1. What is (2, - p/6) in rectangular coordinates?

Question

1. What is (2, - π/6) in rectangular coordinates?

2. What is the vertex of f(x) = x² + 3x + 2?
3. The half-life of carbon 14 is 5,730 years. How old is a bone that has 70% of its original carbon 14?
4. Solve the system: x + y + z = 6, 2x – y + z = 3, x + y – z = 0
5. Solve the system: 2x + 3y = 7, 3x - 5y = 1

Answer 1

Hi Ekene,

 

A nice passel of problems !!

 

We'll consider each one in turn, below:

 

 

Problem #1:

 

(pi/6)(180/pi) = 180/6 = 30 degrees

 

The angle (-pi/6) or (-30 degrees) lies in quadrant IV where x is positive and y is negative.

 

sin(-30) = y/r

-0.5 = y/2

y = 2(-0.5)

y = -1

 

cos(-30) = x/r

√3/2 = x/2

x = √3

 

(2,-pi/6) = (√3,-1)

 

 

Problem #2:

 

f(x) = x² + 3x + 2 (standard form)

 

Convert to vertex form by completing the square:

    x² + 3x + 2 = 0 (set f(x) to 0)

    x² + 3x = -2

    x² + 3x + [1/2(3)]² = -2 + [1/2(3)]²

    x² + 3x + (1.5)² = -2 + (1.5)²

    x² + 3x + 2.25 = -2 + 2.25

    (x + 1.5)² = 0.25

    (x + 1.5)² - 0.25 = 0 (set f(x) to zero)

 

Vertex form:

    f(x) = a(x - h)² + k

             vertex (h,k)

    f(x) = (x + 1.5)² - 0.25

             vertex (-1.5,-0.25)

 

 

Problem #3:

 

Exponential Decay formula:

    F = Ne^kt

          F (final amount) = 70% of N = 0.7N

          N(initial amount) = N

          k(rate of decay) = -0.5 (minus sign, decrease)

          t(time) = unknown

 

Solving for unknown (t):

    0.7N = Ne^-0.5t

    0.7 = e^-0.5t

    ln(0.7) = ln(e^-0.5t)

    ln(0.7) = (-0.5t)ln(e)

    ln(0.7) = (-0.5t)(1)

    ln(0.7) = -0.5t

    t = ln(0.7)/-0.5

    t = -0.3567/-0.5

    t = 0.7134 (fraction of half life time)

 

(5,730)(0.7134) = 4,087.782

4,087.782 ≈ 4,088 years old

 

 

Problem #4:

 

Equation #1:  x + y + z = 6

Equation #2:  2x - y + z =  3

Equation #3:  x + y - z = 0

 

Equation #4:  Equation #1 - Equation #3

Equation #4:  2z = 6

    2z = 6

    z = 6/2

    z = 3

 

Equation #5:  Equation #2 + Equation #3      

Equation #5:  3x = 3

    3x = 3

    x = 3/3

    x = 1

 

Equation #1:

    x + y + z = 6

    1 + y + 3 = 6

    y + 4 = 6

    y = 6 - 4

    y = 2

 

Solution:  (1,2,3)

 

Check (Equation #1):

    x + y + z = 6

    1 + 2 + 3 = 6

    3 + 3 = 6

    6 = 6

 

Check (Equation #2):

    2x - y + z = 3

    2(1) - 2 + 3 = 3

    2 - 2 + 3 = 3

    0 + 3 = 3

    3 = 3

 

Check (Equation #3):

    x + y - z = 0

    1 + 2 - 3 = 0

    3 - 3 = 0

    0 = 0

 

 

Problem #5:

 

Equation #1:  2x + 3y = 7

Equation #2:  3x - 5y = 1

 

Equation #3: 3 times Equation #1

Equation #3:  6x + 9y = 21

 

Equation #4:  2 times Equation #2

Equation #4:  6x - 10y = 2

 

Equation #5:  Equation #3 - Equation #4

Equation #5:  19y = 19

    19y = 19

    y = 19/19

    y = 1

 

Equation #1:

    2x + 3y = 7

    2x + 3(1) = 7

    2x + 3 = 7

    2x = 7 - 3

    2x = 4

    x = 4/2

    x = 2

 

Solution:  (2,1)

 

Check (Equation #1):

    2x + 3y = 7

    2(2) + 3(1) = 7

    4 + 3 = 7

    7 = 7

 

Check (Equation #2):

    3x - 5y = 1

    3(2) - 5(1) = 1

    6 - 5 = 1

    1 = 1

 

 

Thanks for submitting these problems & glad to help.

 

God bless, Jordan (Romans 5:8)