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.

a. 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.)

b. What is the height of the tide at 11:00 a.m?

5. 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.

6. 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

1. In Pascal's triangle, each number is the sum of the two numbers directly above it. The first and last numbers in each row are always 1. Additionally, the diagonals starting from the second row represent the triangular numbers. The middle number(s) in each row represent the central binomial coefficient(s). Finally, the sum of each row corresponds to the powers of 2 in binary form.

2. Ellipses are formed by a plane intersecting a cone at an angle that is not perpendicular to the base. Hyperbolas are formed when a plane intersects both halves of a double cone, resulting in two mirror-image curves. Parabolas are formed by a plane intersecting a cone at an angle parallel to one of the sides of the cone.

3. The inverse of the matrix B is:
[-7/22 2/11 1/22]
[3/22 0 -1/22]
[0 1/7 -1/7]

4.
a. Let t be the time (in hours) after a high tide. Then the function for the height of the tide above low tide is:
h(t) = 5sin(πt/6) + 5

b. To find the height of the tide at 11:00 a.m., we need to find t such that high tide occurred at t=1 and it is now 10 hours later (since the low tide occurs 6 hours after the high tide). So, t=11. Plugging this into the formula from part a, we get:
h(11) = 5sin(11π/6) + 5
≈ 0.4 ft

5. To rotate a point counterclockwise by 190 degrees about the origin, we can use the rotation matrix:
[cos(190°) -sin(190°)]
[sin(190°) cos(190°)]

Multiplying this matrix by the coordinates of the point (7,3), we get:
[-7.99 -2.43]

So the new coordinates of the point are approximately (-7.99, -2.43).

6. The reference angle for the angle of rotation is 1.11 radians (or approximately 63.43 degrees). The point on the terminal arm in the second quadrant with the same reference angle is (-2, 4). The point in the third quadrant is (-2, -4), and the point in the fourth quadrant is (2, -4).