1: Points A(-4,1), B(-1,4) and C(-1,1) are the vertices of triangle ABC. What will be the coordinates of the image of point A if triangle ABC is rotated 90 degrees clockwise about the origin?

2: The water tank in the diagram below is in the shape of an inverted right circular cone. The radius of its base is 16 feet, and its height is 96 feet. The water in the tank is 25% of the tank's capacity. The height of the water in the tank can be written in the form a*cuberoot{b}, where a and b are positive integers and b is not divisible by a perfect cube greater than 1. What is a+b?

Thank you!

AnonymousConfusedGuy May 24, 2018

#1**+1 **

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

MaxWong May 24, 2018