pause for a moment to see which one is easier.  If we choose division we need to divide by 39, which is possible but not fun particulalry in the middle piece because we end up with "m+2600/39" and now we have all sorts of fractions to deal with.  Its much easier to subtract 2600

6890<39m+2600<8840

4290<39m<6240

From here your idea to divide by 39 is spot on.  Our result is

110<m<160

I did

6890<39m+2600<8840    subtract 39 on 3 sides and get 6890m<+2600<8801

then i divide by 39 and get 17.6<+13.9<22.9

Yes, that was the line I was speaking of. And now it all makes sense. Now I am going to try to do the next one.

Am I correct in my assumption that you are talking about this line "-12<-2x<=-9 "

When we subtracted four it killed off the 4 in the 4-2x so what is left over is just the -2x in that middle "chunk"

I have a question. How do we get -2 when we subtract 4 from <4-2x ?

Let the minutes be represented by "t."  So we're trying to find where

6 + .12t < 25 + .05t   Subtract 6 from both sides

.12t < 19 + .05t          Subtract .05t from both sides

.07t < 19        divide both sides by .07

t < 271.4

So basically......if we talk more than about 271 minutes, Plan A is cheaper......

Ill help you set it up and then you can try to solve it.  We know we are trying to figure out when plan b is cheaper so

B<A

6+.12x<25+.05x

try solving the following based on the techniques that I posted in response to one of your earlier questions!  If you get stuck or want to verify your work feel free to post and we can discuss

6890<C<8840

6890<39m+2600<8840

Remember how in basic algebra anything that we do to one side of an equation we do to the other as well to keep it balanced?    The same applies here with one caveat: if you multiply or divide by a negative number you must flip the inequality signs.  Here is an example of why this is important/

take 2<4, an obviously true statement and multiply both sides by -1 so we have -2<-4,  this is now not true!  If we flip the inequality so it says -2>-4 we have preserved out relation however. 

Secondly, its not so much about sides in algebra as it is about hitting every "chunk" of math that is separated by a relation symbol.  With that, back to your initial question.

Let's subtract four, not from both sides, but from all three "pieces"

-12<-2x<=-9      Now we divide by -2, dont forget to flip the signs!

6>x=>9/2      And we are done!

It can also be written as  x<6 and x>=9/2

Here's another way to see this.....if the perimeter = 66, then a side must = 11.

And we have 12 congruent right triangles each having a hypoteneuse of 11 (the radius) and a leg (1/2) the length of a side = 5.5.   And the apothem would form the other leg.

So we have...

apothem^2 = 11^2 - (5.5)^2

apothem^2 = (2^2)(5.5)^2 - (5.5)^2 = (4)(5.5)^2 - (5.5)^2    ....factor out the (5.5)^2......

apothem^2 = (5.5)^2 * (4 -1)

apothem^2 = (5.5)^2 * (3)  ......      take the square root of each side  .....

apothem = 5.5√3  

Just as jboy314 found !!!!

There are a handful of reasons.  Let's start with a relatively basic one and see where the conversation goes.  In geometry we learn that the sine ratio is equal to the ratio of the lengths of the opposide side and hypotenuse of a right triangle.  Let's apply this to an arbitrary right triangle.  When we select the right angle as our angle for the sine function something odd happens.  It turns out that the opposite side, in this case, is the hypotenuse!

sin(90)=o/h

o=h

Therefore o/h=1

If you make it a little bit farther in math you could consider the unit circle.  On the unit circle 90 degrees represents a vector of magnitude 1 pointing straight up from the origin to the point (0,1).  Since sin(x)=y it follows by substitution that since y=1, sin(90)=1

Using the quadratic formula is a method that ALWAYS works.  Isn't it nice when you find something that always works?

There are a few ways to start thinking about this but here is the one that feels most natural to me.  We note that since a hexagon has 6 sides the sum of its interior angles will be 720 degrees and since there are six of them, each must be 120 degrees.  Now let's draw in the radius and apothem somewhere next to eachother on our figure (you should draw this out so you can see).  When we draw the radius and apothem we have created a right triangle.  The radius bisects a 120 degree angle so the interior angle of the triangle at that vertex is 60 degrees, while the final angle of the triangle is 30.  A 30-60-90 triangle!  Lucky us!  Since the hypotenuse(radius) is 11, the shortest leg of the triangle will be half as much.  5.5 in this case.    in a 30-60-90 triangle it always holds the relationship that the longer leg is  greater than the shorter leg by a factor of sqrt(3).  Therefore your apothem has length 5.5sqrt(3)