AnExtremelyLongName

The problem cannot make sense as it ruins the pattern though.

So does it mean it includes EVERY diagonal and EVERY side length? Try to look at the original problem on the screen and match it with mine.

Set Up:

Length = L

Width = W

L = W + 12

Area = LW

Simplify:

W(W+12) = 85

W² + 12W - 85 = 0

Factor:

(W + 17)(W - 5) = 0 

Solve:

W = -17 or W = 5

I will leave the rest up to you. (Hint: lengths can't be negative.)

Assuming the problem means that the basketball has a radius of 4 inches when it is FILLED up, then here we go:

Sphere: $\frac{4}{3}\pi{r^3}$

Substitute and solve for volume:

$\frac{256}{3}\pi\text{ cubic inches}$

We divide:

$\frac{256/6}{3}\pi$

We evaluate:

44.6804288510548372040483

We round:

44.7 seconds

Hmmm, I couldn't find a geometric way to solve this within the 1 minute I looked at this problem. I am very inpatient!

So let us use coordinates!

____________________________________________________________________________________________

Find ST

ST = 40/3 by simple algebra I think you can do at your skill level judging by the difficult of this problem.

Let QR = $a$

Slope of PT: $-\frac{\frac{40}{3}}{a}$

Y-intercept of PT: 20

Equation: $y=-\frac{\frac{40}{3}}{a}x+20$

Slope of QS: (20/a)

Y intercept of QS: 0

Equation: $y=\frac{20}{a}x$

Substitute

$-\frac{\frac{40}{3}}{a}x+20=\frac{20}{a}x$

$20 =\frac{\frac{100}{3}}{a}x$

$20a=\frac{100x}{3}$

$60a=100x$

$3a=5x$

$\frac{5}{3}x=a$

Interpret this:

$x$ is the x-coordinate of the solution of we solved for the location of the intercept at point U. That means $x$ is the length of QV. We know $a$ is the length of QR.

That means QV is three-fifths the length of QR.

We know that QUV is similar to QSR by AA similarity through a series of proofs.

Since we know the proportion of the sides, we can solve for UV:

Solve:

20 * (3/5) = 12

Ta-da! Mathz! 

Hello guest!

Here is a hint: Can you prove triangle QPT and RQS similar? If so, you can use proportions to solve!

I just can't find a way to use proportions to solve it!

Is this what it looks like?

$\text{A regular dodecagon }P_1, P_2, P_3, ... P_12 \text{is inscribed in a circle with radius 1. }$

$\text{Compute } (P_1 P_2)^2 + (P_1 P_3)^2 + ... + (P_{11} P_{12})^2. \text{(The sum includes all terms of the form } (P_i P_j)^2, \text{where 1≤ i < j ≤ 12)}$

In the series, are you sure the second term is $(P_1P_3)^2$? It ruins the pattern!

Never mind! I am running on low-sleep. It is the length of the segments I suppose?

I am still confused by the format.

What is (P_1 P_2)^2 supposed to mean? In other words, what is the operator inside the paranthesis? Scrutinize your question to make sure it is understandable.

Thank you.

I need a membership!