Given the following Axioms of Fano's geometry:

1. There exists at least one line

2. Each line is on exactly 3 points

3. Not all points are on the same line

4. Each pair of points are on exactly one line

5. Each pair of lines are on at least one point

a) Prove every point is on exactly three lines

b) What geometries are possible if you eliminate Axiom 5?

1. Jade has $24,500 to invest. She decides to divide it into the three accounts listed. At the end of the year, she has earned $1,300 in interest. If the amount invested in Hanes Oil was four times the amount of money invested in Gary Games, how much did Jade invest in each account?

1. Write down the system of equations describing this situation. Be sure to define the variables.

b) State the matrix for this system of equations.

c) Write down the reduced-row echelon form (i.e., the “answer matrix”) given by your calculator.

d) Explain in 1-2 sentences the meaning of these results.

create a speaker box with a volume of 14400 cubic inches. The box will be used for speakers of various sizes. The speaker box must have a square base and minimal surface area ( prior to cutting holes for the speakers ). Find the box’s dimensions. Your must construct a diagram [ V = LWH ]

2.) (12 pts.) Show that F = ( xy/(1+x^2y^2) + 1 + arctan(xy))i+ (x^2/(1+x^2y^2-1)j is a conservative vector field. Then use the Fundamental Theorem for Line Integrals to find the Work done by F from point (0,0) to point (2, 1/2).

Find the point of intersection of the lines (x-2)/- 3 = (y-2)/6 = z-3 & (x-3)/2 = y+5 = (z+2)/4. Write the answer as (a, b, c). If they are not cut, write: NO

A) Using a calculus approach, sketch p(x) = -3*x^2 – 300x – 140.

B) What is the largest rectangle that can be inscribed inside the curve, if two of its sides are on the x-axis and the other two lie on the positive portion of the curve.

find the limit as k approaches infinity of [1+(1/10^k)]^10k

step by step using the l'hospital

1. Solve by separation of variables

yy'= xy2 + x, y(1)=0

a. Solve by the integrating factor method

xy'+(x+2)y = ex    

b. Solve the Bernoulli’s Equation

x2y'−3y2 +2xy = 0, y(2) =5

c. Solve by undetermined coefficients

y"− y = x+sinx, y(0)= 2,y'(0) =3

Suppose you have a trapezoid a vertical height of H with bottom and top widths of B and T. Suppose we draw a line of height h that is parallel to the base of the trapezoid. What are the dimensions of the new trapezoid? (i.e. top widths and height)

A company uses TV and magazines for advertising. They know that profit P is related to. the amounts T spent of TV and M spent on magazines by the equation P= 24MT -4M-3T+1. where P, M, T are in hundreds of thousands. Find the maximum profit.

A fast-food restaurant determines the cost and revenue models for its hamburgers.

Find the equation of the tangent plane and the parametric equations for the normal line to the surface x² + y² - z = 0 at the point P(4,-1, 6). Show all steps

The count in a bacteria culture was 800 after 20 minutes and 1100 after 30 minutes. Assuming the count grows exponentially,

What was the initial size of the culture?

Find the doubling period.

Find the population after 70 minutes.

When will the population reach 10000. You may enter the exact value or round to 2 decimal places.

solve using the multiplication principle first. Then add

1/3x+3/2y=5/6

1/4x+1/2y=3/8