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Let   the side of the  hexagon  =  S

Its  area  =   6(1/2)S^2 sin60“    =    3S^2sin 60°

We can find  the  side  of  the  triangle  using the Law of Cosines

This is

sqrt  [ S^2  + S^2  - 2S^2 cos (120°)]  =  sqrt [  2S^2 + S^2]  =  S√3

The area of  this equilateral triangle  is  (1/2)(S√3)^2 sin60° = (3/2)S^2sin60°

So  the ratio  is

(3/2)S^2 sin 60°

_____________   =    (3/2)  / 3   =    1  /  2

     3S^2 sin 60°

P(two red)  =  (2/5)(1/4)  = 2/20  = 1/10

P(two black)   =  (3/5)(2/4)  = 6/20  = 3/10

Add the probabilities =   4/10  =   2 / 5

y  =  2x^2 - 3x + 7

When

x  = -2  y  = 21

x = -1   y  = 12

x = 3  y  =  16

x = 4, y  = 27

The largest  integer in the range  is   27

This can't be so, bioplant......if we  have $25  price increases/decreases, the  rent should  be  750  plus/minus some multiple  of $25.....we won't have  any "partial" dollar amount for rent

Note that triangles  BFG  and  DJI  are isosceles

angle BFG  =angle BGF  =  [180 - 36] /2  =  144/2  = 72°

angle DJI  = angle DIJ  = [ 180 - 39]/ 2  = 70.5°

angles EGF  and DFG   are supplemental to angles  BFG and BGF....so   they measure  108- 72  = 108”  

Likewise  angles FJI  and CIJ  are sumpplementary to  angles DJI and DIJ....so  they measure 180 - 70.5  = 109.5°

And  in pentagon FGHOJ  the sum of its interior angles  = 540°

Therefore

EGF  + DFG  + FJI + CIJ  + GHI   = 540

108 + 108  +  109.5  + 109.5  + GHI  =540

435  + GHI   = 540

GHI  = 540  - 435  =  105°

But angle EHI  is supplemental  to GHI  so its measure    =  180 - 105  =  75°  = x 

There are: 114  133  152  171  190  209  228  247  266  285  399  Total =  11  - such integers.

I think you can but only give out a certain number of points over any given period of time (to anyone, including yourself). After that they do not register. 

Also sometimes they do not get added straight away.

[142 - 136] / 4  + [142 - 130] /4 + [142 - 128] / 4  + [142 - 124] / 4 =1.5 + 3 + 3.5 + 4.5 = 12.5 years difference between the average of oldest and the youngest.

142 / 4 =35.5 years the average of the oldest 4

35.5 - 12.5 = 23 years - the youngest.

\(\frac{3}{1.2} = \frac{30}{12} = \boxed{\frac{5}{2}}\)
to check, we can do the same thing with the \(y\) values:
\(\frac{9}{3.6} = \frac{90}{36} = \frac{15}{6} = \frac{5}{2} \)
yup, the answer is right :D

posted my answer to the original post :)

we first solve for \(f(10)\):

\(\begin{align*} f(10) &= 3(10)^2 - 2(10) + 4 \\ &= 300 - 20 + 4 \\ &= 284 \end{align*}\)

then, we solve for \(g(10)\) in terms of \(k\):

\(\begin{align*} g(10) &= 10^2 - 10k - 6 \\ &= 100 - 10k - 6 \\ &= 94 - 10k \end{align*}\)

plugging in these two values into \(f(10) - g(10) = 10\), we get

\(\begin{align*} 284 - (94 - 10k) &= 10 \\ 284 - 94 + 10k &= 10 \\ 190 + 10k &= 10 \\ 10k &= -180 \\ k &= \boxed{-18}. \end{align*} \)

Have you tried the sum of cubes formula? Try expressing the left side term with the sum of cubes factorization(the 5 term and 1).

Realize that if the value of x is less than 1, that causes the numerator to be the square root of an imaginary(since 0-1 would be -1), which is clearly non-real. The smallest possible integer value for x then becomes 1.

Great solution!!!

Yes that makes sense! Thank you!!

i'm not quite sure i'm interpreting the problem right, but i'm going to solve the problem assuming that the problem is asking you to find the leg length of a 45-45-90 triangle, where one leg is on the ground and another leg is the height of the treehouse, with the 90-foot rope being the hypotenuse.

we know that the hypotenuse of a 45-45-90 triangle with leg length \(x\) is \(x\sqrt2\), so we know that the leg length of a 45-45-90 triangle with hypotenuse \(y\) is \(\frac{y}{\sqrt2} = \frac{y\sqrt2}{2}\), so plugging in \(y = 90\) (since the rope is the hypotenuse), we get 

\(\frac{y\sqrt2}{2} = \frac{90\sqrt2}{2} = \boxed{45\sqrt2}.\)

in general, a line given two vectors can be parametrized as :

a + t*d 

where:

a = starting vector

t = any real number

d = the direction vector

In this case, we have (1,1,2) and (2,3,4)

the direction vector can be found through subtracting the two. We get:

(2,3,4) - (1,1,2) = (1,2,2)

with our starting point being (1,1,2)

We then get the line equation as:

(1,1,2) + t(1,2,2)

this then turns into:

(1,1,2) + (t,2t,2t)

Adding the two, we get:
(t+1,2t+1,2t+2)

a = 1

b = 1

c = 2

d = 1

e = 2

f = 2

c/a = 2/1 = 2

e/a = 2/1 = 2

the ordered pair (c/a, e/a) is then (2,2). Please correct me if I'm wrong, precalc has never been one of my strong suits. 

Great explanation!

just a quick solution for anyone else confused by the question : )
 

the circumfrence of a circle with a radius of \(2\) is \(2\pi r = 4\pi\). the perimeter of a square with side length \(x\) would be \(4x\), so we have 

\(\begin{align*} 4\pi &= 4x \\ x &= \boxed{\pi}. \end{align*}\)

23

Nice solution!

the graph of \(y = f(x)\) has the line \(x=5\) as an axis of symmetry. the graph also passes through the point \((8, -7)\). find anohter point that must lie on the graph of \(y = f(x)\).
 

since \(x=5\) is the line of symmetry, we know that the point \((8, -7)\) reflected over \(x=5\) must also be on the graph of \(y = f(x)\). \((8, -7)\) is 3 units away from \(x=5\) on the \(y\)-axis, so the point 

\((8 - 2(8-5), -7) = (8 - 6, -7) = \boxed{(2, -7)}\)

must also be on the graph ​of \(y = f(x)\).

LOL! I solved this question in my live AOPs Class yesturday