CPhill

First, let's simplify the 3*√20

We have

3*√[(4)(5)]  = 3*√4*√5 = 3*2*√5 = 6*√5

So we have

6*√5 * 6*√5

And since we can multiply in any order, we have

6 * 6 * √5 * √5 =  36 * √25 = 36 * 5  = 180

So, we have

-3 - (1/4)(1/8)

Remember, we must carry out any multiplications before we carry out any subtractions. So, multiplying the two fractions together, we have

-3 - (1/32)   

Note that this is the same thing as

- (3 + 1/32)   

And converting the mixed number to an improper fraction, we have

- (97/32) = -97/32

And that's it.....

So we have

∫x³ √(x⁴ + 9) dx

Let u = x⁴ + 9      du = 4x³ dx   ⇒   du/4 = x³ dx

I'll just continue in terms of "u" instead of "x," so that we don't have to "back-substitute."

So when x = 0, u = 9        And when x = 2, u = 25

So this gives us

(1/4)∫ u^1/2 du = (1/4)(2/3) u^3/2  = (1/6) u^3/2

And evaluating this from u = 9 to u = 25  gives us

(1/6)(25^(3/2)-9^(3/2))  =  49/3

And to get the average value, we divide this by the interval width, i.e., 2. 

This gives us......... 49/6

Note how the "average value" turns a "non-rectangular" area into a "rectangular area!!"

The "width" of the rectangle (2)  times the "height" (49/6) = 49/3 !!

And I think that's it.....

There's no question here, but I'm assuming you may want a graph of this??

Note that it's a line with a slope of (-1/3), a  "y intercept" of 7/3 and an "x intercept" of 7.

Let's call the original price "x."  So, if the sale price was 20% off of the original price, it must just be on sale for 80% of its original price. Therefore....

80%   OF  x  = $68   

Now change the 80% to a decimal - .80 - and let's realize that "OF" means "multiply"

So we have

.80x = 68    Divide both sides by .80, and we have

x = 68/.80 = $85

And that's the original price.

Thanks....I actually didn't really know WHAT to do, at first. I was going to try something "complicated," but then I remembered a problem from a math "puzzles" book that I have that was similar to this. In fact, now that I think of it, I might post it to the forum just so some people can take a stab at it !!

Sisyphus Goes To The Beach......

Using the Chain Rule, we have

G'(x) = aπ cos ( π x)

And

G''(x) =  -aπ² sin (π x)

So we have

1((1.5)¹² -1) / (1.5 - 1) =

((1.5)¹² -1) / (.5 ) =

$$\mathrm{\ }$$$${\mathtt{2}}{\mathtt{\,\times\,}}\left({\left({\mathtt{1.5}}\right)}^{{\mathtt{12}}}{\mathtt{\,-\,}}{\mathtt{1}}\right) = {\mathtt{257.492\: \!675\: \!781\: \!25}}$$

Let's write

4³⁰⁰       ???      3⁴⁰⁰

(4³)¹⁰⁰  ???   (3⁴)¹⁰⁰

64¹⁰⁰    ???       81¹⁰⁰

And it's obvious that the thing on the right is greater than the thing on the left.

So.....

4³⁰⁰       <      3⁴⁰⁰

And there you go....