48,301 views
35 votes
35 votes
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}

User Alliyah
by
2.6k points

1 Answer

8 votes
8 votes

Answer:


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


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

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 (1)/(x) \implies y=(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=(k)/(24)


\implies k=128 \cdot 24


\implies k=3072

Therefore, the equation is:


y=(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=(3072)/(x)


\implies x=(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=(45)/(2700)=(1)/(60)

Therefore, the equation is:


y=(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=(150d^3)/(60)


\implies 3600=150d^3


\implies d^3=(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.

User Vbt
by
3.1k points