Robbie, jennie, willie, and kylie played a numbers game. jennie picked a number but did not tell her friends. robbie and willie both picked a number as well, and jennie described their numbers like this to kylie.

robbie picked a number that is nine more than four times jennie's number.
willie picked a number that is two less than three times robbie's number.

deciding to call her number, n, jennie asked kylie to write an expression that describes willie's number.

kylie came up with two expressions that equal willie's number.

her first expression was
3
(
4n
n +
9
) - 2. simplifying her first expression, her second expression was 12n +
.

The frequency of vibrations of a vibrating violin string is given by f = 1 2l t ρ where l is the length of the string, t is its tension, and ρ is its linear density.† (a) find the rate of change of the frequency with respect to the following. (i) the length (when t and ρ are constant) (ii) the tension (when l and ρ are constant) (iii) the linear density (when l and t are constant) (b) the pitch of a note (how high or low the note sounds) is determined by the frequency f. (the higher the frequency, the higher the pitch.) use the signs of the derivatives in part (a) to determine what happens to the pitch of a note for the following. (i) when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates df dl 0 and l is ⇒ f is ⇒ (ii) when the tension is increased by turning a tuning peg df dt 0 and t is ⇒ f is ⇒ (iii) when the linear density is increased by switching to another string df dρ 0 and ρ is ⇒ f is ⇒

Which articulations allow for greatest range of motion?

Ineed the answers for these promblems i have