Answer the following questions, showing your work at each step:

Question

A. Pascal's triangle:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

B. Matrix to find the inverse of:

[0 -4 0]

[3 7 1]

[0 1 7]

1. The terms of Pascal's triangle consists of different combination arrangements. This special triangle contains triangular arrays of binomial coefficients. There are many interesting patterns in Pascal's triangle. Examine the rows, columns, and diagonals of Pascal's triangle and state the different patterns you see among the numbers.
2. Summarize the requirements that distinguish each of the conic sections (i.e. ellipse, hyperbola, parabola, etc.)
3. Find the inverse of the matrix B.
4. Ocean tides can be modeled by a sinusoidal function. Suppose that there is a low and high tide every 12 hours, and that high tide in Seattle occurs at 1:00 a.m. and 1:00 p.m. with the low tides 6 hours after high tides. Also suppose that the water level at high tide is 10 ft above the water level at low tide.
5. Find a formula for the function y=ℎ(t) that computes the height of the tide above low tide at time t. (In other words, y=0 corresponds to low tide.)
6. What is the height of the tide at 11:00 a.m?
7. In a video game an object represented by the point (7,3) is rotated counterclockwise 190 degrees about an origin. Find the new coordinates that represent the point.
8. The point (2, 4) lies on the terminal arm of an angle of rotation. Name the point in each of the other quadrants which has the same reference angle.

Answer 1

Let me give you some suggestions about how to proceed with these questions, and please feel free to follow up with precise questions if you still have them.

1) There is a pretty cool website "MathIsFun" which offers you reliable information. You can trust what you read and see on that website. Check out this article and see what you can discover on your own as well: https://www.mathsisfun.com/pascals-triangle.html

2) Same thing with conic sections (some really cool graphics to go along with the web page: https://www.mathsisfun.com/geometry/conic-sections.html

3) The matrix you gave is a "3 x 3" matrix. In order to find the inverse, you need to investigate the following: "determinant of a matrix" and "Gauss-Jordan Method" for finding the inverse of a matrix. You need to understand "row operations". If you need help with "row operations" I'd be happy to help with that.

4) I'll give you a hint: You are solving for the values of A, B, C and D in the function f(x) = y = Asin(Bx+C) + D, where the variable "x" is measuring time in hours.

5) Hint: The point (7, 3) is on the edge of a circle centered at (0,0). What is the radius? If you rotate 190 degrees counterclockwise, can you locate the point on the cartesian coordinate system? Start by estimating what the point would be, and then use trigonometry to find the new x and y coordinates. Go.

6) Another hint: Place the point (2, 4) on the edge of a circle centered at (0, 0). What is the radius of the circle? Now use "reflections" to place another point the same distance from the x axis in the other 3 quadrants. What are those coordinates? Reference angle == a measure of distance from the positive or negative x axis.

See if this helps.