All Answers 355923 Answers

Why in the world are you posting copyrighted problems? It's illegal, and it also deprives the whole point of the Geometry class. The whole point of the homework is to practice the problems so you can study for the midterm, and asking others to do it for you won't help at all. Besides, the AoPS Admins already know you are cheating, so doing this won't help at all.

10)The line y = (3x + 20)/4 intersects a circle centered at the origin at A and B. We know the length of chord AB is 20. Find the area of the circle.

We can write the line as

4y  = 3x + 20

3x  - 4y  + 20  = 0

The perpendicular  distance between the center of the circle and the line will bisect the chord

And this distance is

l  3(0) - 4(0)  + 20  l            20

________________  =   ______ =     4

    √[3^2 + 4^2] =                 5

A right triangle will be formed by the bisected chord, the perpendicular distance from the center to the midpoint of the chord and the radius which will form the hypotenuse.

So....the radius  is √[10^2 + 4^2 ]  =  √116  units

And the area  of the circle is

pi *  ( √116)^2  =

116 pi   units^2

f(x)  = 4x + p

g(x)  = px + 1

I think this is  (f ° g )(x)  = 12x + q

f ( g)  =   4(px + 1) + p     =  12x + q

4px + 4 + p   =  12x + q

4p  = 12      so  p  = 3

And

4 +  p  = q

4 + 3  = q  

7  = q

We have

The choice of any two toppings plus  1 with no toppings  =  106

C(n, 2)  +  1  = 106

C(n ,2)  = 105

We need to solve this

     n!

________  =   105

(n - 2)! 2!

     n!

______  =   105 * 2!

( n - 2)!

(n) (n - 1)  =  210

n^2  - n  - 210  = 0      factor

(n -15) ( n + 14)  =  0

Setting both factors to 0 and solving for n produces  n  = 15   or  n  = -14

Since n is positive....there are 15 toppings

What is the northerly component of a wind blowing at 15ms-1

from the South East?

No...I just can't add  !!....LOL!!!

Solve for x :

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

Take the natural logarithm of both sides and use the identities log(a b) = log(a) + log(b) and log(a^b) = b log(a):

log(5) (x + 2) + log(2) = log(2) (x + 5) + log(5)

Expand out terms of the left hand side:

log(5) x + log(2) + 2 log(5) = log(2) (x + 5) + log(5)

Expand out terms of the right hand side:

log(5) x + log(2) + 2 log(5) = log(2) x + 5 log(2) + log(5)

Subtract x log(2) + log(2) + 2 log(5) from both sides:

(log(5) - log(2)) x = 4 log(2) - log(5)

Divide both sides by log(5) - log(2):

x = (4 log(2) - log(5))/(log(5) - log(2))=1.2694123......

We need to solve this

      n!  

________  =   365

(n -4)!* 4!

n!

_____   =   365 * 4!

(n - 4)!

n * ( n - 1) (n -2) (n - 3)  =  365 * 24

n^4 - 6 n^3 + 11 n^2 - 6n  = 8760

n^4  -6n^3 + 11n^2  - 6n - 8760   =   0

Solving this for n produces the positive solution  of  ≈ 11.239

So....we need 12 friends  to assure that the same 4 are not  invited

Proof

C(12,4)   = 495  different groups

C(11,4)  = 330    different groups  [ not enough ]

Minor typo. Should be 56.

8 heads =  C(10,8)   = 45

9 heads  = C(10,9)  = 10

10 heads   = 10

So.....the number of sequences with at least  8 heads  is

45 + 10 + 1  =

56

48. 

arctan (5.67)  =  79.99°   = 80°

49.   I just subbed  "x" for "theta"....same difference  

sin^2 x                1  - cos^2x         (1 + cos x) (1 - cos x)

________  =      _________  =    _________________  =       1  - cos x

1 + cos x             1 + cos x              1  +  cos x

√ [7x ]  ( √x  - 7√7 )

x√7  -  7√[49x]

x√7  - 7*7√x

x√7  - 49√x

He  can choose any 3 of the 12 shirts  =

C (12,3)   = 220 different choices

Oh, so it would be 13.8 Nm as the final answer? It seems like it has more to it then that :\

8)A circle centered at (3,2) is tangent to the line y = (x+1)/3. Find the area of the circle.

y  = (x + 1)  / 3      can be written as

y = (1/3)x + 1/3

3y = x + 1

x - 3y + 1   = 0

The distance from ( 3,2) to this line will be the radius of the circle

l 3 - 3(2) + 1 l               2             √10          √2

____________  =     ____   =      ___  =     ___

√[ 1^2 + 3^2 ]             √10              5            √5

So...the area  of the  circle is

pi * ( √ (2/5) ) ^2  =

2pi / 5   units^2

You have used 5 N instead of 50 N!  Apart from that you have done the calculation correctly.  (Your torque, T, should be 10 times larger than your 1.4 Nm).

I worked  (9)   but my answer disappeared...it is a little lengthy to re-create...but....the answer ends up as  20/3  units^2

Thanks!

For convenience, position the lower left vertex of the equilateral triangle at (0,0)

The side of this triangle is twice the length of the side opposite the 30 degree angle in the other triangle  =  2 * 6   = 12

And the height of the other triangle  is  √3  times the side opposite the 30 degree angle  =   6√3  units =  √108  units

So...the vertex  at the top right of the figure has the coordinates  (12, √108 )

And the distance between this vertex  and  (0, 0)  is

√ [ 12^2  + (√108)^2  ]  =

√ [144  + 108 ] =

√252  =

√ [ 36 * 7 ]  =

6√7  units

Yes, that is correct!

g(x)  = 2x + 5

h(x)  = ax + b

If  h (3)  = 26     we have that   26  =a(3) + b  ⇒    3a + b  = 26   (1)

And   g^-1 (x)   is     [ x - 5] / 2

So  g^-1(21)   is    [ 21 - 5] /2   = 16 /2   =  8

So  g^-1(21)  = 8  =  h(1)   which implies that

8 = a(1) + b    ⇒  a + b  = 8    (2)

So...we have the system

3a + b  = 26

a + b  =  8          subtract the second equation from the first

2a  = 18

a = 9

And    a + b  = 8     so   9 + b  = 8   ⇒   b  = -1

So  the function is

y = 9x  - 1

And a = 9   and  b  = -1

if f(x)=2x/x-5,

find f^-1(x).

That is a bit better :)

You have written

\(if\\ f(x)=\frac{2x}{x}-5\\ find \;\;f^{-1}(x).\)

But I expect you left out the brackets and you actually mean

\(if\\ f(x)=\frac{2x}{x-5}\\ find \;\;f^{-1}(x).\)

Be careful with those brackets. 

Let

  \(​​​​y=\frac{2x}{x-5}\\ \text{x cannot be 5}\\ \text{Now make x the subject}\\ yx-5y=2x\\ yx-2x=5y\\ x(y-2)=5y\\ x=\frac{5y}{y-2}\\ \text{Now for the inverse swap x and y over}\\ y=\frac{5x}{x-2} \)

So if

\(f(x)=\frac{2x}{x-5}\\then\\ f^{-1}(x)=\frac{5x}{x-2}\\ where\;\;x\ne2\)

Here are the graphs

The inverse fof a funtion is the reflection of the funtion in the line y=x

NOW LEARN FROM THIS AND TRY AND DO THE REST BY YOURSELF!

draw a pic and you will see that 

\(tan 20=\frac{x}{250}\\ x=250tan 20 \)

250*tan(20) = 90.9925585665

The wind is blowing at 91km/hour