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

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

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

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

Answer 2

Answer:

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

Step-by-step 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

$\frac{300}{\sqrt{8} } =\frac{x}{\sqrt{72} }\\\\ Now, solve\ for\ x\\\\\\\frac{300*\sqrt{72} }{\sqrt{8} } =x\\x=\frac{300*\sqrt{72} }{\sqrt{8} } \\x=\frac{300*\sqrt{8*9}}{\sqrt{8}} \\x=\frac{300*\sqrt{8}*\sqrt{9}}{\sqrt{8}} \\x=300*\sqrt{9}\\x=300*3\\x=900$

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

have a good day.