## If i have all the angles and one side can i find the other sides

Sure, that's what solving triangles is. Trigonometry has a short menu. Basically we choose between three formulas: Law of Sines, Law of Cosines, Sum of Triangle Angles

The Law of Sines has two sides and two opposite angles; given any three we can solve for the fourth.

The Law of Cosines has three sides and one angle; again given any three we solve for the remaining one.

The Sum of Triangle Angles says all three angles add to 180 degrees, so given two we can find the third.

Here we have all the angles and one side, that's Law of Sines to get the remaining sides.

## Find the difference between the mean and the median of 10.2, 10.6, 11.9, 9.9, 10.6

Let's sort the data

9.9, 10.2, 10.6, 10.6, 11.9

The median is clearly 10.6, the middle value. The sum is 53.2 so the mean is 53.2/5 = 106.4/10 = 10.64

mean - median = 10.64 - 10.6 = 0.04

Answer: **0.04**

## A company buys pens at the rate of $7.50 per box for the first 10 boxes, $5.50 per box for the next 10 boxes, and $4.50 per box for any additional boxes. how many boxes of pens can be bought for $148.00?

So it is really easy to solve firstly we can see how much does the first 10 boxes make which makes around 75$ obviously. Secondly 55$ for the next 10 boxes.

So for now we can simply calculate that we have spent around 130$ which means 20 boxes. The remaining money left is 18$ so we can buy 18/4.5 = 4 only 4 boxes with that money. Hence a total of 24 boxes.

## We say that a point p = (x, y) in r2 is rational if both x and y are rational. more precisely, p is rational if p = (x, y) ∈ q2. an equation f(x, y) = 0 is said to have a rational point if there exists x0, y0 ∈ q such that f(x0, y0) = 0. for example, the curve x2+y2−1=0 has rational point (x0,y0)=(1,0). show that the curve x2 + y2 − 3 = 0 has no rational points

The first curve is the unit circle, which has many rational points, each corresponding to a Pythagorean Triple. To check (1,0) we substitute

That's a rational point, albeit a boring one. The first Pythagorean Triple on the list gives a more interesting point, (x,y)=(3/5, 4/5), which is also on the curve.

Let's assume there are rational points; we write them with a common denominator. Let x=a/d, y=b/d

There are three cases:

Case 1: 3 divides exactly one of a or b

Case 2: 3 divides both a and b

Case 3: 3 divides neither a nor b

In case 1, let's say without loss of generality that it's *a* that's a multiple of 3. So is of course. But if b is not a multiple of 3, i.e for some natural n, its square isn't a multiple of three either:

We see the square of a non-multiple of 3 necessarily has a remainder of 1 when divided by three.

So the sum of squares isn't a multiple of 3 so can't be . That shows case 1 can't be the case.

In case 2, 3 divides both a and b, so each of the squares will be a power of 9 times some other factors, i.e. each square has an even number of factors of 3. So will the sum, because the other non-multiple of three factors when squared and added won't be a multiple of 3 as we showed in case 1. So d will have half of those factors of 3 and since there were an even number there won't be any leftover for the outside factor of 3. So case 2 can't be true.

Case 3. We showed both squares have remainder 1 when divided by 3, so their sum will have remainder 2, so can't be a multiple of 3.

None of the possibilities can give rise to rational points so we conclude there are no rational points on the curve.

## Find the product of 6372×75

the answer is 477900

## Determine whether rolle's theorem applies to the function shown below on the given interval. if so, find the point(s) that are guaranteed to exist by rolle's theorem. f(x)equals=x(xminus−44)squared2; [0,44]

Since the function is continuous between x = 0 and x = 44 then Rolle's theorem applies here.

Differentiating

y' = x * 2(x - 44) + (x - 44)^2

y' = 3x^2 - 176x + 1936 = 0 (at a turning point).

solving we get x = 44 , 14.67

y" = 6x - 176 which is negative for x = 14.67 so this gives a maximum value for f(x)

This maximum is at the point (14.67, 12,619.85)

There is a minimum at ( 44,0)

These are the required points

## The roots of \[z^7 = -\frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}}\]are $\text{cis } \theta_1$, $\text{cis } \theta_2$, $\dots$, $\text{cis } \theta_7$, where $0^\circ \le \theta_k < 360^\circ$ for all $1 \le k \le 7$. find $\theta_1 + \theta_2 + \dots + \theta_7$. give your answer in degrees.

Decoding the LaTeX that didn't render, we seek sum of the angles of the seventh roots of

That's on the unit circle, 45 degrees into the third quadrant, aka 225 degrees.

The seventh roots will all be separated by 360/7, around 51 degrees. The first seventh root has

That's around 32 degrees.

The next angle is

The next one is

and in general

The first sum is just -135° since it's one seventh of the sum of seven -135s.

We have 1+2+3+4+5+6+7 = (1+7)+(2+6)+(3+5) + 4 = 28 so

If I didn't screw it up, that means the answer is

Answer: **1305°**

## Suppose a basketball player is an excellent free throw shooter and makes 94% of his free throws (i.e., he has a 94% chance of making a single free throw). assume that free throw shots are independent of one another. suppose this player gets to shoot three free throws. find the probability that he misses all three consecutive free throws. round to the nearest ten-thousandth.

Probability of a failure on 1 throw = 1 - 0.94 = 0.06 or 6%.

Prob (he misses all 3) = 0.06^3 = 0.000216 = 0.0002 to nearest ten thousandth

## An english teacher needs to pick 11 books to put on his reading list for the next school year. he has narrowed down his choices to 12 novels, 8 plays, and 12 nonfiction books. if he wants to include 5 novels, 4 plays, and 2 nonfiction books, how many ways can he choose the books to put on the list?

That would be 12C5 * 8C4 * 12C2

= 792 * 70 * 66

= 3,659,040 ways

## A rectangle is 3 times as long as it is wide. The perimeter is 60 cm. Find the dimensions of the rectangle. Round to the nearest tenth if necessary.

**Answer:**

**Length = 7.5 , Width = 22.5 units .**

**Step-by-step explanation:**

Given : A rectangle is 3 times as long as it is wide. The perimeter is 60 cm.

To find : Find the dimensions of the rectangle. Round to the nearest tenth if necessary.

Solution : We have given

Let the width of rectangle = x .

According to question

Length of rectangle = 3x

**Perimeter of rectangle = 2 (length + width ) **

Plug the values

60 = 2 ( x + 3x ) .

60 = 2 ( 4x ) .

60 = 8x .

On dividing both sides by 8.

x = 7 .5

Length = 7.5 units

width = 3x = 7.5 *3 = 22.5 units

**Therefore, Length = 7.5 , Width = 22.5 units .**