Question

the pitch, or frequency, of a vibrating string varies directly with the square root of the tension. if a string vibrates at a frequency of 300 hertz due to a tension of 8 pounds, find the frequency when the tension is 72 pounds.

Answer 1

`\bf \qquad \qquad \textit{direct proportional variation}\\\\ \textit{\underline{y} varies directly with \underline{x}}\qquad \qquad y=kx\impliedby \begin{array}{llll} k=constant\ of\\ \qquad variation \end{array}\\\\ -------------------------------\\\\ \begin{array}{llll} \stackrel{frequency}{f}\textit{ of a vibrating string varies directly}\\ \qquad \qquad \textit{with the square root of the tension }\stackrel{tension}{t} \end{array}`


`\bf f=k√(t)\quad \textit{we also know that } \begin{cases} f=300\\ t=8 \end{cases}\implies 300=k√(8) \\\\\\ \cfrac{300}{√(8)}=k\implies \cfrac{300}{√(2^2\cdot 2)}=k\implies \cfrac{300}{2√(2)}=k\implies \cfrac{150}{√(2)}=k \\\\\\ \textit{and we can \underline{rationalize} it to }\cfrac{150√(2)}{2}\implies 75√(2)=k`


`\bf thus\qquad f=\stackrel{k}{75√(2)}√(t)\implies \boxed{f=75√(2t)}\\\\ -------------------------------\\\\ \textit{now, when t = 72, what is \underline{f}?}\qquad f=75√(2(72))`

Answer 2

when the tension is 72 pounds , the frequency is 900 hertz

Explanation:

Hello, I think I can help you with this.

you can easily solve this by using a rule of three.

According to the question data:

the pitch, or frequency, of a vibrating string varies directly with the square root of the tension.in mathematical terms it is:

f ∝ √T

where f is the pitch or frequency and T is the tension

Step 1

if a string vibrates at a frequency of 300 hertz due to a tension of 8 pounds

300 hertz ∝ √8 pounds

300⇔√8

what is the frequency when the tension is 72 pounds

x⇔√72

Step 2

Let

300⇔√8

x⇔√72

the relation is

`(300)/(√(8) ) =(x)/(√(72) )\\\\ Now, solve\ for\ x\\\\\\(300*√(72) )/(√(8) ) =x\\x=(300*√(72) )/(√(8) ) \\x=(300*√(8*9))/(√(8)) \\x=(300*√(8)*√(9))/(√(8)) \\x=300*√(9)\\x=300*3\\x=900`

when the tension is 72 pounds , the frequency is 900 hertz

have a good day.