All practice questions

Question 421Question →

For an angle $\theta$ satisfying $\frac{\pi}{2} < \theta < \pi$ , if $\sec \theta = -\frac{13}{5}$ , what is the value of $12(\cot \theta + \csc \theta)$ ?

Show answer & explanation

Answer: 8

Answer

8

For an angle

\theta

in Quadrant II (

\frac{\pi}{2} < \theta < \pi

), cosine is negative and sine is positive. Given

\sec \theta = -\frac{13}{5}

, the reciprocal identity gives

\cos \theta = -\frac{5}{13}

. The Pythagorean identity

\sin^2 \theta + \cos^2 \theta = 1

yields

\sin \theta = \sqrt{1 - \left(-\frac{5}{13}\right)^2} = \frac{12}{13}

. Then

\cot \theta = \frac{\cos \theta}{\sin \theta} = -\frac{5}{12}

and

\csc \theta = \frac{1}{\sin \theta} = \frac{13}{12}

. Adding these values gives

\cot \theta + \csc \theta = \frac{8}{12} = \frac{2}{3}

. Multiplying by 12 yields the final value of 8.

Step-by-Step Solution

1

Find

\cos \theta

from

\sec \theta

\cos \theta = -\frac{5}{13}

By definition of the reciprocal trigonometric identity,

\cos \theta = \frac{1}{\sec \theta}

.

2

Calculate

\sin \theta

using the Pythagorean identity

\sin \theta = \frac{12}{13}

In Quadrant II (

\frac{\pi}{2} < \theta < \pi

), sine is positive. Applying

\sin \theta = \sqrt{1 - \cos^2 \theta}

gives

\sqrt{1 - \frac{25}{169}} = \frac{12}{13}

.

3

Find

\cot \theta

and

\csc \theta

\cot \theta = -\frac{5}{12} \text{ and } \csc \theta = \frac{13}{12}

Using quotient identity

\cot \theta = \frac{\cos \theta}{\sin \theta}

and reciprocal identity

\csc \theta = \frac{1}{\sin \theta}

.

4

Substitute into the given expression

12(\cot \theta + \csc \theta)

and simplify

8

12\left(-\frac{5}{12} + \frac{13}{12}\right) = 12\left(\frac{8}{12}\right) = 8

.

Key Concept

Pythagorean, quotient, and reciprocal identities with quadrant sign analysis

Question 422Question →

A student measures the electric current, $I$ (in amperes, $\text{A}$ ), passing through a resistor in a closed circuit with a constant voltage. The current is inversely proportional to the resistance, $R$ (in ohms, $\Omega$ ). When the resistance is $4.0\text{ }\Omega$ , the current is $3.0\text{ A}$ . What is the current, in amperes, when the resistance is changed to $6.0\text{ }\Omega$ ?

Show answer & explanation

Answer: 2

Answer

The current is

2.0\text{ A}

when the resistance is changed to

6.0\text{ }\Omega

.

Since current and resistance are inversely proportional, their product remains constant:

I_1 R_1 = I_2 R_2

. Substituting the values gives

(3.0)(4.0) = I_2 (6.0)

, which simplifies to

12.0 = 6.0 I_2

. Solving for

I_2

yields

2.0\text{ A}

.

Step-by-Step Solution

1

State the inverse proportionality relationship between current and resistance.

I \times R = k

, where

k

is a constant.

Since the voltage is constant, current and resistance share an inverse relationship.

2

Calculate the constant value (

k

) using the initial measurements.

k = 3.0\text{ A} \times 4.0\text{ }\Omega = 12.0

Multiplying the known corresponding current and resistance values yields the proportionality constant.

3

Use the constant to calculate the new current at the new resistance.

I = \frac{12.0}{6.0} = 2.0\text{ A}

Dividing the constant by the new resistance value of

6.0\text{ }\Omega

gives the new current.

Key Concept

In an inverse proportionality relationship, the product of the two variables remains constant (

y \times x = k

). If one variable increases, the other must decrease proportionally.

Question 423Question →

During a biology lab, a student measures the volume of a liquid sample to be $2.5 \times 10^{-3}\text{ liters}$ . What is the volume of this sample in milliliters ( $\text{mL}$ )? (Enter only the numeric value.)

Show answer & explanation

Answer: 2.5

Answer

The volume of the sample is

2.5\text{ mL}

.

Since

1\text{ liter} = 1000\text{ milliliters}

, we multiply the volume in liters by

1000

. Thus,

2.5 \times 10^{-3} \times 10^3 = 2.5\text{ mL}

.

Step-by-Step Solution

1

Identify the conversion factor between liters (L) and milliliters (mL).

1\text{ L} = 1000\text{ mL}

To convert from liters to milliliters, we need to know the volumetric ratio between the two units.

2

Convert the volume using scientific notation multiplication.

2.5\text{ mL}

2.5 \times 10^{-3}\text{ L} \times 10^3\text{ mL/L} = 2.5 \times 10^{0}\text{ mL} = 2.5\text{ mL}

.

Key Concept

Converting scientific notation measurements from liters to milliliters

Question 424Question →

In an environmental study monitoring air quality near an industrial site, a scientist uses a flat horizontal collector plate with an active surface area of $5.0 \times 10^{-2}\text{ m}^2$ to gather falling dust particles. Over a sampling period of $1.8 \times 10^4\text{ seconds}$ , a total mass of $2.0 \times 10^{-3}\text{ grams}$ of dust is deposited on the plate. Assuming the dust deposition rate is constant, what is the average dust deposition rate in micrograms per square meter per hour ( $\mu\text{g}/(\text{m}^2\cdot\text{hr})$ )?

Show answer & explanation

Answer: 8000

Answer

The average dust deposition rate is 8000 micrograms per square meter per hour.

The correct rate of 8000 micrograms per square meter per hour is found by converting mass (

2.0 \times 10^{-3}\text{ g} = 2,000\text{ }\mu\text{g}

) and time (

1.8 \times 10^4\text{ s} = 5.0\text{ hours}

), then dividing this mass by the product of the area (

5.0 \times 10^{-2}\text{ m}^2

) and the time (

5.0\text{ hours}

).

Step-by-Step Solution

1

Convert the mass of collected dust from grams (g) to micrograms (\mu g).

2,000 \mu g

The target unit requires mass in micrograms. Since

1\text{ g} = 10^6\text{ }\mu\text{g}

, multiplying

2.0 \times 10^{-3}\text{ g}

by

10^6

gives

2.0 \times 10^3\text{ }\mu\text{g} = 2,000\text{ }\mu\text{g}

.

2

Convert the collection time from seconds (s) to hours (hr).

5.0 hr

The target unit requires time in hours. Since

1\text{ hr} = 3,600\text{ s}

, dividing the total seconds by

3,600

gives

1.8 \times 10^4\text{ s} \div (3.6 \times 10^3\text{ s/hr}) = 5.0\text{ hr}

.

3

Calculate the average deposition rate by dividing mass by the product of area and time.

8,000 \mu g/(m^2\cdot hr)

Deposition rate is given by the formula

\text{Rate} = \frac{\text{Mass}}{\text{Area} \times \text{Time}}

. Substituting the values:

\text{Rate} = \frac{2,000\text{ }\mu\text{g}}{(5.0 \times 10^{-2}\text{ m}^2) \times 5.0\text{ hr}} = \frac{2,000}{0.25} = 8,000\text{ }\mu\text{g}/(\text{m}^2\cdot\text{hr})

.

Key Concept

Multi-step dimensional analysis and calculation with scientific notation

Estimated Time:2m 30s

Question 425Question →

A chemist investigated the reaction rate of a reactant, Substance Y, at various initial concentrations. The initial rate of reaction, $R$ , in millimoles per liter per second ( $\text{mmol}\cdot\text{L}^{-1}\cdot\text{s}^{-1}$ ), was recorded for each concentration, $[Y]$ , in millimoles per liter ( $\text{mmol/L}$ ), at a constant temperature of $298\text{ K}$ . The results are presented in the table below:

Initial Concentration $[Y]$ ( $\text{mmol/L}$ )	Initial Rate of Reaction $R$ ( $\text{mmol}\cdot\text{L}^{-1}\cdot\text{s}^{-1}$ )
$1.5$	$4.5$
$3.0$	$18.0$
$4.5$	$40.5$
$6.0$	$72.0$
$7.5$	$112.5$

Based on the trend shown in the table, what would be the expected initial rate of reaction, in $\text{mmol}\cdot\text{L}^{-1}\cdot\text{s}^{-1}$ , if the initial concentration of Substance Y is increased to $12.0\text{ mmol/L}$ ?

Show answer & explanation

Answer: 288

Answer

The expected initial rate of reaction at a concentration of 12.0 mmol/L is 288.0 mmol*L^-1*s^-1.

The rate of reaction scales quadratically with concentration. Calculating the ratio of the rate to the concentration for each data point reveals that the ratio is equal to 2.0 times the concentration, yielding the equation R = 2.0 * [Y]^2. Substituting the target concentration of 12.0 mmol/L gives R = 2.0 * (12.0)^2 = 288.0 mmol*L^-1*s^-1. Alternatively, using the method of finite differences, the second difference between successive values is constant at 9.0, and continuing this sequence to 12.0 mmol/L also results in 288.0.

Step-by-Step Solution

1

Calculate the ratio of the rate R to the concentration [Y] for each data point.

The ratios are 3.0, 6.0, 9.0, 12.0, and 15.0.

To determine whether a direct proportional or higher-order relationship exists.

2

Formulate the mathematical model that represents this trend.

The ratio R/[Y] increases by 3.0 for every 1.5 mmol/L increase in [Y], which corresponds to R/[Y] = 2.0 * [Y], or R = 2.0 * [Y]^2.

To establish the quadratic relationship governing the dataset.

3

Substitute the target concentration value of 12.0 mmol/L into the derived quadratic equation.

R = 2.0 * (12.0)^2 = 288.0.

To calculate the extrapolated reaction rate.

Key Concept

Extrapolation of Quadratic Trends

Question 426Question →

An agricultural scientist measured the water absorption capacity of three different types of superabsorbent polymer gels (Gel A, Gel B, and Gel C) used in soil conditioning. Six trials were conducted for each gel type, and the mass of water absorbed (in grams per gram of gel) was recorded in the table below:

Gel Type	Trial 1 (g)	Trial 2 (g)	Trial 3 (g)	Trial 4 (g)	Trial 5 (g)	Trial 6 (g)
Gel A	42.1	44.5	41.8	43.2	45.0	42.6
Gel B	51.5	48.2	53.4	49.8	50.6	52.1
Gel C	33.7	35.2	34.0	32.9	36.1	34.5

Based on the table, what is the median water absorption capacity, in grams, for Gel B across the 6 trials?

Show answer & explanation

Answer: 51.05

Answer

The median water absorption capacity for Gel B across the 6 trials is 51.05 grams.

The correct answer is 51.05. To find the median value for Gel B, the 6 water absorption capacities must first be ordered from least to greatest: 48.2, 49.8, 50.6, 51.5, 52.1, and 53.4. Because there is an even number of values, the median is the average of the two middle values (50.6 and 51.5), which is 51.05.

Step-by-Step Solution

1

Identify the water absorption capacity values for Gel B.

51.5, 48.2, 53.4, 49.8, 50.6, and 52.1

To find the median, we must start with the complete set of data points for the specified group.

2

Sort the data points in ascending order.

48.2, 49.8, 50.6, 51.5, 52.1, 53.4

Calculating a median requires the values to be ordered from least to greatest.

3

Locate the middle values and calculate their average.

Middle values: 50.6 and 51.5. Average: 51.05

For an even number of data points, the median is the arithmetic mean of the two middle values.

Key Concept

Calculating the median of a dataset with an even number of values.

Question 427Question →

A student investigated the electrical properties of a negative temperature coefficient (NTC) thermistor. The thermistor was placed in a temperature-controlled water bath, and its electrical resistance, $R$ (in kilohms, $\text{k}\Omega$ ), was measured at various temperatures, $T$ (in degrees Celsius, $^\circ\text{C}$ ). The data from this experiment are presented in the table below.

Temperature ( $T$ , $^\circ\text{C}$ )	Resistance ( $R$ , $\text{k}\Omega$ )
10	48.0
30	24.0
50	12.0
70	6.0

Based on the trend shown in the table, what is the predicted electrical resistance, in $\text{k}\Omega$ , of the thermistor at a temperature of $110^\circ\text{C}$ ?

Show answer & explanation

Answer: 1.5

Answer

The predicted electrical resistance of the thermistor at

110^\circ\text{C}

is

1.5\text{ k}\Omega

.

The correct calculation identifies that the resistance decreases by a factor of 2 for every

20^\circ\text{C}

increase in temperature. Following this exponential trend, the resistance at

90^\circ\text{C}

is

3.0\text{ k}\Omega

, and at

110^\circ\text{C}

it is half of that, which equals

1.5\text{ k}\Omega

.

Step-by-Step Solution

1

Analyze the pattern of temperature changes in the table.

The temperature increments are constant at

\Delta T = 20^\circ\text{C}

(e.g.,

30 - 10 = 20^\circ\text{C}

,

50 - 30 = 20^\circ\text{C}

,

70 - 50 = 20^\circ\text{C}

).

Establishing a constant independent variable interval simplifies trend extrapolation.

2

Analyze the corresponding ratio of resistance values at each temperature step.

At each interval, the resistance value is divided by 2:

\frac{24.0}{48.0} = 0.5

,

\frac{12.0}{24.0} = 0.5

, and

\frac{6.0}{12.0} = 0.5

. This indicates a non-linear, exponential decay trend.

Determining the mathematical relationship allows for precise calculation of values outside the dataset range.

3

Extrapolate the trend to

90^\circ\text{C}

by applying the factor of

0.5

to the resistance at

70^\circ\text{C}

.

R(90^\circ\text{C}) = R(70^\circ\text{C}) \times 0.5 = 6.0 \times 0.5 = 3.0\text{ k}\Omega

.

Since

90^\circ\text{C}

is exactly

20^\circ\text{C}

above

70^\circ\text{C}

, the pattern dictates that the resistance halves.

4

Extrapolate the trend further to

110^\circ\text{C}

by applying the factor of

0.5

to the resistance at

90^\circ\text{C}

.

R(110^\circ\text{C}) = R(90^\circ\text{C}) \times 0.5 = 3.0 \times 0.5 = 1.5\text{ k}\Omega

.

Since

110^\circ\text{C}

is exactly

20^\circ\text{C}

above

90^\circ\text{C}

, the resistance halves once more.

Key Concept

Extrapolating non-linear (exponential) relationships by identifying constant ratios over equal intervals of the independent variable.

Question 428Question →

A physicist studies the rate of heat transfer, $H$ (in watts, $\text{W}$ ), through cylindrical metal rods. The researcher determines that $H$ is directly proportional to both the cross-sectional area of the rod and the temperature difference ( $\Delta T$ , in kelvins, $\text{K}$ ) between its two ends, and inversely proportional to the length of the rod ( $L$ , in meters, $\text{m}$ ). For Rod 1, the radius is $0.020\text{ m}$ , the length is $0.80\text{ m}$ , the temperature difference is $50.0\text{ K}$ , and the rate of heat transfer is $100.0\text{ W}$ . Rod 2 is made of the same metal and has a radius of $0.040\text{ m}$ , a length of $0.40\text{ m}$ , and a temperature difference of $30.0\text{ K}$ . What is the rate of heat transfer, in watts, for Rod 2?

Show answer & explanation

Answer: 480

Answer

The rate of heat transfer for Rod 2 is 480 W.

The correct answer is 480 W because the rate of heat transfer is proportional to the square of the radius and the temperature difference, and inversely proportional to the length. The radius is doubled (scaling factor of

2^2 = 4

), the temperature difference is multiplied by 0.6, and the length is halved (scaling factor of

\frac{1}{0.5} = 2

). This yields a combined factor of

4 \times 0.6 \times 2 = 4.8

, and multiplying 100.0 W by 4.8 results in 480 W.

Step-by-Step Solution

1

Relate the cross-sectional area to the radius of the rod.

The area

A

is directly proportional to the square of the radius

r

:

A \propto r^2

.

The cross-section of a cylinder is a circle with area

A = \pi r^2

.

2

Formulate the complete proportionality expression for the rate of heat transfer.

H \propto \frac{r^2 \cdot \Delta T}{L}

.

Heat transfer is directly proportional to area (

r^2

) and temperature difference (

\Delta T

), and inversely proportional to length (

L

).

3

Set up a ratio to compare Rod 2's heat transfer rate to Rod 1's rate.

\frac{H_2}{H_1} = \left(\frac{r_2}{r_1}\right)^2 \cdot \left(\frac{\Delta T_2}{\Delta T_1}\right) \cdot \left(\frac{L_1}{L_2}\right)

.

Using a ratio cancels out the constant of proportionality.

4

Calculate the ratio multiplier by inserting the known values.

Multiplier =

2^2 \cdot 0.6 \cdot 2 = 4 \cdot 0.6 \cdot 2 = 4.8

.

The radius doubles (factor of 4), the temperature difference is multiplied by 0.6, and the length is halved (factor of 2).

5

Multiply the original heat transfer rate by the calculated factor.

H_2 = 100.0 \text{ W} \times 4.8 = 480 \text{ W}

.

To find the final heat transfer rate of Rod 2.

Key Concept

Combining direct and inverse proportionalities to calculate a new value using scaling factors.

Question 429Question →

In a biochemistry laboratory experiment, a microfluidic device is calibrated to pump an enzyme solution into a reaction chamber at a constant rate of $4.8 \times 10^{-5}$ liters per minute ( $\text{L/min}$ ). What is this pump rate in microliters per second ( $\mu\text{L/s}$ )?

Show answer & explanation

Answer: 0.8

Answer

The correct pump rate is 0.8 microliters per second.

To convert

4.8 \times 10^{-5}\text{ L/min}

to

\mu\text{L/s}

, first convert liters to microliters by multiplying by

10^6\ \mu\text{L/L}

, which gives

48\ \mu\text{L/min}

. Next, convert minutes to seconds by dividing by

60\text{ s/min}

to get

0.8\ \mu\text{L/s}

.

Step-by-Step Solution

1

Convert the volume flow rate from liters per minute to microliters per minute.

48 \mu\text{L/min}

Since

1\text{ L} = 10^6\ \mu\text{L}

, multiplying

4.8 \times 10^{-5}\text{ L/min}

by

10^6\ \mu\text{L/L}

yields

48\ \mu\text{L/min}

.

2

Convert the flow rate from microliters per minute to microliters per second.

0.8 \mu\text{L/s}

Since

1\text{ minute} = 60\text{ seconds}

, dividing

48\ \mu\text{L/min}

by

60\text{ s/min}

yields

0.8\ \mu\text{L/s}

.

Key Concept

Scientific Notation and Unit Conversions

Question 430Question →

A student measured the speed of sound in a chamber filled with pure carbon dioxide ( $CO_2$ ) gas at various temperatures. The measured speed of sound, in meters per second ( $\text{m/s}$ ), at each temperature, in degrees Celsius ( $^\circ\text{C}$ ), is shown in the table below:

Temperature ( $^\circ\text{C}$ )	Speed of Sound ( $\text{m/s}$ )
$0$	$259$
$20$	$268$
$40$	$277$
$60$	$286$

Assuming the speed of sound continues to change at a constant rate with respect to temperature, what is the predicted speed of sound in $CO_2$ gas, in meters per second ( $\text{m/s}$ ), at a temperature of $100^\circ\text{C}$ ?

Show answer & explanation

Answer: 304

Answer

The predicted speed of sound in carbon dioxide gas at

100^\circ\text{C}

is

304\text{ m/s}

.

The speed of sound increases linearly by

9\text{ m/s}

for every

20^\circ\text{C}

increase in temperature, which is a rate of

0.45\text{ m/s}

per

1^\circ\text{C}

. The target temperature of

100^\circ\text{C}

is

40^\circ\text{C}

higher than the highest data point in the table (

60^\circ\text{C}

). The speed of sound will therefore increase by

40 \times 0.45 = 18\text{ m/s}

beyond the

60^\circ\text{C}

speed. Adding this to

286\text{ m/s}

yields

304\text{ m/s}

.

Step-by-Step Solution

1

Determine the constant rate of change of the speed of sound per

1^\circ\text{C}

temperature increase.

The speed of sound increases at a rate of

0.45\text{ m/s}

per

1^\circ\text{C}

.

This establishes the linear trend shown in the experimental data.

2

Find the temperature interval between the highest measured data point and the target temperature.

The difference is

40^\circ\text{C}

(from

60^\circ\text{C}

to

100^\circ\text{C}

).

This determines how far outside the measured data range the extrapolation must extend.

3

Multiply the temperature interval by the rate of change and add it to the speed of sound at the highest measured temperature.

286\text{ m/s} + (40^\circ\text{C} \times 0.45\text{ m/s/}^\circ\text{C}) = 304\text{ m/s}

.

This completes the linear extrapolation to predict the final value.

Key Concept

Linear extrapolation relies on determining a constant rate of change from the given data points and applying it to a target value outside the experimental range.

Question 431Question →

A student conducted a series of trials to investigate Fick's first law of diffusion using a synthetic membrane. The rate of diffusion of a solute, $J$ (in milligrams per second, $\text{mg/s}$ ), is directly proportional to both the surface area of the membrane, $A$ (in square centimeters, $\text{cm}^2$ ), and the concentration difference of the solute across the membrane, $\Delta C$ (in moles per liter, $\text{mol/L}$ ), and is inversely proportional to the thickness of the membrane, $x$ (in millimeters, $\text{mm}$ ).

The parameters for Trial 1 and Trial 2 are shown in the table below:

Trial	Membrane thickness, $x$ ( $\text{mm}$ )	Membrane surface area, $A$ ( $\text{cm}^2$ )	Concentration difference, $\Delta C$ ( $\text{mol/L}$ )	Diffusion rate, $J$ ( $\text{mg/s}$ )
1	$0.20$	$3.0$	$0.06$	$27.0$
2	$0.60$	$8.0$	$0.04$	?

Based on the table, what was the resulting diffusion rate of the solute in Trial 2, in $\text{mg/s}$ ?

Show answer & explanation

Answer: 16

Answer

The diffusion rate of the solute in Trial 2 is 16.0 mg/s.

By setting up the proportionality equation

J = k \frac{A \cdot \Delta C}{x}

, we find the constant

k = 30

using the parameters from Trial 1. Substituting the parameters from Trial 2 yields a diffusion rate of exactly 16.0 mg/s.

Step-by-Step Solution

1

Set up the algebraic relationship for the variables based on proportionality rules.

J = k \frac{A \cdot \Delta C}{x}

The rate of diffusion (

J

) is directly proportional to the surface area (

A

) and concentration difference (

\Delta C

), meaning they appear in the numerator. It is inversely proportional to membrane thickness (

x

), meaning it appears in the denominator. Here,

k

represents the constant of proportionality.

2

Calculate the constant of proportionality,

k

, using the values provided for Trial 1.

k = 30

Substitute

J_1 = 27.0

,

A_1 = 3.0

,

\Delta C_1 = 0.06

, and

x_1 = 0.20

into the equation:

27.0 = k \frac{3.0 \times 0.06}{0.20} \implies 27.0 = 0.9k \implies k = 30

.

3

Calculate the diffusion rate for Trial 2,

J_2

, by substituting the new parameters and the calculated constant

k

into the equation.

J_2 = 16.0

Substitute

k = 30

,

A_2 = 8.0

,

\Delta C_2 = 0.04

, and

x_2 = 0.60

into the equation:

J_2 = 30 \times \frac{8.0 \times 0.04}{0.60} = 30 \times \frac{0.32}{0.60} = 16.0

.

Key Concept

Direct and Inverse Proportionality

Question 432Question →

Aerodynamic drag force ( $F_d$ ) acts on vehicles as they move through the air. A group of students measured the drag force, in newtons ( $\text{N}$ ), acting on a scale model of a sports car in a wind tunnel at various wind velocities ( $v$ ), in meters per second ( $\text{m/s}$ ). The data from their trials are recorded in the table below:

Velocity ( $v$ , $\text{m/s}$ )	Drag Force ( $F_d$ , $\text{N}$ )
$10$	$12$
$20$	$48$
$30$	$108$
$40$	$192$

Based on the trend shown in the table, what is the expected aerodynamic drag force, in newtons ( $\text{N}$ ), acting on the scale model when the wind velocity is $50\text{ m/s}$ ?

Show answer & explanation

Answer: 300

Answer

The expected aerodynamic drag force is 300 N.

The correct answer is 300 N because the drag force scales quadratically with velocity according to the relation

F_d = 0.12 v^2

. Plugging in

v = 50\text{ m/s}

yields

0.12 \times 2500 = 300\text{ N}

.

Step-by-Step Solution

1

Calculate the ratio of drag force to the square of the velocity for the given data points.

For all data points,

F_d / v^2 = 0.12

. This establishes the quadratic trend

F_d = 0.12 v^2

.

Identifying the mathematical relationship between the variables is necessary to accurately extrapolate beyond the measured data range.

2

Substitute the target velocity of

50\text{ m/s}

into the identified quadratic formula.

F_d = 0.12 \times (50)^2 = 300\text{ N}

.

Applying the mathematical trend allows for the calculation of the drag force at the extrapolated velocity.

Key Concept

Extrapolation of a quadratic relationship between velocity and aerodynamic drag force.

Estimated Time:1m 30s

Question 433Question →

Geologists drilled a deep borehole into the Earth's crust at a research site and measured the rock temperature at various depths. The recorded temperatures are shown in the table below:

Depth ( $m$ )	Temperature ( $^\circ\text{C}$ )
$0$	$12.0$
$250$	$19.5$
$500$	$27.0$
$750$	$34.5$
$1,000$	$42.0$

Assuming the temperature continues to increase linearly with depth at the same rate observed between $0\text{ m}$ and $1,000\text{ m}$ , what will the rock temperature, in degrees Celsius ( $^\circ\text{C}$ ), be at a depth of $1,800\text{ m}$ ?

Show answer & explanation

Answer: 66

Answer

The projected rock temperature at a depth of 1,800 m is 66.0°C.

The correct calculation determines that the temperature increases by 7.5°C for every 250 m (a rate of 0.03°C/m). Extrapolating linearly to 1,800 m, the temperature increases by 54.0°C (0.03°C/m * 1,800 m) from the surface baseline of 12.0°C, yielding a final temperature of 66.0°C.

Step-by-Step Solution

1

Calculate the rate of temperature change per meter of depth.

0.03^\circ\text{C/m}

Using the interval from

0\text{ m}

to

250\text{ m}

, the temperature increases by

19.5^\circ\text{C} - 12.0^\circ\text{C} = 7.5^\circ\text{C}

. The rate of change is

\frac{7.5^\circ\text{C}}{250\text{ m}} = 0.03^\circ\text{C/m}

.

2

Calculate the total temperature change over the depth interval of 1,800 m.

54.0^\circ\text{C}

Multiplying the constant rate of temperature change (

0.03^\circ\text{C/m}

) by the target depth (

1,800\text{ m}

) yields the total increase in temperature from the surface:

0.03 \times 1,800 = 54.0^\circ\text{C}

.

3

Determine the final temperature by adding the increase to the baseline surface temperature.

66.0^\circ\text{C}

Adding the

54.0^\circ\text{C}

increase to the baseline temperature at the surface (

12.0^\circ\text{C}

) gives the projected temperature at depth:

12.0^\circ\text{C} + 54.0^\circ\text{C} = 66.0^\circ\text{C}

.

Key Concept

Extrapolation of linear data trends

Estimated Time:1m 30s

Question 434Question →

A student conducted a series of trials using a rotating mass apparatus to investigate the relationship between centripetal force ( $F_c$ ) and the radius of rotation ( $r$ ) for a constant mass moving at a constant speed. The results of the trials are shown in the table below:

Trial	Radius ( $r$ , $\text{m}$ )	Centripetal Force ( $F_c$ , $\text{N}$ )
1	$0.80$	$12.0$
2	$2.40$	?

Given that the centripetal force is inversely proportional to the radius of rotation under these conditions, what is the centripetal force, in newtons, for Trial 2?

Show answer & explanation

Answer: 4

Answer

The centripetal force in Trial 2 is

4.0\text{ N}

.

Since centripetal force is inversely proportional to the radius of rotation, their product must remain constant (

F_c \times r = k

). Using Trial 1, we find

k = 12.0 \times 0.80 = 9.6

. For Trial 2, we set up the equation

F_c \times 2.40 = 9.6

, which yields

F_c = 4.0\text{ N}

.

Step-by-Step Solution

1

Identify the mathematical relationship between the variables.

F_c \times r = k

The problem states that centripetal force is inversely proportional to the radius of rotation.

2

Calculate the constant of proportionality (

k

) using the data from Trial 1.

k = 12.0 \times 0.80 = 9.6

Both the radius (

r = 0.80\text{ m}

) and centripetal force (

F_c = 12.0\text{ N}

) are known for Trial 1.

3

Calculate the unknown centripetal force for Trial 2.

F_c = \frac{9.6}{2.40} = 4.0

The constant of proportionality is

9.6

and the radius for Trial 2 is

2.40\text{ m}

.

Key Concept

Inverse Proportionality

Question 435Question →

A student set up an experiment to investigate the mechanical properties of a simple two-gear system consisting of a driver gear and a driven gear. The driver gear rotates at a constant speed, while various driven gears with different numbers of teeth, $N$ , are swapped into the system. The student measured the rotational speed, $S$ (in revolutions per minute, rpm), of each driven gear. The table below shows the results of the experiment:

Driven Gear	Number of Teeth ( $N$ )	Rotational Speed ( $S$ , rpm)
Gear 1	8	270
Gear 2	12	180
Gear 3	16	135
Gear 4	18	?

Given that the rotational speed of the driven gear is inversely proportional to its number of teeth, what is the rotational speed, in rpm, of Gear 4?

Show answer & explanation

Answer: 120

Answer

The rotational speed of Gear 4 is 120 rpm.

Since rotational speed is inversely proportional to the number of teeth, the product of these two values must remain constant. For all measured gears,

S \times N = 2160

. Dividing this constant by the 18 teeth of Gear 4 yields a rotational speed of 120 rpm.

Step-by-Step Solution

1

Determine the proportionality constant

Constant

k = 2160

Because rotational speed (

S

) and number of teeth (

N

) are inversely proportional, their product is constant (

S \times N = k

). Using Gear 2 data,

12 \times 180 = 2160

.

2

Calculate the speed for Gear 4

Rotational speed = 120

Using the constant

k = 2160

and the number of teeth for Gear 4 (

N = 18

), the speed is calculated as

S = \frac{2160}{18} = 120

.

Key Concept

Direct and Inverse Proportionality

Question 436Question →

A group of students conducted a physics experiment to study the relationship between the mass of a glider and its acceleration on a horizontal air track. A constant net force was applied to the glider during all trials. The students observed that the acceleration of the glider, $a$ , is inversely proportional to its mass, $m$ . During Trial 1, a glider with a mass of $0.20\text{ kg}$ was measured to have an acceleration of $5.0\text{ m/s}^2$ . During Trial 2, a different glider was used under the same constant net force. If the mass of the glider in Trial 2 is $0.50\text{ kg}$ , what is the acceleration of the glider, in $\text{m/s}^2$ ?

Show answer & explanation

Answer: 2

Answer

The acceleration of the glider in Trial 2 is 2.0 m/s^2.

Since acceleration is inversely proportional to mass, their product remains constant under a constant net force. Using the data from Trial 1, the constant is calculated as

0.20\text{ kg} \times 5.0\text{ m/s}^2 = 1.0\text{ N}

. In Trial 2, with a mass of

0.50\text{ kg}

, the acceleration is found by dividing the constant by the new mass:

1.0 / 0.50 = 2.0\text{ m/s}^2

.

Step-by-Step Solution

1

Set up the inverse proportionality equation.

a \times m = k

Since acceleration is inversely proportional to mass, their product must equal a constant value.

2

Calculate the constant of proportionality using Trial 1 data.

k = 1.0

Substitute

m = 0.20\text{ kg}

and

a = 5.0\text{ m/s}^2

into the equation:

5.0 \times 0.20 = 1.0

.

3

Calculate the new acceleration for Trial 2.

a_2 = 2.0\text{ m/s}^2

Substitute the constant

k = 1.0

and the new mass

m_2 = 0.50\text{ kg}

into the equation:

a_2 \times 0.50 = 1.0

, so

a_2 = 2.0

.

Key Concept

Inverse proportionality relates two variables such that their product is constant. If one variable increases by a factor, the other must decrease by the same factor.

Question 437Question →

A community library has a collection of 250 books. If $44\%$ of these books are classified as fiction, how many fiction books are in the library's collection?

Show answer & explanation

Answer: 110

Answer

There are 110 fiction books in the library's collection.

To find

44\%

of 250 books, first convert

44\%

to a decimal by dividing by 100, which is

0.44

. Then, multiply the decimal by the total number of books:

0.44 \times 250 = 110

. This represents the number of fiction books in the library.

Step-by-Step Solution

1

Convert the percentage of fiction books to a decimal.

0.44

To write a percent as a decimal, divide by 100 or move the decimal point two places to the left.

2

Multiply the decimal by the total number of books to find the number of fiction books.

110

Multiplying the decimal representing the fraction of the total by the total number of items gives the portion representing fiction books.

Key Concept

Calculating a percentage of a total amount

Estimated Time:45s

Question 438Question →

A student took a history exam containing $80$ questions. If the student answered $15\%$ of the questions incorrectly, how many questions did the student answer correctly?

Show answer & explanation

Answer: 68

Answer

The student answered

68

questions correctly.

First, find the percentage of correct answers by subtracting the percentage of incorrect answers (

15\%

) from

100\%

, which gives

85\%

. Convert

85\%

to a decimal (

0.85

) and multiply it by the total number of questions (

80

):

80 \times 0.85 = 68

.

Step-by-Step Solution

1

Calculate the percentage of questions answered correctly.

85\%

Subtract the percentage of incorrect questions (

15\%

) from the total (

100\%

) to find the percentage of correct questions.

2

Convert the percentage of correct questions to a decimal.

0.85

Divide the percentage by

100

to express it as a decimal.

3

Multiply the total questions by the decimal value.

68

Multiplying the total number of questions by the proportion of correct answers gives the total number of correct questions:

80 \times 0.85 = 68

.

Key Concept

Calculating percentages of a whole number

Question 439Question →

On a standard number line, points $A$ , $B$ , $C$ , and $D$ have distinct integer coordinates $a$ , $b$ , $c$ , and $d$ , respectively, such that $a < b < c < d$ . The distance between $A$ and $B$ is equal to the distance between $C$ and $D$ . The distance between $B$ and $C$ is $\frac{2}{3}$ of the distance between $A$ and $B$ . If the average (arithmetic mean) of the four coordinates is $0$ and $|a - d| = 24$ , what is the coordinate of $B$ ?

Show answer & explanation

Answer: -3

Answer

The coordinate of

B

is

-3

.

The coordinate of

B

is found by setting the segment lengths to

3k

,

2k

, and

3k

based on the given ratio. Using the absolute value distance

|a - d| = 24

, we find

k = 3

, meaning the segments are

9

,

6

, and

9

. Using the coordinate sum of

0

, we get

4a + 48 = 0 \implies a = -12

. Substituting this back gives the coordinate of

B

as

-3

.

Step-by-Step Solution

1

Define segment lengths using a variable

k

based on the given ratio.

Let the distance

AB = CD = 3k

and

BC = 2k

.

This allows us to write all distances as integer multiples of a single variable since the ratio of

BC

to

AB

is

\frac{2}{3}

.

2

Set up an equation for the total distance from

A

to

D

using the absolute value

|a - d| = 24

.

Since

a < d

,

d - a = |a - d| = 24

. The sum of the segments is

3k + 2k + 3k = 8k

. Solving

8k = 24

yields

k = 3

.

The absolute value of the difference between the outermost points represents the total length of the number line segment containing all four points.

3

Express the coordinates of

b

,

c

, and

d

in terms of

a

using the calculated segment lengths.

b = a + 9

,

c = a + 15

, and

d = a + 24

.

Since the points are in order

a < b < c < d

, we add the segment lengths successively to find the coordinates.

4

Apply the average condition to solve for the coordinate

a

.

The sum of the coordinates is

4 \times 0 = 0

, so

a + (a + 9) + (a + 15) + (a + 24) = 4a + 48 = 0

, giving

a = -12

.

An average of

0

for four numbers means their sum must be

0

.

5

Find the coordinate of

B

by substituting the value of

a

into the expression for

b

.

b = -12 + 9 = -3

.

This yields the specific coordinate requested by the question.

Key Concept

Using absolute value as distance on a number line and expressing relationships between ordered coordinates algebraically.

Question 440Question →

A positive integer $K$ is a multiple of $10$ and has exactly $8$ positive factors (including $1$ and $K$ ). What is the least possible value of $K$ ?

Show answer & explanation

Answer: 30

Answer

The least possible value of

K

is

30

.

The correct answer is

30

because it is the smallest multiple of

10

that has exactly

8

positive factors (

1, 2, 3, 5, 6, 10, 15,

and

30

).

Step-by-Step Solution

1

Determine the prime factors required for a multiple of

10

.

K

must have at least one factor of

2

and at least one factor of

5

.

Since

10 = 2 \times 5

, any multiple of

10

must be divisible by both

2

and

5

.

2

Set up the factor counting formula.

The number of factors is

(a+1)(b+1)(c+1)\cdots = 8

, where

a, b, c, \dots

are the exponents of the prime factorization.

The number of positive divisors of

p_1^a p_2^b \cdots

is found by adding

1

to each exponent and multiplying the results.

3

Evaluate the smallest multiples of

10

to find the first one with exactly

8

factors.

10

has

4

factors (

1, 2, 5, 10

);

20

has

6

factors (

1, 2, 4, 5, 10, 20

);

30

has

8

factors (

1, 2, 3, 5, 6, 10, 15, 30

).

Testing the positive multiples of

10

in ascending order ensures that the first number with exactly

8

factors is the smallest possible value.

Key Concept

Factors, Multiples, and Prime Factorization

Answer

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