NotThatSmart

Well, it's been a while since I've answered a question on this website, and what better way to celebrate with a geometry question :)

So, we don't have a right triangle, so we're gonna have to settle with Law of Cosines. 

Also, note that we have 

\((cos PMQ) = (- cos PMR )\)

Now, let's apply Law of Cosines and add subsitute using equation above. 

\(PQ^2 = QM^2 + PM^2 - 2(QM * PM) * (-cos PMR) \\ PR^2 = RM^2 + PM^2 - 2 (RM * PM) * ( cos PMR)\)

Simplify

\(5^2 = 5.5^2 + PM^2 + 2(5.5 * PM)cos(PMR) \\ 8^2 = 5.5^2 + PM^2 - 2(5,5 * PM) cos (PMR) \)

Now, we're stuck...but we can add these two equations to keep progressing. 

\(5^2 + 8^2 = 2 * 5.5^2 + 2PM^2 \\ 89 = 60.5 + 2PM^2 \\ 28.5 / 2 = PM^2\\ 14.25 = PM^2 \\ \sqrt {14.25} = PM ≈ 3.77\)

And we're done!

Also, side note, this is gonna be my 1000 answer on this website which is a weird flex but whatever. 

I might be gone for another while, but hopefully be back in the summer. 

~NTS

This problem actually has a very simple answer. 

Let's note the two sidelengths given, We have \(WX = 2\)  and \(XY = 3\)

In order to find the area of WXYZ, we must find the height and base. 

The base of the rectange is \(WX\) and the height of rectangle is \(XY\)

The problem gave all the dimensions we need to already solve this problem. We have

\(WX \cdot XY = 2 \cdot 3 = 6\)

So our final answer is 6. 

Thanks! :)

There are a few ways you can tackle this problem. 

The first way is reccomended. We find that after distributing, how many pairs create a coefficient of x. 

We first have x and 14. \(14 \cdot x = 14x\)

Next, we have 1 and -23x. \(1 \cdot -23 = -23x\)

These are the only two ways to achieve x. Now, we simply add then up. 

\(14x - 23x = -9x\)

Orrrrrrrrrrrrrrrrrrrrrrrrrrr, you can distribibute EVERYTHIG and take 100 years, but here u go :)

\(x^{7}-7x^{6}+10x^{5}-13x^{4}+8x^{2}-9x+14\)

Thanks! :)

According to the Triangle Angle Sum Theorem, the sum of all interior angles of a triangle is always 180 degrees. 

Using this information, we can write the equation

\(2x + 3y + 8x + 15y + 4x - 2y = 180\\ 14x + 16y = 180\\ 7x+8y = 90\)

Now, if we were to say that x and y can be any number, then there an infinite amount of choices. 

However, if we were to only limit x and y to integers, then \(x=6; y= 6\) is the only solution. 

Hope this helps. 

~NTS

First, let's calculate the distance Will travels before grace even leaves. 

Since he traveled for 45 minutes, we have

\((45 min)(50 m/min) = 2250 m\)

So when Grace starts, she and Will have

\((2800 m) – (2250 m) = 550 m\) to cover. 

Let's let t be the amount of time it takes for the two to catch up to each other after Grace starts rowing. 

We have the equation

\((50)(t) + (30)(t) = 550 \\ 80t = 550 \\ t = 550/80 = 6.875\)

This is approximately 6 minutes and 52 seconds. 

So the clock time they meet is 6 min 52 sec after Grace starts.

Grace started at 2:45, so add 6 min 52 sec and the clock will read 2:51:52 pm when they meet  

Thus, our final answer is \(2:51:52\)

Thanks! :)

So for this problem, let's use the arithmetic series formula.

We find the sum is \(\frac{(140+1)(70)}{2}\), which is \(141*35.\) Now that we have a factorization, we can easily find the greatest prime divisor.

The prime factorization is \(3*5*7*47\), so the greatest prime divisor is 47.

Thanks! :)

Let's know that XZW will form a triangle

\(\angle ZXW = 90^\circ \\ \angle XWZ = 60^\circ\)

Thus, \(\angle XZW = 180 - 90 - 60 = 30\)

30 degrees is our answer. 

Thanks! :)

An obtuse angle is an angle that measures in between \(90^\circ\) and \(180^\circ\)

Here's an image that lists all types of angles.

Notice that B is the only one that fits for the obtuse angle. 

A would be an acute angle since it is less than 90 degrees. 

C and D would be reflexive angles since they are more than 180 degrees, less than 360 degrees. 

The hour hand will move at a rate of 

\(360°  \space \text{in} \space 12 \space \text{hours} \\ 30° \space \text{in}  \space 1 \space \text{hour} \\ 30°/ 60  =  0.5° \space \text{in}  \text{ one minute}\)

So  at   4:15, the hour  hand has  moved   \(4 (30°) +  15(.5°)  =  [120+ 7.5]° = 127.5°\)from 12 o'clock

The minute hand  will move  at a rate of \(360°/ 60  = 6°\) per minute

So at 4:15 it will have moved  \([15* 6]°  = 90°\)  from the top of the hour

So...the smaller angle formed by the hands\(  = 127.5 - 90 = 37.5^\circ\)

May have made an error....

Thanks! :)