Question

1. the pitch of a musical note varies inversely as its wavelength. if the tone has a pitch of 440 vibrations per second and a wavelength of 2.4 feet, find the pitch of a note that has a wavelength of 1.6 feet.
2. in kick - boxing, it is found that the force needed to break a board, varies inversely with the length of the board. if it takes 5 lbs of pressure to break a board 2 feet long, how many pounds of pressure will it take to break a board that is 8 feet long?
3. when air is pumped into an automobile tire, the pressure is inversely proportional to the volume. if the pressure is 35 pounds when the volume is 120 cubic inches, what is the pressure when the volume is 140 cubic inches?

Explanation:

Question 10

Step1: Establish inverse - variation formula

Let $p$ be the pitch and $w$ be the wavelength. The inverse - variation formula is $p=\frac{k}{w}$, where $k$ is the constant of variation. We know that when $p = 440$ and $w = 2.4$, we can find $k$.
$440=\frac{k}{2.4}$, so $k = 440\times2.4=1056$.

Step2: Find the new pitch

Now we want to find $p$ when $w = 1.6$. Using the formula $p=\frac{k}{w}$ with $k = 1056$ and $w = 1.6$, we have $p=\frac{1056}{1.6}=660$.

Step1: Establish inverse - variation formula

Let $F$ be the force and $L$ be the length of the board. The inverse - variation formula is $F=\frac{k}{L}$. When $F = 5$ and $L = 2$, we find $k$.
$5=\frac{k}{2}$, so $k=5\times2 = 10$.

Step2: Find the new force

Now we want to find $F$ when $L = 8$. Using the formula $F=\frac{k}{L}$ with $k = 10$ and $L = 8$, we have $F=\frac{10}{8}=1.25$.

Step1: Establish inverse - variation formula

Let $P$ be the pressure and $V$ be the volume. The inverse - variation formula is $P=\frac{k}{V}$. When $P = 35$ and $V = 120$, we find $k$.
$35=\frac{k}{120}$, so $k=35\times120 = 4200$.

Step2: Find the new pressure

Now we want to find $P$ when $V = 140$. Using the formula $P=\frac{k}{V}$ with $k = 4200$ and $V = 140$, we have $P=\frac{4200}{140}=30$.

Answer:

660

Question 11