CPhill

It's used to indicate a ratio of two numbers    .....for example  7:5, 3:4..........etc.

x/a = 1+ x/b

 Subtract x/b from both sides.

x/a - x/b = 1       get a common denominator on the left

[bx - ax] / ab  = 1     factor out x on the left

[ x (b - a) ] / ab= 1      multiply both sides by ab / [b-a]

x = (ab) / (b - a]

And that's it.....!!!

This one is a little tough!!

The slope of a tangent line to the given parabola at any point is just y' = 6.96x

Now, what we're looking for is at least one point on the parabola where the line through  (-5.67, - 44.75) is tangent to that point (or points).

Let's call the point(s) on the parabola (x, 3.48x^2). And the slope of the tangent line at that point is just 6.96x.

So, using this point on the parabola and the point (-5.67 , - 44.75), we have that, using the slope "formula,"

(3.48x^2 + 44.75) / (x + 5.67) = 6.96x

(3.48x^2 + 44.75)  /(x + 5.67) - 6.96x = 0

And solving this equation using the onsite calculator, we have....

$${\frac{\left({\mathtt{3.48}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{44.75}}\right)}{\left({\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{5.67}}\right)}}{\mathtt{\,-\,}}{\mathtt{6.96}}{\mathtt{\,\times\,}}{\mathtt{x}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{3\,406\,662\,741}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{49\,329}}\right)}{{\mathtt{8\,700}}}}\\
{\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{3\,406\,662\,741}}}}{\mathtt{\,-\,}}{\mathtt{49\,329}}\right)}{{\mathtt{8\,700}}}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = -{\mathtt{12.378\: \!807\: \!301\: \!025\: \!932}}\\
{\mathtt{x}} = {\mathtt{1.038\: \!807\: \!301\: \!025\: \!932}}\\
\end{array} \right\}$$

And the equation of the line that goes through (-5.67, -44.75) and touches the parabola at 1.038807301025932 is given by

y +44.75 = 6.96(1.038807301025932)(x + 5.67)

y = 7.23009881514048672x + 40.9946602818465597024 - 44.75

y = 7.23009881514048672x -3.7553397181534402976 ........ and rounding, we have

y = 7.23x - 3.755

And the equation of the line that touches the graph at - 12.378807301025932 is given by

y + 44.75 = 6.96( - 12.378807301025932)(x + 5.67)

y = -86.15649881514048672x -488.5073482818465597024 - 44.75 

y = -86.15649881514048672x - 533.2573482818465597024  .....and rounding, we have

y = -86.156x - 533.257

6ft 7in = (6 + 7/12) ft

So.....

[(6 + 7/12) ft] * [8 ft ] =  [48 + 56/12]sq ft. = [48 + 4 + 8/12]sq ft = [52 + 2/3] sq ft

Good answer, DS.  And if we wanted to leave it in an irrational form, we have;

√156 = √(52*3) = √(4*13*3) = √4*√13*√3 = 2√39

No...integers are, informally, all the real positive and negative "whole" numbers plus 0.

K = {0, 2, 4, 6, 8, 10}

There are 12 inches ina foot so......

3.28 * 12 = 39.36 inches

1 - sin(x)^2  factors as a difference of two squares thusly....(1 + sin(x)) (1 - sin(x))

So we have.....

[(1 + sin(x)) (1 - sin(x))] / (1 + sin(x))  =  1 - sin(x)

Thanks for that detailed explanation, ND.........this is something that gives a lot of students trouble!!!!

Maybe you should add this to "Great AnswersTo Learn From".......just a suggestion......