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Performance Task A
Modern scuba-diving equipment allows divers to stay underwater for long periods of time. Underwater instructors use mathematics to explore safety issues related to scuba diving. An instructor gives scuba-diving students the handout shown at the right. YOU WILL USE THE
INFORMATION THROUGH THE WHOLE ASSESSMENT

Use the information in the handout.
Part A
Divers' Information

At the water's surface, the air around a
diver exerts 1 atmosphere
(atm) of pressure.
Underwater, the pressure
P(atm) around the diver increases. It varies with the diver's depth
d, in feet, according to the equation
P=(d)/( 33)+1.
The volume
V of a given amount of air varies inversely with the pressure
P around it, so the volume of air in the diver's lungs increases as she ascends.
Suppose that a certain amount of air takes up a volume of
4qt in a diver's lungs at a depth of
66ft, where the pressure is
3atm. As the diver changes depth, the volume taken up by this fixed amount of air changes. Write an equation that defines the volume
V, in quarts, taken up by that amount of air at a given depth
d, in feet.

Performance Task A $\newline$ Modern scuba-diving equipment allows divers to stay underwater for long periods of time. Underwater instructors use mathematics to explore safety issues related to scuba diving. An instructor gives scuba-diving students the handout shown at the right. YOU WILL USE THE $\newline$ INFORMATION THROUGH THE WHOLE ASSESSMENT $\newline$ $1$ . Use the information in the handout. $\newline$ Part A $\newline$ Divers' Information $\newline$ - At the water's surface, the air around a $\newline$ diver exerts $1$ atmosphere $(\mathrm{atm})$ of pressure. $\newline$ - Underwater, the pressure $P(\mathrm{~atm})$ around the diver increases. It varies with the diver's depth $d$ , in feet, according to the equation $P=\frac{d}{33}+1$ . $\newline$ - The volume $V$ of a given amount of air varies inversely with the pressure $P$ around it, so the volume of air in the diver's lungs increases as she ascends. $\newline$ Suppose that a certain amount of air takes up a volume of $4 \mathrm{qt}$ in a diver's lungs at a depth of $66 \mathrm{ft}$ , where the pressure is $3 \mathrm{~atm}$ . As the diver changes depth, the volume taken up by this fixed amount of air changes. Write an equation that defines the volume $V$ , in quarts, taken up by that amount of air at a given depth $d$ , in feet.

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Math Problems

Grade 8

Evaluate a linear function: word problems

Full solution

Q. Performance Task A

\newline

Modern scuba-diving equipment allows divers to stay underwater for long periods of time. Underwater instructors use mathematics to explore safety issues related to scuba diving. An instructor gives scuba-diving students the handout shown at the right. YOU WILL USE THE

\newline

INFORMATION THROUGH THE WHOLE ASSESSMENT

\newline

1

. Use the information in the handout.

\newline

Part A

\newline

Divers' Information

\newline

- At the water's surface, the air around a

\newline

diver exerts

1

atmosphere

(\mathrm{atm})

of pressure.

\newline

- Underwater, the pressure

P(\mathrm{~atm})

around the diver increases. It varies with the diver's depth

d

, in feet, according to the equation

P=\frac{d}{33}+1

\newline

- The volume

V

of a given amount of air varies inversely with the pressure

P

around it, so the volume of air in the diver's lungs increases as she ascends.

\newline

Suppose that a certain amount of air takes up a volume of

4 \mathrm{qt}

in a diver's lungs at a depth of

66 \mathrm{ft}

, where the pressure is

3 \mathrm{~atm}

. As the diver changes depth, the volume taken up by this fixed amount of air changes. Write an equation that defines the volume

V

, in quarts, taken up by that amount of air at a given depth

d

, in feet.

Find Pressure at $66$ ft: First, we need to find the relationship between the volume $V$ and the pressure $P$ at the starting depth of $66$ ft. We use the given pressure equation $P = (d / 33) + 1$ .
Calculate Constant $k$ : Plug in $d = 66$ ft into the pressure equation to find the pressure at that depth: $P = (66 / 33) + 1 = 2 + 1 = 3$ atm.
Derive General Volume Equation: Since the volume $V$ of air varies inversely with the pressure $P$ , we can write $V = \frac{k}{P}$ , where $k$ is a constant. At $66\,\text{ft}$ , $V = 4\,\text{qt}$ and $P = 3\,\text{atm}$ .
Derive General Volume Equation: Since the volume $V$ of air varies inversely with the pressure $P$ , we can write $V = \frac{k}{P}$ , where $k$ is a constant. At $66\,\text{ft}$ , $V = 4\,\text{qt}$ and $P = 3\,\text{atm}$ .Now, solve for $k$ using $V = 4\,\text{qt}$ and $P = 3\,\text{atm}$ : $P$ $0$ .
Derive General Volume Equation: Since the volume $V$ of air varies inversely with the pressure $P$ , we can write $V = \frac{k}{P}$ , where $k$ is a constant. At $66$ ft, $V = 4$ qt and $P = 3$ atm. Now, solve for $k$ using $V = 4$ qt and $P = 3$ atm: $P$ $0$ qt $P$ $1$ atm $P$ $2$ qt $P$ $3$ atm. With $k$ found, we can write the general equation for $V$ at any depth $P$ $6$ : $V = \frac{k}{P}$ . Substitute $P$ $8$ qt $P$ $3$ atm and the pressure equation $V = \frac{k}{P}$ $0$ into this to get $V = \frac{k}{P}$ $1$ .
Derive General Volume Equation: Since the volume $V$ of air varies inversely with the pressure $P$ , we can write $V = \frac{k}{P}$ , where $k$ is a constant. At $66$ ft, $V = 4$ qt and $P = 3$ atm. Now, solve for $k$ using $V = 4$ qt and $P = 3$ atm: $P$ $0$ qt $P$ $1$ atm $P$ $2$ qt $P$ $3$ atm. With $k$ found, we can write the general equation for $V$ at any depth $P$ $6$ : $V = \frac{k}{P}$ . Substitute $P$ $8$ qt $P$ $3$ atm and the pressure equation $V = \frac{k}{P}$ $0$ into this to get $V = \frac{k}{P}$ $1$ . Simplify the equation for $V$ : $V = \frac{k}{P}$ $3$ .