MaxWong

\((y-1)^2 = 16(x+3)\\ 4(4)(x-(-3)) = (y-1)^2\\ p = 4\)

The focus of the parabola is 4 units on the right of the vertex.

Therefore the directrix of parabola is x = -3-4, i.e., x = -7.

First you need to complete the square.

 \(y=-\dfrac{1}{6}x^2+7x-80\)

\(y = \dfrac{-1}{6}(x^2-42x+480)\\ y = \dfrac{-1}{6}((x-21)^2+39)\\ y = \dfrac{-1}{6}(x-21)^2-\dfrac{13}{2}\\ (y-(-\dfrac{13}{2})) = -\dfrac{1}{6}(x-21)^2\\ -6(y-(-\dfrac{13}{2})) = (x-21)^2\\ 4(\dfrac{-3}{2})(y-(-\dfrac{13}{2})) = (x-21)^2\\\)

We can see from the equation that vertex of parabola = (21,-13/2) and focus is 3/2 units below the vertex.

Therefore the equation of directrix is y = (-13/2 + 3/2), i.e., y = -5.

\(\begin{cases} 3x+2y-2=4x+y-1\\ 3x-2y+2=2x+y-1\\ \end{cases}\\ \begin{cases} x-y+1=0 ---(1)\\ x-3y+3=0 --- (2)\\ \end{cases}\\ (1)-(2): 2y-2 = 0\\ y = 1 ---(3)\\ \text{Substitute (3) into (1)}\\ x-1+1=0\\ x=0\\ \therefore \begin{cases}x=0\\y=1\end{cases}\)

1) If A is rotated 90 degrees clockwise, the coordinates of the image of point A will be (1,4).

2)

Tank's capacity = \(\dfrac{1}{3}\times\pi\times(16)^2\times96 = 8192\pi\text{ cubic feet}\)

Volume of water = 25% * 8192pi = \(2048\pi\) cubic feet

By similar figures, base radius : height = 1 : 6

Base radius of water cone = Height/6

Let h feet be the height of the water cone.

\(\dfrac{1}{3}\times\pi\times(\dfrac{h}{6})^2*h=2048\pi\\ \dfrac{h^3}{108} = 2048\\ h^3 = 221184 = 2^{13}\cdot3^3\\ h = \sqrt[3]{2^{13}\cdot3^3} = 2^4\cdot3\cdot\sqrt[3]{2}=48\sqrt[3]{2}\)

Therefore a = 48, b = 2, a+b = 50 :)

~Smartest living cookie in the world :P

log 10 + log 10 + log 10 + log 10 = 4, and 10 - log 10 - log 10 - log 10 = 7 perhaps?

:P

log is a function, not a number.

Still 0.

In this case, you think of a number that when it multiplies itself, it gives out 0.

0 * 0 = 0, so \(\sqrt0\) = 0.

\(11011_b = (b^4 + b^3 + b + 1)_{10}\\ \quad 11011_b \times (b-1)\\ =\left((b^4+b^3+b+1)(b-1)\right)_{10}\\ =(b^5 - b^3 + b^2 - 1)_{10}\\ \\ (b^5-b^3+b^2-1) + (b^3+1) = b^5 + b^2\\ (b^5+b^2)_{10} = 100100_b\)

Done! Yay!

f(x) = ax + b

x = a*f^-1(x) + b

f^-1(x) = (1/a) * x - b/a

so you get 1/a = b, -b/a = a

because -b/a = a, b = -a^2

Only solution is a = -1, b = -1.

so a + b = -2.

ab + a + b = 524

ab + a + b + 1  = 525

(a+1)(b+1) = 525.

bc + b + c = 146

(b+1)(c+1) = 147.

cd + c +d = 104

(c+1)(d+1) = 105.

525 = 3 * 5^2 * 7

147 = 3 * 7^2

105 = 3*5*7

Most likely b+1 = 21, c+1 = 7 by finding GCD.

so b = 20, c = 6.

Now we find a and d using the values of b and c and the equations bolded.

21(a+1) = 525

a+1 = 25

a = 24

7(d+1) = 105

d+1 = 15

d = 14

So, a-d = 24 - 14 = 10