Question

PLEASE HELP ME !!!!!

1 ) The rate of vibration of a string under constant tension varies inversely with the length of the string. If a string is 24 inches long and vibrates 128 times per second, what is the length of a string that vibrates 64 times per second?

48 inches
46 inches
42 inches
55 inches

------------------------------------------------------------------

2 ) The horsepower (hp) that a shaft can safely transmit varies jointly with its speed (in revolutions per minute - rpm) and the cube of the diameter. If the shaft of a certain material 3 inches in diameter can transmit 45hp at 100 rpm, what must the diameter be in order to transmit 60hp at 150rpm?

24
$\sqrt[3]{150}$
$2\sqrt[3]{3}$
$4\sqrt[3]{6}$

Answer 1

Answer:

$\textsf{1)\quad 48\;inches}$

$\textsf{2)\quad 2\sqrt[3]{3} \; inches}$

Step-by-step explanation:

Question 1

Define the variables:

• Let x be the length of the string.
• Let y be the rate of vibration of a string under constant tension.

The rate of vibration of a string under constant tension varies inversely with the length of the string:

$\boxed{y \propto \dfrac{1}{x} \implies y=\dfrac{k}{x}\quad\text{for a constant k }}$

Given values:

• x = 24 inches
• y = 128 times per second

Substitute the given values of x and y into the formula and solve for k:

$\implies 128=\dfrac{k}{24}$

$\implies k=128 \cdot 24$

$\implies k=3072$

Therefore, the equation is:

$y=\dfrac{3072}{x}$

To find the length of a string that vibrates 64 times per second, substitute y = 64 into the equation and solve for x:

$\implies 64=\dfrac{3072}{x}$

$\implies x=\dfrac{3072}{64}$

$\implies x=48$

Therefore, the length of a string that vibrates 64 times per second is 48 inches.

---------------------------------------------------------------------------------------------

Question 2

Define the variables:

• Let y be the horsepower (hp) that a shaft can safely transmit.
• Let v be the speed of the shaft (in rpm).
• Let d be the diameter of the shaft (in inches).

The horsepower that a shaft can safely transmit varies jointly with its speed and the cube of the diameter:

$\boxed{y \propto vd^3 \implies y=kvd^3\quad\text{for a constant k }}$

Given values:

• y = 45 hp
• v = 100 rpm
• d = 3 inches

Substitute the given values of y, v and d into the formula and solve for k:

$\implies 45=k \cdot 100 \cdot 3^3$

$\implies 45=k \cdot 100 \cdot 27$

$\implies 45=2700k$

$\implies k=\dfrac{45}{2700}=\dfrac{1}{60}$

Therefore, the equation is:

$y=\dfrac{vd^3}{60}$

To find the diameter of the shaft in order to transmit 60 hp at 150 rpm, substitute y = 60 and v = 150 into the equation and solve for d:

$\implies 60=\dfrac{150d^3}{60}$

$\implies 3600=150d^3$

$\implies d^3=\dfrac{3600}{150}$

$\implies d^3=24$

$\implies d=\sqrt[3]{24}$

$\implies d=\sqrt[3]{8 \cdot 3}$

$\implies d=\sqrt[3]{8} \sqrt[3]{3}$

$\implies d=\sqrt[3]{2^3} \cdot \sqrt[3]{3}$

$\implies d=2\sqrt[3]{3}$

Therefore, the diameter of a shaft that transmits 60 hp at 150 rpm is 2³√3 inches.