Logarhythm

lol I was debating whether or not to post that because there was a similar question, however this was asking for area of the shaded region instead of the perimeter of it... 

https://gyazo.com/0bb2e632441b774f3d412d086fa55957

I made an equilateral triangle using the centers of the triangle. 
Since an equilateral triangle has angles of 60^o, each sector part is just \(\frac{1}{6}\) of the circle.

Circumference of each circle = 36 units 

Radius = \(\frac{18}{π}\)

Sector areas = \(3 \cdot \frac{1}{6} \cdot (\frac{18}{π})^2 π\)

= \(\frac{1}{2} \cdot \frac{324}{π^2}π\)

= \(\frac{1}{2} \cdot \frac{324}{π}\)

= \(\frac{162}{π}\)

Now the area of the equilateral triangle is: 

\(\frac{\sqrt{3}}{4}\cdot (\frac{36}{π})^2\)

= \(\frac{\sqrt{3}}{4} \cdot \frac{1296}{π^2}\)

= \(\frac{1296\sqrt{3}}{4π^2}\)

Which means the area of the shaded region is: 

\(\frac{1296\sqrt{3}}{4π^2} - \frac{162}{π}\)

= \(\frac{1296\sqrt{3}}{4π^2} - \frac{162π}{π^2}\)

= \(\frac{1296\sqrt{3}}{4π^2} - \frac{648π}{4π^2}\)

= \(\frac{1296\sqrt{3} - 648π}{4π^2}\)

= \(\frac{4(324\sqrt{3} - 162π)}{4π^2}\)

= \(\frac{324\sqrt{3} - 162π}{π^2}\)

please tell me if I did anything wrong...

perimeter = 2l + 2w 

144 = 2l + 2w

72 = l + w

but we know that l = 4w, since 4 widths make up one side of the square, and the length does too

72 = 4w + w

72 = 5w

w = 14.4

72 = l + w

72 = l + 14.4

l = 57.6

perimeter of square

= 4l 

= 4(57.6)
= 230.4 units

\(\frac{1}{4} x= \frac{3}{5}\)

can also be written as:
\(\frac{x}{4} = \frac{3}{5}\)

cross multiply

\(5x = 12\)

\(x = \frac{12}{5}\)

4. a) since we know RT and TS are the same, we know that triangle RTS is an isosceles triangle, meaning angle RST is the same as TRS; being 32^o

    b) angle RTS is --> 180 - 32 - 32 = 116^o

    c) we know angle SPQ is 45^o, and now that we know angle QSP or RST is 32^o, we can find angle PQS. 
        180 - 45 - 32 = 103^o

5. a) angle DBC 72^o because of corresponding angles

    b) angle C is the same as angle DBC because it's an isosceles triangle. (72^o)

    c) angle CDB is 180 - 72 - 72 = 36^o

    d) angle DEB  is 180 - 90 - 36 = 54^o

    e) angle BEA is 36^o, and we know that because of alternate angles (equal to angle DBE)

    f) angle EBA is 180 - 36 - 72 = 72^o 

tell me if I did anything wrong :)

You and your brother took a piece each, which means you took 2 pieces. Since there are only 4 pieces, there would only be 2 left after you both took one each. This means there are \(\frac{2}{4}\) (2 out of the four pieces) left, which you can also write as \(\frac{1}{2}\) when simplified :)

Let x = Jacob
John has 9x 

9x - 12 = x + 12

8x = 24

x = 3

This means Jacob has 3 toys and John has 3 x 9 = 27 toys.

3 + 27 = 30 toys altogether

let x = percentage of cream in final mixture

6 * 0.1 + 3 * 0.15 = 9x

0.6 + 0.45 = 9x

1.05 = 9x

x = 0.116666666...

which means the percentage is about 11.67%

(tell me if I did something wrong since I haven't done these types of problems in a while)

(x³ - 4x² + 5x - 5)(x² + tx - 7)
Find the pairs that make x² terms:
(-4x²)(-7) + (5x)(tx) + (-5)(x²)
= 28x² + 5tx² - 5x²

= 23x² + 5tx²

Since you're trying to find no x² term:

23x² + 5tx² = 0

23 + 5t = 0

5t = -23

t = \(-\frac{23}{5} \)   or   -4.6