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(a)

the amount of money after  0  years   =   500

the amount of money after  1  year     =   500  *  1.015

the amount of money after  2  years   =   500 * 1.015 * 1.015

the amount of money after  t  years   =   500 * 1.015^t

f(t)   =   500 * 1.015^t

(b)

We want to find the value of  t  when  f(t) = 800

800   =   500 * 1.015^t

1.6  =   1.015^t

log( 1.6 )  =   log( 1.015^t )

log( 1.6 )   =   t log( 1.015 )

log( 1.6 ) / log( 1.015 )   =   t

31.568   ≈   t

Since interest is added annualy, it would take  32  years for the account to have  $800 .

( After  31  years, it  has  $793.26  .  After 32 years, it has  $805.16 .)

In order to find the points where the parabola intersects, we set the equations to equal to each other in order to find a common x term. We get:

-x^2-x+1=2x^2-1 

Simplfying, we get 3x^2+x-2=0, and using the quadratic formula, we get that x can be equal to 2/3 or -1.

These are the x-values of the coordinates of the intersection points, or a and c. SInce c is greater than a, we get that c=2/3, and a=-1. 2/3-(-1) is equal to 1 2/3, or 5/3, which is our answer.

Ans=5/3

a)  The function we need is this :

P(t)  =  5655 (1.014)^t

b) In 2022, t  = 12......so we have

P(12)  = 5655 (1.014)¹²      ≈  6682  

If the number that is being raised to the power of  t is in the interval  (0, 1) , it is exponential decay.

If the number that is being raised to the power of  t  is greater than  1 , it is exponential growth.

For example,

y  =  3.7(0.2)^t       Here,  0.2  is being raised to the power of  t .

0.2  is in the interval  (0, 1)  so this equation represents exponential decay.

See if you can figure out the others.

And for this one...

g(x)  =  2.1(x)

Here, there is no exponent. So this equation is neither exponential growth nor exponential decay.

8/6=1.3

magnitude 8 earthquake is 1.3 times stronger than magnitude 6 earthquake.

did not see image as it did not load.

The normal  graph is shifted to the left by 4/3 units  and is up by 3

So....the correct graph is  the one on the bottom left

question 1(a) second part:

1.4% of 5655=79.17 increase in population

2022-2010=12 years

79.17*12 years=950.04

5655+950.04=6605.04

roughly 6055 population at 2022

Just go to the calculator here  and hover your cursor over the "log"  button

The "Ln"  function wil appear

Note...."log"  uses base 10

"Ln"  uses the constant "e" ≈  2.718.......

f(x)   =   -0.25(6)^x+2 - 1

The y-intercept is the value of the function when  x = 0 .

Plug in  0  for  x  .

f(0)   =   -0.25(6)⁰⁺² - 1

f(0)   =   -0.25(6)² - 1

f(0)   =   -0.25(36) - 1

f(0)   =   -9 - 1

f(0)   =   -10

The y-intercept is  -10

y  =  log ( 7x + 3) - 2

To find the y intercept, let x  = 0 

y  = log (3)  -  2  ≈   -1.52

To find the x intercept, let y  = 0

0   =  log ( x + 3)  - 2       add 2 to both sides

2  =  log (x + 3)    this  says that

10^2  =  x + 3

100 = x + 3        subtract 3 from both sides

97  =  x

thanks cphill i was not sure and how do you do natural log?

the number of bacteria after  0  hours   =   250

the number of bacteria after  1  hour   =   250 * 4

the number of bacteria after  2  hours   =   (250 * 4) * 4

the number of bacteria after  3  hours   =   ( (250 * 4) * 4 ) * 4   =   250 * 4 * 4 * 4   =   250 * 4³

the number of bacteria after  x  hours   =   250 * 4^x

y   =   250 * 4^x

x=0

y=4(f)=3

y=3

The general equation of a parabola with a vertex at  (-4, 0) is...

y   =   A(x + 4)² + 0

We want the parabola to pass through the point  (1, -75) , so we want a value of  A  such that...

-75   =   A(1 + 4)² + 0

-75  =  A(5)²

-75   =   A(25)

-75/25   =   A

-3  =  A

So the equation of a parabola with a vertex at  (-4, 0)  that passes through the point  (1, -75)  is...

y   =   -3(x + 4)² + 0

y   =   -3(x + 4)(x + 4)

y   =   -3(x² + 8x + 16)

This is the "natural log"

ln (121)   ≈  4.796

log(121)=2.08278537

=2.0828 (nearest ten thousandth)

17.

√[2x + 13 ]  - 5  = x         add 5 to both sides

√[ 2x + 13 ]   =  x + 5        square both sides

2x + 13    = x^2 + 10x + 25       subtract 2x, 13 from both sides

x^2 + 8x + 12  =  0   factor

(x + 6)  ( x + 2)  = 0

Set each factor to 0 and solve for x and we have that

x =  -6     or   x = - 2

The first solution does not solve the original equation

Only the second is good   

16.

3⁵√ [(x + 2)³ ]  +  3  = 27       subtract 3 from both sides

3⁵√ [(x + 2)³ ]    = 24              divide both sides by 3

⁵√ [(x + 2)³ ]   =   8                 take each side to the 5/3 power

(x + 2)  =  8^5/3                       and we can write

(x + 2)  = ( 8^1/3)⁵    

(x + 2)  =  2⁵      

x + 2    =  32                subtract 2 from both sides

x  = 30

4x+y

explanation to tertre's answer

√2x+13 -5=x

+5               +5

√2x+13= x+5

2x+13=(x+5)^2

2x+13=(x+5)(x+5)

2x+13=x^2+5x+5x+25

2x+13=x^2+10x+25

2x=x^2+10x+25-13

2x=x^2+10x+12

x=x^2+10x+12/2

x=0.5x^2+5x+6

question 20:

√3*√3=√9

√9=3

Assuming that we can have "non-sense" words

We could have  a word with an "A"  and this letter could appear in any of 4 positions......and we could choose any of the other three consonants to occupy a second positon, then any two of the consonants to occupy a third position, and the last consonant would appear in the last position by default.......so we have

4 * 3 * 2 *1  =      24 words using just the A  and the other three consonants

The same result would occur if we just used the "E" and the other three consonants

If we used the "A"  and the "E," we would have 24 words using any two of the other consonants......but, we could choose any 2 of the 3 remaining consonants.....so   the total "words"   would be  24 * (3C2)  = 24 * 3  =  72

So....the total number of words would  be 24 + 24 + 72  =  120 "words"

10.

f(x)  =  \(\sqrt[3]{x-4}\)

                               Instead of  f(x)  write  y .

y   =   \(\sqrt[3]{x-4}\)

                               Raise both sides to the power of  3 .

y³   =   x - 4

                               Add  4  to both sides.

y³ + 4  =   x

                               Now we have solved the original function for  x , so the inverse function is...

f^-1(x)   =   x³ + 4