CPhill

We can write this as

log₅ ( 24z² * x^4/3 y^5/3).......and by a log property we can write

log₅ ( 24z² )  +  log₅ ( x^4/3 y^5/3 ) .......and by applying the property again, we have

log₅(24)   +  log₅ (z²)  +   log₅(  x^4/3 ) +   log₅ ( y^5/3 )... .....we can simpify it further using another log property.....

log₅(24)  + 2 log₅ (z)    + (4/3) log₅ (x)  + (5/3) log₅ (y) ......this is linear (without exponents)......

(Using the "change of base" rule, you could write the first term as a decimal approximation, but I might just keep it as is)

Notice that, if we add the equations, we can "eliminate" X....so we have

      3X - 4Y = 9

+ (-3X + 2Y = 9)    This gives us

           -2Y = 18     divide by -2 on both sides

              Y = -9

To find X, just substitute for Y in one of the equations....I'll use the first one

3X - 4(-9) = 9

3X + 36 = 9        Subtract 36 from both sides

3X= -27              Divide  by 3 on both sides

X= -9

And here's the solution     ......  (X , Y)  = (-9 ,  -9)

You should check this answer in both equations to verify for yourself.

Using the distance formula, SQRT[(x₂- x₁)^2 + (y₂- y₁)^2], we have

((8-3)^2 + (12-4)^2))^(.5) =

(5^2 + 8^2)^(.5) = (25 + 64)^(.5) = (89)^(.5) =

$${\left({\mathtt{89}}\right)}^{\left({\mathtt{0.5}}\right)} = {\mathtt{9.433\: \!981\: \!132\: \!056\: \!603\: \!8}}$$

It appears that you've asked two questions....

-12-6 = -18

-12-8 = -20

Note that .10 = 10%

And .01 = 1%

So it appears that the "rule" is........move the decimal point two places to the right and add a percent sign...

Can you do these two, now???

Let me just add to what reinout said....if we took the second derivative of the first function, we get "2," which indicates that the curve is concave upward, and the second derivative of the second function = -2, which indicates the curve is concave downward.

And as Melody indicated, both parabolas share the same vertex - (0,1).

Compared to what....The Great Pyramid ????

Let the number of litres of 30% concentration = x

And since we have 120 litres as a final amount, then the number of litres of the 60% solution is just

(120-x).    So we have

.30x + .60(120-x) = .40(120)

.30x + 72 - 60x = 48   Subtract 72 from both sides and simplify the left side

-.30x  = -24        

.30x = 24      Divide both sides by .30

x = 80  And that's the number of litres of the 30% solution we need

And (120 - x) = (120 - 80) = 40 And that's the number of litres of the 60% solution we need

The cost of each ticket  - with the discount - equals (d - x)

So....$500 / (price of each ticket) =  the number of tickets purchased

So.......

$500 /( d - x)  = the number of tickets purchased

Assuming your points of intersection are correct - I haven't checked them- we can use the distance formula to find the length of chord AB....we have

( (2.68 - .52)^2 + (1.36 + 2.96)^2)^(.5) =

$${\left({\left({\mathtt{2.68}}{\mathtt{\,-\,}}{\mathtt{0.52}}\right)}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\left({\mathtt{1.36}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2.96}}\right)}^{{\mathtt{2}}}\right)}^{\left({\mathtt{0.5}}\right)} = {\mathtt{4.829\: \!906\: \!831\: \!399\: \!545\: \!7}}$$

We'll just call it 4.83 to simplify things

Next, we can use the Law of Cosines to find the (minor) angle formed by the two radii drawn to the points of intersection.  This will help us find the arc length.  So we have

AB^2 = 3^2 + 3^2 - 2(3)(3)cos (theta)

(4.83)^2 = 18 - 18cos (theta)

cos(theta) = -( (4.83)^2 -18)/18

cos (theta) =  $${\mathtt{\,-\,}}{\frac{\left({\left({\mathtt{4.83}}\right)}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{18}}\right)}{{\mathtt{18}}}} = -{\mathtt{0.296\: \!05}}$$

Now, taking the cosine inverse of this will give us the angle....

$$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{cos}}^{\!\!\mathtt{-1}}{\left(-{\mathtt{0.296\: \!05}}\right)} = {\mathtt{107.220\: \!510\: \!644\: \!933^{\circ}}}$$

=  about 107.22 degrees....this seems reasonable.....

Now, we need to convert this to radians to find the arc length

 This is given by  .....107.22 x (pi)/180 =

$${\frac{{\mathtt{107.22}}{\mathtt{\,\times\,}}{\mathtt{\pi}}}{{\mathtt{180}}}} = {\mathtt{1.871\: \!342\: \!023\: \!988\: \!320\: \!2}}$$

= about 1.87 radians

And the "formula" for the arc length (s) is given by

s = r x (theta in radians)   =     3 x 1.87 = 5.61

Now, to find the area of the shaded region, we can find the area of the (minor) sector formed by the two radial lines less the area of the triangle formed by the two radii and AB.

We have..... (1/2)(r^2) (theta in radians) - (1/2)(r^2)sin(theta in rads)

= (1/2)(r^2) (theta in radians - sin(theta in radians))=

(1/2)(9) (1.87 - sin(1.87)) = about 4.11 square units

I think that's it....if I haven't made any errors!!