Question

How do you solve `500=\frac{1}{2}(20)(v)^{2}`?

Answer 1

`v=+-5sqrt2`

Explanation:

`"multiply both sides by 2"`

`(2xx500)=cancel(2)xx1/cancel(2)xx20v^2`

`rArr20v^2=1000`

`"divide both sides by 20"`

`(cancel(20) v^2)/cancel(20)=1000/20`

`rArrv^2=50`

`color(blue)"take the square root of both sides"`

`rArrv=+-sqrt50larr" note plus or minus"`

`color(white)(rArrv)=+-sqrt(25xx2)`

`color(white)(rArrv)=+-sqrt25xxsqrt2`

`color(white)(rArrv)=+-5sqrt2`

Answer 2

Once you get used to these you start using shortcuts and get much faster at solving them.

`v=5sqrt(2) color(white)("d")` Exact value

`v=7.071 color(white)("d")` to 3 decimal places: Approximate value

Explanation:

`color(blue)("Step 1 - simplify the number on the right")`

`1/2xx20color(white)("d")->color(white)("d")1/2xx2xx10color(white)("d")->color(white)("d")2/2xx10=10` giving:

`500=10v^2`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Step 2 - isolate the "v^2)`

If we can change the 10 into 1 we then have `1xxv^2->v^2`

Multiply each side by `1/10` giving:

`color(green)(500=10v^2color(white)("ddd")->color(white)("ddd")500color(red)(xx1/10)color(white)("d")=color(white)("d")10color(red)(xx1/10)xxv^2)`

`color(white)("dddddddddddd")->color(white)("ddddd")50color(white)("dddd,")=color(white)("dddd")v^2`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Step 3 - isolate the "v)`

Square root both sides

`color(white)("dddddddddddd")->color(white)("ddddd")sqrt(50)color(white)("dddd,")=color(white)("dddd")sqrt(v^2)`

Note that `50color(white)("d") ->color(white)("d") 5xx10color(white)("d") ->color(white)("d")5xx2xx5 = 2xx5^2` giving:

`color(white)("dddddddddddd")->color(white)("ddddd")sqrt(2xx5^2)color(white)("dddd,")=color(white)("dddd")sqrt(v^2)`

`color(white)("dddddddddddd")->color(white)("dddddd")5sqrt(2)color(white)("dddd.dd,")=color(white)("dddddd")v`

`v=5sqrt(2) color(white)("d")` Exact value

`v=7.071 color(white)("d")` to 3 decimal places: Approximate value