Solving for p and q:

Given: [ x = \sqrt{a}(\sin u + \cos v) ] [ y = \sqrt{a}(\cos u - \sin v) ] [ z = 1 + \sin(u - v) ]

We need to express ( u ) and ( v ) in terms of ( x, y, z, ) and ( a ). Let’s start by squaring the equations for ( x ) and ( y ):

[ x^2 = a(\sin u + \cos v)^2 = a(\sin^2 u + 2\sin u\cos v + \cos^2 v) = a(1 + 2\sin u\cos v) ] [ y^2 = a(\cos u - \sin v)^2 = a(\cos^2 u - 2\cos u\sin v + \sin^2 v) = a(1 - 2\cos u\sin v) ]

Adding these two equations gives: [ x^2 + y^2 = 2a(1 + \sin(u - v)) = 2a(1 + z - 1) = 2az ]

This simplifies to: [ z = (x^2 + y^2)/(2a) ]

Now, let’s find an expression for ( p = (\tan u)/(\tan v) ):

Dividing the equation for ( x ) by the equation for ( y ), we get: [ p = (\tan u)/(\tan v) = (\sin u + \cos v)/(\cos u - sin v) = (x/\sqrt{a})/(y/\sqrt{a}) = x/y]

Similarly, let’s find an expression for ( q = (\cotan(u))/(\cotan(v))) :

Dividing the equation for ( x) by the equation for ( y) gives: [ q=(\cotan(u))/(\cotan(v))=(\cos(v)+\sin(u))/(-\sin(v)+\cos(u))=-(x/\sqrt{a})/(y/\sqrt{a})=-x/y=-p]

Therefore, p=-q.

Hence, the values of p and q are such that p=-q.

Solve the equation for t:

Solve the following equations for x:

6x - 3(3x + 2) = 3(x + 7)

Simplify the numerical expression and write in scientificnotation.

(0.0042)(280,000)

4,900,000

Simplify the expression using Order of Operations:

14 - 8

Vivian bucycles 160 miles at the rate of r mph. The same trip would have taken 2 hours longer if she had decreased her speed by 4mph. Find the r.

(b) Using the formula "distance = rate * time ", write an equation for the trip taken at a rate of r miles per hour, including a variable to represent the time, with the distance as stated in the description.

(c) Using the variable r for the rate, write an expression for this rate decreased by 4 miles per hour

(d) Using the time variable from (b), write an expression for this time increased by 2 hours

(e) Using the "distance = rate * time " formula, with the expression in (c) and (d), write an equation for the trip taken at the slower speed, with the distance as stated in the description.

(f) Subtract the equation in (e) from the equation in (b), to get a new equation (subtract both sides of the equations)

(g) Solve the equation in (b) for the time variable, to express the time variable as an expression in terms of the rate r

(h) Replace the time variable in the equation in (f) with the expression for the time variable obtained in (g), to get new equation with only the r variable (the time variable is eliminated)

(i) Clear the equation in (h) of fractions, by multiplying each term by the r variable, to get a quadratic equation for the r variable

(j) Use the quadratic formula to solve the equation in (i) for the value of the r variable