"To tune your violin, you first tune the A string to the correct pitch of $440 \text{~Hz}$, and then you bow both it and an adjoining string simultaneously, all the while listening for beats. While bowing the A and E strings, you hear a beat frequency of $3.00 \text{~Hz}$ and note that the beat frequency increases as the tension on the E string are increased. (The E string is to be tuned to $660 \text{~Hz}$.) $(a)$ Why are beats produced by these two strings when bowed simultaneously?"

If h(t) is a person's heart rate in beats per minute t minutes into an exercise session, what does

$$\int_{0}^{30} h(t) d t$$

represent?

Two tuning forks are producing sounds of wavelength $34.40 \mathrm{~cm}$ and $33.94 \mathrm{~cm}$ simultaneously. How many beats do you hear each second?

Recommended heart rates during exercise vary with age and physical condition. For a healthy person doing moderate to intense exercise, such as hiking, the inequality $0.5(220-a) \leq R \leq 0.9(220-a)$ gives a target range for the heart rate R (in beats per minute), based on age a (in years). a. What is the target range for heart rates for a person 15 years old? b. How old is a person whose target range is between 99 and 178.2 beats per minute?

To play a $\mathrm{C}$ major chord on the piano, you hit the $\mathrm{C}, \mathrm{E}$, and $\mathrm{G}$ keys simultaneously. When you do so, you hear no beats. Why?

Assuming the heart beats once every second, how many times does the heart beat in 1 year? How many times does it beat in 40 years? 70 years?

Use a case 1-case 2 ratio box to solve this problem. Don's heart beats 225 times in 3 minutes. At that rate, how many times will his heart beat in 5 minutes?

The function $P(t)=145 e^{-0.092 t}$ models a runner's pulse, $P(t)$, in beats per minute, $t$ minutes after a race, where $0 \leq t \leq 15$. Graph the function using a graphing utility. $\boxed{\text{TRACE}}$ along the graph and determine after how many minutes the runner's pulse will be $70$ beats per minute. Round to the nearest tenth of a minute. Verify your observation algebraically.

Identify the following as an observation(O), a hypothesis (H), an experiment (E), or a conclusion (C): During an assessment in the emergency room, a nurse writes that the patient has a resting pulse of 30 beats/min.

A $2.00 \mathrm{MHz}$ sound wave travels through a pregnant woman's abdomen and is reflected from the fetal heart wall of her unborn baby. The heart wall is moving toward the sound receiver as the heart beats. The reflected sound is then mixed with the transmitted sound, and $85$ beats per second are detected. The speed of sound in body tissue is $1500 \mathrm{~m} / \mathrm{s}$. Calculate the speed of the fetal heart wall at the instant this measurement is made.

Read the poem aloud, and scan it line by line, marking the stressed syllables and the unstressed syllables. How many strong beats do you hear in each line? Where do you hear variations in the meter?

The hummingbird has an average wing speed of $70$ beats per second. The expression $70 t$ gives the number of wingbeats in $t$ seconds.

Calculate the number of wingbeats in 1 hour for the hummingbird.

A piano tuner listens to a tuning fork vibrating at 440 Hz to tune the string of a piano. He hears beats between the tuning fork and the piano string. Is the string in tune? Explain your answer.

Heart Rate Suppose that a person's heart rate, $x$ minutes after vigorous exercise has stopped, can be modeled by

$$f(x)=\frac{4}{5}(x-10)^2+80 .$$

The output is in beats per minute, where the domain of $f$ is $0 \leq x \leq 10$. (a) Evaluate $f(0)$ and $f(2)$. Interpret the result. (b) Estimate the times when the person's heart rate was between 100 and 120 beats per minute, inclusive.