CPhill

Heck, EP.....you beat me to it....LOL!!!!

3^(2x + 2)  + 27^(x + 1)  = 36      we can write

[ 3^2] ^(x + 1)  + [3^3]^(x + 1)] = 36       factor the left side

[3^2]^(x + 1) [ 1 + 3] = 36

[3^2]^(x + 1) [4] = 36       divide both sides by 4

[3^2]^(x + 1) = 9   

[9]^(x + 1) = 9^1

Since the bases are the same, solve for the exponents

x + 1    = 1      subtract 1 from both sides

x = 0

LOL!!!!....me neither....!!!!

 3^(2x+2)+27^(x+1)=36

Same here. Guest....

By inspection, the only integer solution is   x = 0

Thanks EP....here's another way using symmetry about the y axis

a.  Let the ends of the arch be at  (-10, 0)  and (10,0)

Let the apex of the arch be at (0,80)

So....we   just need to find    "a" in this function

y = ax^2 + 80 

We know the point (10, 0) = (x, y) is on the graph...so....filling in what we know, we have

0 = a(10)^2 + 80     

0 = 100a + 80     subtract 80 from both sides

-80 = 100a     divide both sides by 100

-80/ 100 = a = - 4/5

So....the function is

y = (-4/5)x^2 + 80

b.   At 20 ft above the ground, we have

20 = (-4/5)x^2 + 80       subtract 80 from both sides

-60 = (-4/5)x^2        multiply both sides by -5/4

75 = x^2      take the positve root

sqrt (75) = x =  5sqrt(3) ft

But this is only half the width...so multiplying by 2 we have

Width = 10sqrt(3) ≈ 17.3 ft

To have a domain of all real numbers

1x^2 + 4x + c     must not = 0

We have the form  ax^2 + bx + c

We need to find the  x root that makes this 0.... to do this,set the discriminant to 0

b^2 - 4ac  =   0     

4^2 - 4(1)(c) = 0

4^2 - 4c = 0

And   when c = 4....we will have one root that makes the denominator = 0

So.... c = 5  is the smallest value that produces a non-zero denominator for all x values because the value makes the discriminant < 0 and we will have no real roots 

Look at the graph here to confirm this : https://www.desmos.com/calculator/4qtwwfaidb

When c is an integer < 4, we have two zeroes

When c = 4, we have one zero

When c is an integer > 4, we have no real zeroes

E(3)  =   5

At Disneyland they sell candy to fundraise for outside charities. They sell about 5500 candy bars at a profit of $0.75/each. Each $0.50 increase in the price will increase the profit of each candy bar by $0.50. However, they will sell about 1000 fewer candy bars. 

A is correct

B    Let x be the number of .50 increases....and for each N, they sell 1000 fewer candy bars

So we have

Profit =  Quantity * Price

Profit  =  ( 5500 - 1000x) (.75 + .50x)

C   If they increase the price by 1.00, x = 2   and we have

Profit =  (5500 - 1000 (2) ) ( .75 + .50(2) ) =

              (3500) ( 1.75)  =  $6125

D   Let us simplify the function as

Profit = 4125 - 750x + 2750x - 500x^2 =    - 500x^2 + 2000x + 4125 

We    have the form     Ax^2 + Bx + C

A = -500    B = 2000

The number of x increases that maximize the profit is given by

-B / [ 2A]  =     -2000 / [ 2 * -500] =  2000 / 1000 =   2  increases

E    So...a  $ 1.00 increase maximizes the profit...and we have  seen that this profit is $6175

Here's a graph to show this : https://www.desmos.com/calculator/lhabxwm2yz

First one

C(30, 3)    = 4060 different combinations

Second one

30 possibilities * 29possibilities * 28 possibilities * ......* 3 * 2 * 1    =   30!

Let x be the horizontal distance that the ladder is from the wall = 6 ft

At that time....the top of the ladder is sqrt (r^2 - [6ft]^2) = sqrt (r^2 - 36ft^2) ft from the floor  

We have this

r^2  = x^2 + y^2 

Take the derivative of each term with respect to time

2r dr/dt = 2x dx/ dt   + 2y dy/dt  

y = sqrt (r^2 - 6^2)   = sqrt (r^2 - 36)

dr/dt = 0      [ the ladder length is constant ]

So when x  = 6 ft , the distance the bottom of the ladder is from the wall is increasing and the distance the top of the ladder is from the flooor is decreasing

So dx / dt = 2 ft / s     and dy / dt =   -3ft / s

So we have 

0 =    2(6ft) (2 ft/s) + 2sqrt (r^2 - 36ft^2)ft *(-3 ft/s)     

Divide out ft/s  

0 = 24ft   -  6sqrt (r^2 - 36ft^2) ft

6 sqrt (r^2 - 36ft^2) ft = 24 ft     divide both sides by 6

sqrt (r^2 - 36ft^2 ) =  4 ft        square both sides

r^2  - 36ft^2  = 16ft^2

r^2 =  (36 + 16) ft^2       take the square root of both sides

r = sqrt (52) ft  ≈  7.2 ft =  the ladder length