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Algebra: Geometric Progression and Acceleration

1. a) Transpose the following formula to make v the subject:
f = uv/(u + v)

b) Solve the folowing equation to find the value of x:
(3.4)^2x+3 = 8.5

c) In the formula theta = Ve^(-Rt/L), the value of theta = 58, V = 255, R = 0.1 and L = 0.5. Find the corresponding value of t.

d) w = 1/h ln (L/L_0 - 1) Find L if w = -2.6, L_0 = 16 and h = 1.5.

2. a) Use polynomial long division to determine the quotient when
3x^3 - 5x^2 + 10x + 4 is divided by 3x +1.

b) Show, by polynomial long division, that
(x^3 - 3x^2 + 12x - 5) / (x - 2) = (x^2 - x + 10) + 15 / (x - 2)

3. A ball is thrown down at 72 km/h^-1 speed from the top of a building. The building is 125m tall. The distance travelled before it reached the ground is as follows,
s = u_0t + 1/2*gt^2
u_0 = inital velocity (m*s^-1)
g = acceleration due to gravity (10 m*s^-2)
t = time (s)

a) Find the time for the ball to drop to fifth of the height of the building.
b) Find the time for the ball to reach the ground.

4. Water fills a tank at a rate of 150 litres during the first house, 350 litres during the second hour, 550 litres during the third hour and so on. Find the number of hours necessary to fill a rectangular tank 16m x 9m x 9m.

5. a) A firm starts work with 110 employees for the 1st week. The number of employees rises by 6% per week. How many persons will be employed in the 20th week if the resent rate of expansion continues?

b) A contractor hires out machinery. In the first year of hiring out one piece of equipment the profit is $6000, but this diminishes by 5% in successive years. Show that the annual profits form a geometric progression and fin the total of all the profits for the first five year.

Solution Preview

Please see the attachment for full solutions.

1. a) v = fu / (u - f)