Archive for the ‘General Questions’ Category

Problem 710

 

The letters x and y represent rectangular coordinates. Write the following equation using polar coordinates (rθ).

y = x

 

Solution:-

 

x = r * cosθ

y = r * sinθ

give the equation y = x

r cosθ = r sinθ (divide both sides by r cosθ)

1 = tanθ

tanθ = 1

 

Problem 709

Find z*w . Leave your answer in simplified polar form. Express all angles in degrees rounded to the nearest tenth.

z = 9(cos45+I sin45° )

w = 3(cos10+isin10°)

 

Solution:-

 

z.w = 9(cos45 + i sin45)*3(cos10+i sin10)

z.w=9*3(cos(45+10)+isin(45+10))

z.w=27(cos55°+i sin55°)

 

Problem 701

A woman walks 200 yards west along a straight shoreline and then swims 50 yards north into the ocean on a line that is perpendicular to the shoreline. Using her starting point as the pole and the east direction as the polar axis, give her current position polar coordinates. Round the coordinates to the nearest hundredth. Express θ in degrees.

 

Solution:-

(206.16,165.96^{\circ})

 

 

Problem 700

Find (x3-6x2+10)÷ (x-3) using synthetic division.

 

Solution:-

 

x-3 | 1  -6  0   10

| 0   3  -9   -27

—————

1  -3   -9   -17
x^{3}-6x^{2}+10 = x^{2} -3x -9 + \frac{-17}{x-3}

 

Problem 699

Solve

x2 + 6x -7 = 0 by completing the square.

 

Solution:-

 

Solve for x by first isolating the constant and adding the square of half of the coefficient of x to both sides of the equation.

 

x^2 +6x  -7 + 16 – 16  = 0

x^2 +6x + 9 – 16 =0

(x+3)^{2} = 16

(x+3)^{2} = 4^{2}

x +3 =± 4

x = -3 ± 4

x = -3+4 or -3 – 4

x = 1 ,-7

 

Problem 698

The value of a particular investment follows a pattern of exponential growth. You invested money in a money market account. The value of your investment t years after your initial investment is given by the exponential growth model   A = 10,800 e0.125t . How much will be in the account after 12 years? Round the answer to the nearest dollar.

 

Solution:-

A = pe^{rt}

t = 12 years

A = 10800e^{0.125*12}

A = 48402

 

Problem 695

Evaluate the expression: log100

 

Solution:-

 

log 100

log 10^{2}= 2 log 10

we know log 10 = 1

log 100 = 2(1) = 2

 

Ludo and dices

Suppose in a game of ludo only two throws of dice are allowed. What is the most probable distance moved and what is the probability of the same? Hint:- Distance moved is the sum of numbers obtained on dices.

Problem 18

Problem – 18

What is the probability that Bo, Colleen, Jeff, and Rohini win the first, second, third, and fourth prizes, respectively, in a drawing if 51 people enter a contest and

a) no one can win more than one prize?

b) winning more than one prize is allowed?

 

Solution :

Probability when no one can win the prize = \frac{1}{51\times50\times49\times48}= \frac{1}{5997600}

Probability when more than one prize is allowed = \frac{1}{51^4}= \frac{1}{6765201}

 

Problem 17

Problem – 17

Find the probability of selecting none of the correct six integers in a lottery, where the order in which these integers are selected does not matter, from the positive integers not exceeding67.

 

Solution :

Total ways to selecting the 6 numbers from 67 = C(67,6)

Total ways to selecting the 6 incorrect numbers from (67-6=61) 61 numbers = C(61,6)

So probability = \frac{c(61,6)}{c(67,6)} = 0.56