CPhill

1 + sin xcos^2x = sinx

We need to be careful about dividing through by a trig function, YEEEEEET...we might be throwing away a possible solution!!!

We can wrtite this as

1 + sin x ( 1 - sin ^2 x)  = sin x

1 +sin x  - sin ^3 x = sin x

1 - sin^3 x =  0

sin^3x  - 1 = 0    factor as a difference of cubes

(sin x - 1) (sin^2x + sin x + 1) = 0      

The second factor produces no real solutions for   x

For the first factor

sin x - 1 = 0

sin x = 1

And this happens when x = 90°

The graph here confirms this

https://www.desmos.com/calculator/lljppoztzo

The exterior angles of a k-sided polygon form an arithmetic sequence. The smallest and largest interior angles of the polygon are 136 and 176. What is k?

The sum of the exterior angles of any polygon = 360°

So....the exterior angle of the 176°   interior angle = 180 - 176 =  4°

And the exterior angle of the 136° interior angle = 180 - 136  = 44°

So.........using the sum for an arithmetic series we have that

360 = (k/2) (4 + 44)

360 = k  (48/2)

360 =  24k

15 = k

A square tile, a regular hexagon tile, and a regular n-sided polygon tile share a common vertex. There are no gaps or overlaps between the tiles. What is n?

The interior angle of the hexagon =120°

The interior angle of the square = 90°

So....the interior angle of the remaining regular polygon = 360 - 120 - 90  =   150“

So  we can solve this

(n - 2)180

________  =      150

     n

(n - 2) 180  = 150n

180n - 360 =  150n

180n - 150n = 360

30n =  360

n = 12

We can use your formula from yesterday to answer this

1/4 + 1/T = 1/3         where T is the time (in hours) for the small snowplow to clear the lot

Subtract 1/4 rom both sides

1/T =  1/3 - 1/4

1 /T = 1/12       "flip" both fractions

T = 12   =    12 hours

x^3 - 7x + 6x = 0    simplify

x^3 - x = 0             factor

x ( x^2 - 1)     =

x(x + 1) (x - 1) = 0

Setting each factor to 0 and solving for x   produces  

x = 0        x = -1       and    x = 1

Here's one more way

Let x =  rcos(theta)    and y = rsin(theta)

Since the circle ha a radius of 1......these become x = cos(theta)  and y = sin (theta)

So     let  A =  (x  + y)^2 =   x^2 + 2xy + y^2  =   cos^2(theta) + 2sin(theta)os(theta) + sin^2 (theta)  =

2sin(theta)cos(theta) + 1          take the derivative of this and set to 0

A' = 2sin^2(theta) - 2cos^2(theta) = 0

sin^2(theta) - cos^2(theta) = 0        factor

[sin (theta)  + cos(theta) ] [ sin (theta - cos(theta) ] = 0

Set both factors to 0 and solve for theta

sin (theta) + cos(theta) = 0                   sin (theta) - cos(theta) = 0

The solutions of the first = 3pi/4  and 7pi/4       these do not produce maxes

The solutions for the second =  pi/4 and 5pi/4

And x, y at these angles are   ( √2/2, √ 2/2)  and (-√2/2, -√2/2)

And these will produce the max for A  =    2

I remember having such a problem in my Calculus class, but I've slept since then and I don't remember the procedure for solving it....however.....I believe the surface area is minimized when each dimension is = ∛36 units

So...the minimum surface area is

2  [  (∛36)^2 + (∛36)^2 + (∛36)^2 ]  ≈ 65.416 units^2

Maybe someone else on here knows the procedure

OK....!!!

Using a little trial and error (and some logic)

Note that 72 is divisible by 3....so....the sum of the digits of the total price must also be divisible by 3

Also...since we have 72 turkeys.....the final digit in the price must be an even number 

If we assume that the last digit of the price is 0

We have these possibilities for the price

$267. 90

$567.90

$867.90   however....none of these are evenly divisible by 72

If we assume that the final digit in the price is a 2....we have the following :

$367.92

Dividing this by 72 gives us the price per turkey = $5.11

x^2 + y^2= 1      is a circle with a radius of 1 centered at the origin

Using implicit differentiation

(x + y)^2 =

x^2 + 2xy + y^2

2x  + 2y + 2xy' + 2yy'

y' [ 2x + 2y]  =  -2x - 2y

y'  =  -2 (x + y) / [ 2 (x + y ) ]

y' =  -1      

Ths is the slope of the tangent line to the circle at the points where the function is maxed

And this line will be perependicular to  the  line through the origin with a slope of 1  ( y = x )

But in a circle with a radius of 1 centered at the origin, this line will intersect the circle at  (√2/2, √2/2) and (-√2/2, -√2/ 2)

And using impiicit differentiation again on the circular function we get that

2x + 2yy' = 0

2yy' = - 2x

y' = -2x/2y  = -x/y

And this gives the slope of any  tangent line to the circle

Plugging in the given points will give us the slope of -1   which is what we want

So   (x + y)^2 is maxed at either (√2/2, √2/2) or (-√2/2, -√2/ 2)

And the max is    ( √2/2 +√2/2 )^2  =  ( √2 )^2  =   2