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Use symmetry here.  

1.  There is nothing to distinguish x from y, so set x = y.  (You should get x + y = 4)

2.  Set  25z^3 = 5y^3 = x^3   (You should find a minimum value of 15).

Thanks Alan.  :)

Hi Melody,  I'm not 100% sure I've got all of them.  However, there can't be any higher polynomials as any extra multiplying coefficients would turn out to be zero. I certainly can't think of any other forms that would fit!

Find a monic quartic with real coefficients where two of the roots are 3 - 2i and 5 + i.

The roots occur in complex conjugate pairs:

Note that \(27=3^3 \text{ and } 8=2^3\)

So  \(3^x = 3^{3(y-5)} \text{ and }2^y=2^{3(2x-4)}\)

Now equate powers so that you have two equations for x and y:

x = 3(y - 5)   and   y = 3(2x - 4)

Can you take it from here?

As follows:

You have the correct f(x).

To find the sum of the even coefficients of f(x)^16 simply calculate (f(1)^16  +  f(-1)^16)/2.

A rocket traveled 12,000 miles in the first four hours. Then the rocket traveled 8000 miles in the next four hours. What was the rocket’s average speed, in miles per hour, for these 8 hours?

Hello Guest!

\(average\ speed=\frac{(12000+8000)miles}{(4+4)hours}=\color{blue}2500\ miles\ per\ hour\)

  !

How many integers x satisfy x^2 < 7x?  

x² is the same as x • x so since (x • x) must be less than (7 • x) then x has to be less than 7..  

The interval 0 < x < 7 contains 6 integers that comply with this requirement, namely 1, 2, 3, 4, 5, and 6. 

You can't use negative integers because then x² would be positive, thus larger than 7x which would be negative. 

_.

To estimate the height of a radio tower, Jody steps 25 feet away from the center of the tower's base, until his line of sight to the top of the tower and his line of sight to the center of its base form a right angle. His eyes are 6 feet above the ground. How tall is the radio tower to the nearest foot?

I'll let you do the math.  

Thnkyou Alan that is awesome! Very detailed, muchly appreciated

x = 7 and y = -31

164*7 + (-31 *37) =1,148 - 1,147 = 1

Can you have two sixes such as: 66123, or 60645, or 601246........etc? if so, I calculate that there are:

3,528 such numbers. If you are not allowed to have more than 1 six at the beginning such as: 612245, 60323,

65523......etc, then there are: 1,512 such numbers.

yes and we get the 6 degrees by doing this

60 minutes/hour

360 degrees

360/60 = 6.

Beautifully explained, Guest. :)

2x+1 > 7x-3

1+3 > 7x-2x

4> 5x

4/5 > x

x<4/5

If you dont understand anything feel free to ask!

180º - 116º = 64º

Angle RPG = 64º  

48 - 12 = 36 minutes. Each minute is 6 degrees, so:

36 x 6 = 216 degrees.

Given that the longer leg of the right triangle is one foot less than twice the shorter leg, we can set up an equation that will help find the length of the shorter and longer leg.

a = x ft

b = 2x - 1 ft

c = 17 ft

Using the pythagorean theorem, a² + b² = c², you can find the length of both the shorter and longer leg.

a² + b² = c²

x² + (2x -1)² = 17²

x² + 2x² + 1 = 289

               -1       -1

3x² / 3 = 288 / 3

x² = 96

sqrt(x²) = sqrt(96)

x = 9.8

The length of the shorter leg is 9.8 ft. 

To find the length of the longer leg, substitute 9.8 for x in 2x - 1.

2(9.8) - 1 =

19.6 - 1 

18.6

The length of the longer leg is 18.6 ft.

To find the area of a triangle, multiply the base by the height, and then divide by 2.

area = ( 9.8 x 18.6 ) / 2 

area = 182.28 / 2

area = 91.12 ft

The area of the triangle is 91.12 ft.

Thanks Alan,

I had no idea how to do these and it did not occur to me to just try linear.

It does say find all functions.  Are you sure you have found all of them?

I need to ponder on your answer.

Actually, I got it. The right answer was 10. 

Thank you!

the answers 15

(x,y) = (28,-15)