CPhill

Nice work, OneDirection and Solveit.....points from me.....!!!

Here's  a way to confirm this......

x + y + z  = 13

x * y * z > 30

Now ....assume z = 9.....but this means that x and y could only be 1 and 3  since 1 + 3 + 9 = 13......but 1 * 3 * 9  = 27

And that's < 30

So......let's assume that  z = 7

So  z = 13 - x - y  →  7  = 13 - x - y  →   x + y   = 13 - 7  →  x + y  = 6

Thus.....either x = 1 and y = 5   or   vice-versa

So......if we write the number as xyz, the only possibility is  xyz  =  157

1)   The tangent  of the angle at any time is given by:

tan(θ)  = y / 120     differentiating this with respect to  time, t, we have

sec^2(θ) dθ/dt  =   (1/120) dy/dt    (1)

When the helicopter is  90m above the ground, θ  is given by

arctan(90/120)  = about .6435 rads   ....so....filling in the missing pieces in (1), we have

sec^2(.6435) dθ/dt  = (1/120m) (2.5m/s)

dθ/dt  =  (1/120m) (2.5m/s)/ sec^2(.6435)  = about .01334 rads/s

2)  The area  of the square  = x^2

The perimeter  of the circle = 100 - 4x  = 2*pi*r

Therefore.....the radius of the circle is   [50 -2x] /pi

So....the total area is given by :

A  = x^2  +  pi ( [50 -2x] /pi )^2  =   x^2 + (1/pi) *(50-2x)^2

Take the derivative of this and set it to 0

A' =  2x  - 2* (2/pi) [ 50 - 2x]  =

2x + 8[x -25] /pi   =  0

x + 4[x - 25]/ pi  = 0

pi*x + 4x - 100  =0

x (pi + 4)  = 100

x = 100 / (pi + 4)  = about 14.002 cm

The second derivative =

1  + 4/pi   .....since this is always positive, the curve is concave up at all points....so x = 14.002cm is a minimum

And this is a minimum as verified by the curve shown here : https://www.desmos.com/calculator/orqtlce6ju

The minimum total area  is about  350.062 cm^2

Frankly, OneDirection, I find this calculator on the website a little clunky for solving equations....

I like WolframAlpha   ........http://www.wolframalpha.com/input/

Just type your equation in the box and hit "Enter"

Thanks for that, heureka.....I've never seen it before  !!!!

Thanks, rarinstraw.....if we just wanted to leave it in radical form, we would have....

√129+√216+√68+√169  =

√(43 * 3) +√(36 * 6) +√4 * 17) +√(13*13)   =

√129  + 6√6  + 2√17  + 13

Not really sure if it's much of an improvement.......!!!!!

Area  of top and bottom  = 2[ pi*R^2  - pi*r^2]  =  2pi[ R^2  - r^2]  = 2pi(R + r) (R - r)

Area  of inside of smaller cylinder  = 2*pi*r*h

Area of outside of larger cylinder = 2*pi*R*h

Total Lateral (side) surface area  = 2*pi*h*[R + r]

Total surface area = 2*pi(R + r) (R - r) + 2*pi*h*[R + r]  = 2*pi*[R + r] [ h + R - r] =

2*pi * [ (6 + 3)mm] [(18 + 6 - 3) mm]  = 1187.52 mm^2

The sine of an angle only takes on values from -1 to + 1   ....so.....sin(alpha)  = 78   is impossible

Note, Yura_chan......

t - 4     can be factored  as

(√(t)  - 2) (√(t)  + 2) 

Because

√(t) * √(t)   - 2√(t)  + 2√(t)  (-2)*(+2)   =

t  -  4

Thanks, TR.....I'll do it by substitution just so you can have two different methods to look at

4x-y=-5   →    y  = 4x + 5     (1)

-2x+3y=10   (2)

Substitute  (1)  into (2)  for y

-2x + 3(4x + 5)   = 10    simplify

-2x + 12x + 15  = 10

10x + 15  = 10      subtract 15 from both sides

10x  = -5           divide both sides by 1

x = -5/10  = -1/2

And using (1)    y = 4(-1/2)  + 5   =  -2 + 5   =  3

Thanks for that, TR.......couldn't have said it better, myself.....