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radius = 38 mm --> 3.8 cm
thickness = 35 mm --> 3.5 cm

volume = πr²h
volume = π(3.8 cm)²(3.5 cm)
volume = 50.54 cm³

density = 1.5 g/cm³

density = mass / volume
1.5 g/cm³ = mass / 50.54 cm³ [multiply 50.54 cm³ on both sides]
75.81 g = mass [cm³ cancels out]

since an angle bisector divides an angle into two equal parts, the two parts must be equal to each other.
so, set angles TUV and TUW equal to each other and solve for x:
13x - 5 = 7x + 31
6x = 36
x = 6
 

Thanks!!!

Thank you so much for the help.

By looking at images of symmetrically placed points and the lines connecting them I see that the number of intersections in the interior of the circle given n points is given by

\(ni = 2^{n-2}-3,~\forall n \geq 4 \\ \text{and for }n=9\text{ this results in 125 intersections.}\)

have fun verifying this.

Here is my naive approach to this:
Weight of apples =99% moisture + 1%"F", for "fibre". Originally
After 2 days =98% moisture + 1%F =99 kg of their original weight.
The loss of 1 kg was solely due to moisture loss in the sun!!.

I think 98 kg for the 3rd one 

If we center a circle with a radius of 1 at "A"...the area of the circle  sector contained by the triangle will be :

 pi * (1)^2  (60 / 360)  =    pi/ 6  ....   (1)

And the area of the triangle is   (1/2)(3)^2 sqrt (3)/2   =  9sqrt(3)/4   .....(2)

So.....the  probability is  (1) / (2)   =  [  pi/6 ]  / [ 9 sqrt (3) / 4 ]   ≈  13.44%

Here's the first one....if we consider the points to be A, B ,C, D...etc, we can choose any two of the points without regard to  order.....this will result in   C(9,2)   ⇒   36  segments being drawn

Alternatively, we can factor out the common term on the numerator. 
(16a⁴-40a²+24a)/12a³

Factor out 8a:
8a(2a³-5a+3)/12a³

From here we can separate the fraction into 8a/12a³ times (2a³-5a+3)
8a/12a³ --> 2/3a²

Put them back together --> 2(2a³-5a+3)/3a²

Distribute the 2 --> (4a³-10a+6)/3a²
 

Thank you so much! ^-^ have an amazing day!

aˣ/aʸ = aˣ⁻ʸ

For (16a⁴/12a³)], you can separate out the numbers and the variable so you get:
16/12 times a⁴/a³. 
Think of 16/12 as a fraction and reduce it to its lowest terms --> 4/3
Same for a⁴/a³ --> a⁴⁻³ --> a
Put them back together and you get 4a/3

For [(40a^2/12a^3)], separate to 40/12 times a²/a³.
40/12 --> 10/3
a²/a³ --> a²⁻³ --> a⁻¹ --> 1/a
Put them together --> 10/3a

For [(24a/12a^3)], separate to 24/12 times a/a³.
24/12 --> 2
a/a³ --> a¹⁻³ --> a⁻² --> 1/a²
Put them together --> 2/a²

So, we get
(4a/3) - (10/3a) + (2/a²)

We can make it neater by keeping the denominators the same. The LCM is 3a².
Multiply 4a/3 by a²/a² to get 4a³/3a².
Multiply 10/3a by a/a to get 10a/3a².
Multiply 2/a² by 3/3 to get 6/3a².

Now, we can turn the expression into a single fraction.
(4a³/3a²) - (10a/3a²) + (6/3a²) --> (4a³-10a+6)/(3a²)

Notice: In the cancellation of a common factor a, we canceled 2a and 3a^3, yielding us with: 2 and 3a^2.

Wait, I can try 20!

20. \(\frac{16a^4-40a^2+24a}{12a^3}\), we can factor out a common term 8a from the numerator, so we get: \(8a\left(2a^3-5a+3\right)\).

Now, we are left with \(\frac{8a\left(2a^3-5a+3\right)}{12a^3}\) . We can factor a common term 4, and cancel \(a\) , a common factor. 

Cancel common factor 4: \(\frac{2a\left(2a^3-5a+3\right)}{3a^3}\).

Cancel common factor \(a\): \(\frac{2\left(2a^3-5a+3\right)}{3a^2}\).

So, \(\boxed{\frac{2\left(2a^3-5a+3\right)}{3a^2}} \)

is our answer!

Oh darn >.< I just needed 20. It's alright 

18. \(4(2-3w)(w^2-2w+10)\), we take a good look at the last two terms, excluding the 4: (2-3w) and (w^2-2w+10)

We can expand or distribute the first two terms(2 and -3w) with the rest of the terms. So, we yield: \(2w^2+2\left(-2w\right)+2\cdot \:10+\left(-3w\right)w^2+\left(-3w\right)\left(-2w\right)+\left(-3w\right)\cdot \:10\) . We simplify this, to get: \(-3w^3+8w^2-34w+20\).

Now, finally, we multiply this terms by 4: \(\:4\left(-3w^3+8w^2-34w+20\right)= \boxed{-12w^3+32w^2-136w+80}\), and that's our answer!

I have to go now, bye!

Oops, didn't see your reply, sorry.

Nvm I got 18 but still need help with 20

Thanks so much for your help!

I got the set up part done but not sure how to do this >.<

[(16a^4/12a^3)]-[(40a^2/12a^3)]+[(24a/12a^3)]

21. SA = 2(LW + LH + WH)
2[(x+3)(x+7)+(x+3)(x-1)+(x+7)(x-1)]
2[(x² + 7x + 3x + 21) + (x² - x + 3x - 3) + (x² - x + 7x - 7)]
2[(x² + 10x + 21) + (x² + 2x - 3) + (x² + 6x - 7)]
2(3x² + 18x + 11)
6x² + 36x + 22

22. A = (bh)/2
14x⁵ + 63x² = (7x²h)/2
28x⁵ + 126x² = 7x²h
(28x⁵ + 126x²) / 7x² = h
(28x⁵ / 7x²) + (126x² / 7x²) = h
4x³ + 18 = h

16

(a + b)(3a - b) (2a + 7b)      expand the first two factors
 

( 3a *a + 3ab - ab - b^2) (2a  + 7b) =     simplify the terms in the left parentheses

(3a^2 + 2ab - b^2)(2a + 7b)  =    expand the first terms over the second

3a^2 * 2a  +  2ab * 2a -  b^2* 2a   + 3a^2* 7b +  2ab*7b - b^2*7b   =

6a^3 + 4a^2b - 2ab^2 + 21a^2b + 14ab^2 - 7b^3   =

6a^3 + 25a^2b + 12ab^2 - 7b^3

Thank you! ^-^

16. (a + b)(3a - b)(2a + 7b) = 
(3a² - ab + 3ab - b²)(2a + 7b) = 
(3a² + 2ab - b²)(2a + 7b) = 
6a³ + 21a²b + 4a²b + 14ab² - 2ab² - 7b³ = 
6a³ + 25a²b + 12ab² - 7b³
Explanation: Multiply any two terms first, in this case I multiplied (a+b) and (3a-b) first. Then with the result, multiply it with the last term. 
17/18 follows the same procedure as 16.

19. (-42x¹⁰y⁵+12x⁸y³-6x²y) / (6x²y) = 
[(-42x¹⁰y⁵) / (6x²y)] + [(12x⁸y³) / (6x²y)] + [(-6x²y) / (6x²y)]
(-7x⁸y⁴) + (2x⁶y²) + (-1)
-7x⁸y⁴ + 2x⁶y² - 1
Explanation: Separate the fraction into 3 fractions. Simplify using division and laws of exponents. 
20 follows the same procedure as 19. 

I'm working on 18 right now can you gelp with 19 and 20?

Working on them  one at a time, RP....give me a few....