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Using similar triangles:

1 ft / 10 ft  =   x ft / (5280 x 4)ft      x = 2112   PLUS the 7 ft observer's 'eye-height' = 2119 ft high mountain

Using trig fxns:

For the surveyor's pole:

    Tan theta = 1/10     theta = 5.71 degrees

For the mtn:

    tan 5.71  = opp/adj  = opp/(4 x 5280 ft)    opp = 2112     2112 + observer's height (7 ft .....very tall surveyor) = 2119 ft high

Happy Anniversary , Chris !        I know I've learned a lot from your responses !

Any sum of 5 + 6 both raised to an ODD power is divisible by the sum of 5 + 6 =11

Example: 5^3 + 6^3 =341 / 11 = 31......and so on. So, the smallest prime divisor of {5^1999 + 6^1999} is 11.

Or use the EQUATION for the sum of a finite arithmetic series

SUM =  n (a1 + an)/2     n = 33   a1 = -3    a33 = - 131       results in  -2211 

Using synthetic division, we have

4  [   1      -k       2           1         ]

                4     16 - 4k    72 -16k

       ______________________

        1    4-k     18 -4k    73  - 16k

So   73 -16k  is the remainder.....and this =  -7....so....

73  - 16k  =  -7       rearrange as

80  = 16k     divide both sides by 16

5   = k

Solve this open sentence and graph the inequality 

9 - |3-b| > 2

**This is absolute value in open sentences.... Algebra 1

Since  AD  = CD

Then the angles opposite these sides are also equal

So angle DAC  = angle DCA

And if AB = CB, then the angles opposite thes sides are also equal

So angle CAB  = angle ACB

So angle DAC + angle CAB  = angle DCA + angle ACB

So angle DAB   = angle CDB

Then, by  SAS....triangle  DAB  = triangle DCB

Then angle ADB  = angle CDB.....so angle ADC is bisected

And angle  DAC  = angle DCA

AD = CD

And angle DAC  = angle DCA

So....by ASA....triangle EAD  = triangle   ECD

So angle AED  = angle CED

And, by Euclid.....a line standing upon a line that makes adjacent angles equal means that the lines are perpendicular....so AC  and Db are perpendicular

∑[(-4n+1), n, 1, 33] =-2,211

WHAT do yo need?      Do you have values for 'x' ?    Do you need to find 'zeroes' ?  Do you need to factor these expressions?

Thanks Tiggsy!

Here is a START for you.....every power of 5 ends in 5  and every power of 6 ends in 6.....what happens when you add these together.......     will it be an odd number?  (not divisible by 2)   .....it probably won't be divisible by 5 will it?    Maybe 7?

What do YOU think???

By looking at the first few terms I see  11 is divisible in the 1st 3rd  5th 7th   9th  ....terms

                while 2 3 5 7 are not.....I would guess 11.

When the polynomial P(x) is divided by x − 2, the remainder is 7.

Which of the following must be true?

P(2) = 7

Example:

\(\begin{array}{|rcll|} \hline x^3 - x^2 + 2x - 1 : x- 2 &=& x^2 + x + 4 + \dfrac{7}{x - 2} \\\\ P(2) = 7 \\\\ P(2) = 2^3 - 2^2 + 2\cdot 2 - 1 &=& 7 \\ 8-4+4 - 1 &=& 7 \\ \mathbf{ 7 } & \mathbf{=} & \mathbf{7} \\ \hline \end{array}\)

Yes and yes (I confess I didn’t notice the “per wheel” bit!).

What is the value of

\(k\) if \(x^3 - kx^2 + 2x + 1\) is divided by \(x - 4\)

and gives a remainder of \( -7\) ?

polynomial long division:
\(x^3 - kx^2 + 2x + 1 : x - 4 = x^2 - x·k + 4·x - 4·k + 18 - \dfrac{16·k - 73}{x - 4}\)

remainder  is -7:

\(\begin{array}{|rcll|} \hline -(16·k - 73) &=& -7 \\ 16·k - 73 &=& 7 \\ 16·k &=& 7+73 \\ 16·k &=& 80 \\ \mathbf{ k } & \mathbf{=} & \mathbf{5} \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline x^3 - 5x^2 + 2x + 1 : x-4 &=& x^2 - x - 2 -7 \\ \hline \end{array}\)

or:

\(\begin{array}{|rcll|} \hline P(4) = -7 \\\\ P(4) = 4^3-k\cdot 4^2 + 2\cdot 4 + 1 &=& - 7 \\ 64-16k + 8 + 1 &=& - 7 \\ 73-16k &=& - 7 \\ 16k &=&73+7 \\ 16k &=& 80 \\ \mathbf{ k } & \mathbf{=} & \mathbf{5} \\ \hline \end{array}\)

the number of eggs collected each day on a farm.  Yes, if it’s a large egg farm

the number of yolks in randomly selected eggs.   No, the number of yolks in each egg will only be a small number like 1 (mostly), a few with 2 and possibly the occasional 3.

the weights of eggs in the kitchen of a restaurant.  Yes, if there are a large number of eggs.

the number of eggs in cartons sold at a supermarket. No, this will be a few discrete numbers like, 6, 12 etc.

≈≈≈Soccer balls:

Number of standard deviations above the mean = (0.45 - 0.43)/0.01 → 2

A proportion of ≈ 0.023 lie above 2 standard deviations (from tables)

0.023*800 ≈ 18

Ambulances:

Stdevs below mean = (400 - 450)/50 = -1

Stdevs above mean = (500 - 450)/50 = 1

Proportion between these: 0.843 - (1-0.843) = 0.686

Expect 0.686*97 = 67

Thanks so much alan!

So I was making it much too complicated!

I just wanted to confirm with you, for question 4. once inputing the units m/s^2 for g,  k would = kg/s^2 ?

also as we're looking for the spring stiffness at each wheel, we're looking for k/4?

Here are parts (3) and (4):

I'll leave you to put in a value for g and hence work out k (make sure your units are consistent).

(0 + i) + x^4 = 0

Subtract i from both sides:

x^4 = -i

Taking 4^th roots gives -(-1)^(7/8) times the 4^th roots of unity:

x = -(-1)^(3/8)    or    x = (-1)^(3/8)    or    x = -(-1)^(7/8)    or    x = (-1)^(7/8)

well..I'm sorry...I just did not understand logarithms that much...But thanks, EP!!!