All practice questions

Question 221Question →

An industrial cardboard baler utilizes a closed hydraulic system to compress recyclable materials. The system features an input pump piston with a cross-sectional area of $3 \text{ in}^2$ and a large output driving piston with a cross-sectional area of $45 \text{ in}^2$ . If the input piston is pushed downward for a single stroke distance of $15 \text{ inches}$ , how many inches will the output piston move? (Assume no fluid is lost and the fluid is incompressible.)

Show answer & explanation

Answer: 1

Answer

1

In a hydraulic system, the volume of fluid displaced must be conserved. The output piston has 15 times the cross-sectional area of the input piston (

45 \div 3 = 15

). Because it is 15 times larger, it will only travel 1/15th of the distance to displace the identical volume of fluid. Therefore, an input distance of 15 inches results in an output distance of exactly 1 inch.

Step-by-Step Solution

1

Identify the relationship for fluid displacement in a closed hydraulic system.

A_1 \times d_1 = A_2 \times d_2

Because liquids are incompressible, the volume of fluid pushed out by the input piston (

A_1 \times d_1

) must equal the volume of fluid pushing up the output piston (

A_2 \times d_2

).

2

Substitute the given values for the input piston (

A_1 = 3

,

d_1 = 15

) and output piston (

A_2 = 45

).

3 \times 15 = 45 \times d_2

This sets up the equation with all known variables to solve for the unknown distance (

d_2

).

3

Calculate the total volume displaced by the input piston.

45 = 45 \times d_2

Multiplying the input area by its travel distance gives a total displaced volume of 45 cubic inches.

4

Solve for the output distance (

d_2

) by dividing the total volume by the output piston's area.

d_2 = 1

Dividing 45 cubic inches by 45 square inches yields a travel distance of 1 inch.

Key Concept

Conservation of fluid volume and the distance-force tradeoff in hydraulics

Question 222Question →

A bucket is being raised from a well using a windlass mechanism (a type of wheel and axle). The cylindrical drum (axle) around which the rope winds has a diameter of $8 \text{ inches}$ . The crank handle used to turn the drum has a length (radius) from the center of rotation of $20 \text{ inches}$ . If the bucket is raised vertically by exactly $15 \text{ feet}$ , what is the total distance, in feet, that the crank handle must travel along its circular path? (Assume the rope does not overlap itself on the drum).

Show answer & explanation

Answer: 75

Answer

75

In any simple machine, assuming 100% efficiency, the work put in equals the work put out. The Ideal Mechanical Advantage (IMA) for a wheel and axle is the ratio of the effort radius to the load radius. Here, the load radius (the drum) is

4 \text{ inches}

(half of the

8 \text{ inch}

diameter) and the effort radius (the crank) is

20 \text{ inches}

. Thus, the IMA is

20 / 4 = 5

. This geometric relationship dictates that the effort force must travel

5

times further than the load to conserve work. If the load is lifted

15 \text{ feet}

, the effort distance is

15 \times 5 = 75 \text{ feet}

.

Step-by-Step Solution

1

Calculate the drum's radius.

4 \text{ inches}

The Ideal Mechanical Advantage compares radii, but the drum's dimension was provided as a diameter.

2

Calculate the Ideal Mechanical Advantage (IMA).

IMA =

5

Dividing the effort radius (

20 \text{ inches}

) by the load radius (

4 \text{ inches}

) gives the mechanical advantage of the system.

3

Calculate the total distance traveled by the crank handle.

75 \text{ feet}

By the principle of conservation of energy, the effort distance must be equal to the load distance (

15 \text{ feet}

) multiplied by the IMA (

5

).

Key Concept

Ideal Mechanical Advantage (IMA) and Force-Distance Tradeoff

Estimated Time:1m 30s

Question 223Question →

An industrial pneumatic press system utilizes trapped gas to generate force. The gas is initially at an absolute pressure of $30 \text{ psi}$ and occupies a volume of $200 \text{ in}^3$ . During the operation cycle, a mechanical ram pushes down, reducing the gas volume to $50 \text{ in}^3$ while the temperature remains constant. If the face of the piston has a cross-sectional area of $5 \text{ in}^2$ , what is the total upward force, in pounds, exerted by the compressed gas on the piston at this new volume?

Show answer & explanation

Answer: 600

Answer

600

Since the temperature is constant, Boyle's Law (

P_1V_1 = P_2V_2

) applies. The volume is reduced from

200 \text{ in}^3

to

50 \text{ in}^3

, which is one-quarter of its original size. Because pressure and volume are inversely proportional for a fixed amount of gas at a constant temperature, the absolute pressure must increase by a factor of four (

30 \text{ psi} \times 4 = 120 \text{ psi}

). The total force exerted by a pressurized fluid on a surface is the product of its pressure and the surface area (

F = P \times A

). Multiplying the new pressure of

120 \text{ psi}

by the piston area of

5 \text{ in}^2

gives a total upward force of

600 \text{ lbs}

.

Step-by-Step Solution

1

Calculate the final pressure of the gas after compression.

P_2 = 120 \text{ psi}

According to Boyle's Law, when temperature is constant, the product of absolute pressure and volume is constant (

P_1V_1 = P_2V_2

).

2

Calculate the total upward force exerted by the gas.

F = 600 \text{ lbs}

Force is equal to pressure multiplied by the area over which it is applied (

F = P \times A

).

Key Concept

Combining Boyle's Law for isothermal compression with the basic relationship between pressure, force, and area.

Question 224Question →

A contractor uses a $15\text{-foot}$ ramp and a motorized winch to load a $600\text{-lb}$ generator. If the winch can pull with a maximum steady force of $120\text{ lbs}$ , what is the maximum vertical height, in feet, that the ramp can reach assuming a frictionless surface?

Show answer & explanation

Answer: 3

Answer

The maximum vertical height the ramp can reach is 3 feet.

An inclined plane multiplies the applied effort force by spreading the required work over a longer distance. First, determine the Mechanical Advantage needed by dividing the load (

600\text{ lbs}

) by the available effort force (

120\text{ lbs}

), which gives an IMA of

5

. This means the ramp must be

5

times longer than it is high. Given a ramp length of

15\text{ feet}

, divide this length by the IMA of

5

to find the maximum vertical height, which is

3\text{ feet}

.

Step-by-Step Solution

1

Determine the required Ideal Mechanical Advantage (IMA).

\text{IMA} = \frac{600}{120} = 5

The IMA represents how many times the machine multiplies the effort force. It is found by dividing the load force by the effort force.

2

Use the inclined plane IMA formula to set up an equation for the height.

5 = \frac{15}{\text{Height}}

For an inclined plane, the

\text{IMA}

is also equal to the ratio of the

\text{Ramp Length}

to the

\text{Vertical Height}

.

3

Solve the equation for the vertical height.

\text{Height} = \frac{15}{5} = 3

Rearrange the equation to isolate the height, dividing the ramp length by the required IMA.

Key Concept

Ideal Mechanical Advantage (IMA) of an inclined plane and determining vertical height from force ratios.

Question 225Question →

A forestry worker uses a high-impact plastic felling wedge to direct the fall of a large tree. The wedge has a length (penetration depth) of $15\text{ cm}$ and a maximum heel thickness of $3\text{ cm}$ . If the worker strikes the wedge with a sledgehammer, applying an average effort force of $450\text{ N}$ , what is the theoretical maximum lifting force (in Newtons) exerted by the wedge? Assume $100\%$ efficiency.

Show answer & explanation

Answer: 2250

Answer

2250 Newtons

The correct theoretical maximum lifting force is found by first determining the mechanical advantage of the wedge (length divided by thickness), which is 5. Multiplying this mechanical advantage by the input effort force (450 N) yields an output force of 2250 N.

Step-by-Step Solution

1

Identify the formula for the Ideal Mechanical Advantage (IMA) of a wedge.

\text{IMA} = \frac{\text{Length}}{\text{Thickness}}

The mechanical advantage of a wedge is determined by the ratio of its penetration length to its maximum separation thickness.

2

Calculate the IMA using the given dimensions.

\text{IMA} = \frac{15\text{ cm}}{3\text{ cm}} = 5

Substituting the length (

15\text{ cm}

) and thickness (

3\text{ cm}

) into the formula gives a mechanical advantage multiplier of 5.

3

Calculate the theoretical maximum output force.

\text{Output Force} = \text{IMA} \times \text{Input Force} = 5 \times 450\text{ N} = 2250\text{ N}

The output force is the product of the mechanical advantage and the applied effort force, assuming ideal conditions.

Key Concept

Ideal Mechanical Advantage (IMA) of Wedges

Question 226Question →

An automated logistics drone is transporting a sealed supply crate. On Earth, where the gravitational acceleration is considered to be $10 \text{ m/s}^2$ , the crate has a measured weight of $1400 \text{ N}$ . The crate is then delivered to a controlled low-gravity research habitat where the local gravitational acceleration is maintained at $3.5 \text{ m/s}^2$ .

What is the weight of the crate inside the low-gravity habitat in Newtons?

Show answer & explanation

Answer: 490

Answer

490

To find the new weight, first determine the crate's mass, which does not change. Dividing the Earth weight (

1400 \text{ N}

) by Earth's gravity (

10 \text{ m/s}^2

) gives a mass of

140 \text{ kg}

. Then, multiply this invariant mass by the habitat's gravity (

3.5 \text{ m/s}^2

) to find the new weight:

140 \text{ kg} \times 3.5 \text{ m/s}^2 = 490 \text{ N}

.

Step-by-Step Solution

1

Calculate the mass of the crate on Earth.

m = \frac{1400 \text{ N}}{10 \text{ m/s}^2} = 140 \text{ kg}

Mass is an intrinsic property of the object that remains constant regardless of location.

2

Calculate the weight of the crate in the low-gravity habitat.

W = 140 \text{ kg} \times 3.5 \text{ m/s}^2 = 490 \text{ N}

Weight is the force exerted by gravity on a mass, which changes depending on the local gravitational acceleration.

Key Concept

Mass is invariant, while weight depends on the local gravitational field.

Question 227Question →

A compact utility tractor applies a constant net forward force of $1,200 \text{ N}$ to push a stalled passenger vehicle on a level, paved road. If the vehicle accelerates at a rate of $1.5 \text{ m/s}^2$ , what is the mass of the vehicle?

Show answer & explanation

Answer: 800

Answer

800 kg

According to Newton's Second Law (

F=ma

), the mass of an object is equal to the net force acting upon it divided by its acceleration. By dividing the applied net force of

1,200 \text{ N}

by the acceleration of

1.5 \text{ m/s}^2

, we determine the mass of the vehicle is

800 \text{ kg}

.

Step-by-Step Solution

1

Identify the known variables provided in the problem.

Net Force (

F

) =

1,200 \text{ N}

, Acceleration (

a

) =

1.5 \text{ m/s}^2

.

Extracting the given information is the first step to determine which physics formula to apply.

2

State the formula for Newton's Second Law of Motion.

F = ma

This formula establishes the mathematical relationship between net force, mass, and acceleration.

3

Rearrange the formula to solve for the unknown variable, mass (

m

).

m = \frac{F}{a}

To find the mass, it must be isolated algebraically on one side of the equation.

4

Substitute the known values into the rearranged formula and compute the result.

m = \frac{1,200}{1.5} = 800

Executing the division calculation provides the final mass in kilograms.

Key Concept

Newton's Second Law of Motion (F=ma)

Question 228Question →

A $2,000 \text{ kg}$ pickup truck is coasting along a level road at a velocity of $15 \text{ m/s}$ . What is the magnitude of the truck's momentum?

Show answer & explanation

Answer: 30000

Answer

30000

Momentum is directly proportional to both mass and velocity. By multiplying the

2,000 \text{ kg}

mass by the

15 \text{ m/s}

velocity, we determine the total momentum is

30,000 \text{ kg} \cdot \text{m/s}

.

Step-by-Step Solution

1

Identify the known variables from the text.

Mass (

m

) is

2,000 \text{ kg}

and velocity (

v

) is

15 \text{ m/s}

.

Momentum calculation requires both the mass and the velocity of the object.

2

Apply the formula for momentum.

p = m \times v

The linear momentum of an object is defined as the product of its mass and its velocity.

3

Calculate the final value.

p = 2,000 \times 15 = 30,000

Multiplying the mass by the velocity yields the total magnitude of momentum.

Key Concept

Momentum is the product of an object's mass and its velocity (

p = mv

).

Question 229Question →

A $120$ -pound crate rests on a sturdy $12$ -foot plank spanning two sawhorses placed at opposite ends. The center of the crate is located $3$ feet from the left sawhorse. Assuming the weight of the plank is negligible, what is the downward force, in pounds, exerted by the crate on the left sawhorse?

Show answer & explanation

Answer: 90

Answer

The downward force exerted on the left sawhorse is

90

pounds.

Because the crate is positioned

3

feet from the left sawhorse and

9

feet from the right sawhorse on a

12

-foot span, the left sawhorse must support

\frac{9}{12}

(or

75\%

) of the crate's weight. Multiplying

120

pounds by

0.75

yields exactly

90

pounds.

Step-by-Step Solution

1

Calculate the distance from the crate to the right sawhorse.

12 - 3 = 9

feet.

The weight supported by one end of a beam is proportional to the distance from the load to the opposite end.

2

Determine the fraction of the total weight carried by the left sawhorse.

\frac{9}{12}

, which simplifies to

\frac{3}{4}

or

0.75

.

This fraction represents the ratio of the opposite distance to the total span, dictating how much load is distributed to the left support.

3

Calculate the downward force on the left sawhorse.

120 \times 0.75 = 90

pounds.

Multiplying the total crate weight by the calculated proportion gives the specific load carried by that support.

Key Concept

Load Distribution and Balancing

Estimated Time:45s

Question 230Question →

A barber chair operates using a closed hydraulic lifting system to raise a customer. The input pump piston has a cross-sectional area of $4 \text{ in}^2$ , and the output lifting piston has a cross-sectional area of $60 \text{ in}^2$ . If the barber pushes the input piston down a distance of $7.5 \text{ inches}$ , how many inches does the output piston move upward?

Show answer & explanation

Answer: 0.5

Answer

0.5

In a hydraulic system, the mechanical advantage that multiplies the output force must correspondingly divide the output distance. The output piston has an area 15 times larger than the input piston (

60 / 4 = 15

). Therefore, while it can lift 15 times the force, it will only travel

1/15\text{th}

of the input distance (

7.5 / 15 = 0.5 \text{ inches}

).

Step-by-Step Solution

1

Identify the relevant principle for hydraulic piston travel distances.

The volume of fluid displaced by the input piston equals the volume gained by the output piston:

A_1 \times d_1 = A_2 \times d_2

.

Because hydraulic fluid is essentially incompressible, the total volume in the closed system remains constant.

2

Set up the equation using the given values.

4 \times 7.5 = 60 \times d_2

Input area (

A_1

) is 4, input distance (

d_1

) is 7.5, and output area (

A_2

) is 60.

3

Calculate the total fluid volume displaced.

4 \times 7.5 = 30 \text{ in}^3

Multiplying area by distance gives the volume of fluid moved.

4

Solve for the output piston's upward distance (

d_2

).

30 / 60 = 0.5 \text{ inches}

Dividing the displaced volume by the output piston's larger area yields the smaller travel distance.

Key Concept

Conservation of volume in hydraulic systems dictates that distance and area are inversely proportional (Mechanical Advantage trades distance for force).

Question 231Question →

Inside a wind-up toy, a main spring rotates a driving gear with 60 teeth at a speed of 10 RPM. This gear directly meshes with a smaller driven gear that powers the toy's legs. If the legs require the driven gear to spin at 40 RPM to function properly, how many teeth must the driven gear have?

Show answer & explanation

Answer: 15

Answer

15 teeth

The correct answer is derived using the principle that the product of the number of teeth and rotational speed must be equal for two meshed gears (

60 \times 10 = 15 \times 40

). This confirms the driven gear must have 15 teeth to rotate at 40 RPM.

Step-by-Step Solution

1

Identify the relationship between gear teeth and rotational speed.

The governing formula is

T_1 \times S_1 = T_2 \times S_2

, where T represents the number of teeth and S represents the speed in RPM.

Gear speed is inversely proportional to the number of teeth.

2

Substitute the given numerical values into the equation.

60 \times 10 = T_2 \times 40

Applying the known driving gear parameters (60 teeth, 10 RPM) and the target speed for the driven gear (40 RPM).

3

Calculate the unknown number of teeth for the driven gear.

600 = T_2 \times 40

, which gives

T_2 = 15

Isolating the variable

T_2

yields the required tooth count for the mechanism to function correctly.

Key Concept

Gear Ratio and Speed Calculations

Question 232Question →

A sealed industrial holding tank contains water with a pocket of compressed air trapped above the water surface. The total depth of the water in the tank is $8.0\text{ m}$ . A pressure gauge located at the absolute bottom of the tank reads a gauge pressure of $130\text{ kPa}$ . A secondary discharge valve is located on the side of the tank, exactly $5.0\text{ m}$ above the tank's bottom. Assuming the density of water is $1,000\text{ kg/m}^3$ and the acceleration due to gravity is $g = 10\text{ m/s}^2$ , what is the gauge pressure, in kilopascals (kPa), at the secondary discharge valve?

Show answer & explanation

Answer: 80

Answer

80

The pressure at any point in a static fluid is equal to the pressure at a lower point minus the hydrostatic pressure of the fluid column between them. Since the valve is

5.0\text{ m}

above the bottom, the pressure there is less than the bottom pressure by

\rho g \Delta h = 1,000 \times 10 \times 5.0 = 50\text{ kPa}

. Subtracting this from the bottom pressure (

130\text{ kPa}

) gives the correct pressure of

80\text{ kPa}

. Alternatively, calculating the trapped air pressure (

130 - 80 = 50\text{ kPa}

) and adding the

3.0\text{ m}

water column above the valve (

30\text{ kPa}

) yields the same correct result.

Step-by-Step Solution

1

Determine the vertical distance between the pressure gauge at the bottom and the secondary valve.

\Delta h = 5.0\text{ m}

Hydrostatic pressure changes linearly with vertical height within a continuous fluid.

2

Calculate the hydrostatic pressure difference exerted by this

5.0\text{ m}

column of water.

\Delta P = \rho g \Delta h = 1,000 \times 10 \times 5.0 = 50,000\text{ Pa} = 50\text{ kPa}

This represents the amount of pressure lost as you move upward from the bottom to the valve.

3

Subtract the pressure difference from the gauge pressure at the bottom of the tank.

130\text{ kPa} - 50\text{ kPa} = 80\text{ kPa}

The pressure at the valve must be lower than the pressure at the bottom by exactly the weight of the water column between them.

Key Concept

Hydrostatic pressure varies directly with depth. The pressure difference between any two points in a continuous static fluid depends only on the vertical distance between them, regardless of surface air pressure.

Question 233Question →

Calculate the exact value of

n

that balances the following equation:

\frac{2}{3}(6n - 9) + 4 = 3(n - 2) - n

Show answer & explanation

Answer: -2

Answer

The correct value for n is -2.

The solution is found by correctly applying the distributive property to both sides of the equation, combining like terms to simplify the expression to

4n - 2 = 2n - 6

, and then isolating the variable

n

using inverse operations to get

n = -2

.

Step-by-Step Solution

1

Distribute the fraction on the left side of the equation.

4n - 6 + 4 = 3(n - 2) - n

To eliminate the parentheses on the left side and isolate the individual terms.

2

Distribute the coefficient on the right side of the equation.

4n - 6 + 4 = 3n - 6 - n

To expand the expressions before combining them.

3

Combine like terms on both sides.

4n - 2 = 2n - 6

To simplify the equation into the standard linear form ax + b = cx + d.

4

Group the variable terms on one side and the constants on the other.

2n = -4

By subtracting 2n and adding 2 to both sides, we isolate the variable term.

5

Divide by the coefficient of n.

n = -2

To solve for the isolated variable n.

Key Concept

Solving multi-step linear equations involving fractions, distribution, and combining like terms.

Question 234Question →

Calculate the exact value of the algebraic expression $3x^2y - \frac{12y}{x}$ when the variables are assigned the values $x = -2$ and $y = -5$ .

Show answer & explanation

Answer: -90

Answer

The correct evaluated result is -90.

By accurately substituting the values, correctly squaring the negative number

(-2)^2 = 4

, evaluating the fractional term as

\frac{-60}{-2} = 30

, and completing the final subtraction

-60 - 30

, the expression correctly evaluates to -90.

Step-by-Step Solution

1

Substitute the given variable values into the algebraic expression.

3(-2)^2(-5) - \frac{12(-5)}{-2}

This sets up the expression with numerical values for evaluation.

2

Evaluate the exponent applied to the negative base.

(-2)^2 = 4

, making the expression

3(4)(-5) - \frac{12(-5)}{-2}

According to the order of operations (PEMDAS), exponents must be evaluated before multiplication and division.

3

Perform the multiplication in the first term.

3(4)(-5) = 12(-5) = -60

Evaluating the first term completely from left to right.

4

Perform the multiplication and division in the second term (the fraction).

\frac{12(-5)}{-2} = \frac{-60}{-2} = 30

Evaluating the fractional term by first multiplying the numerator, then dividing by the negative denominator.

5

Subtract the result of the fraction from the first term.

-60 - 30 = -90

The final operation is the subtraction of the two evaluated terms.

Key Concept

Evaluating algebraic expressions using substitution and the order of operations with signed numbers.

Question 235Question →

Determine the value of $h$ that satisfies the equation below:

5 - 2(3h - 4) = 6h - 4(h + 4) - 3

Show answer & explanation

Answer: 4

Answer

The correct value is 4.

The equation is solved by first distributing the coefficients across the parentheses, which yields

5 - 6h + 8 = 6h - 4h - 16 - 3

. Combining like terms on both sides simplifies the equation to

13 - 6h = 2h - 19

. Adding

6h

and

19

to both sides isolates the variable, resulting in

32 = 8h

. Dividing by

8

gives the final answer of

4

.

Step-by-Step Solution

1

Apply the distributive property.

5 - 6h + 8 = 6h - 4h - 16 - 3

Multiply the terms inside the parentheses by their outside coefficients, carefully distributing the negative signs.

2

Combine like terms on the left side.

13 - 6h = 6h - 4h - 16 - 3

Combine the constant terms

5

and

8

.

3

Combine like terms on the right side.

13 - 6h = 2h - 19

Combine the variable terms (

6h - 4h = 2h

) and the constant terms (

-16 - 3 = -19

).

4

Isolate the variable

h

.

13 + 19 = 2h + 6h

Add

6h

to both sides and add

19

to both sides to group the variables and constants.

5

Solve for the final value.

32 = 8h \rightarrow h = 4

Simplify the addition and divide by

8

to find

h

.

Key Concept

Solving One-Variable Linear Equations

Estimated Time:1m 30s

Question 236Question →

Evaluate the algebraic expression $\frac{3a^2b - 4bc}{a - c}$ given that $a = -2$ , $b = 3$ , and $c = -5$ . What is the final numerical value?

Show answer & explanation

Answer: 32

Answer

The numerical value of the expression is 32.

After properly substituting the given values, evaluating the exponents, and following the order of operations, the numerator calculates to 96 and the denominator simplifies to 3. The final division of 96 by 3 yields 32.

Step-by-Step Solution

1

Substitute the assigned values of

a

,

b

, and

c

into the given algebraic expression.

\frac{3(-2)^2(3) - 4(3)(-5)}{-2 - (-5)}

Direct variable substitution is required to compute the numerical value.

2

Evaluate the exponent applied to the negative base.

(-2)^2 = 4

According to the order of operations (PEMDAS), exponents must be resolved prior to multiplication.

3

Carry out the multiplication operations in the numerator.

3(4)(3) = 36

and

4(3)(-5) = -60

Multiplication is processed from left to right within the grouping of the numerator.

4

Complete the subtraction operation in the numerator.

36 - (-60) = 36 + 60 = 96

Subtracting a negative quantity is algebraically identical to adding its positive counterpart.

5

Evaluate the subtraction operation in the denominator.

-2 - (-5) = -2 + 5 = 3

The entire denominator acts as a single grouping and must be fully simplified before division.

6

Divide the evaluated numerator by the evaluated denominator.

\frac{96}{3} = 32

This final arithmetic step produces the precise evaluation of the fractional expression.

Key Concept

Evaluating complex rational algebraic expressions using substitution and the order of operations with signed integers.

Question 237Question →

Consider the following algebraic equation:

5(2m - 1) - 3m = 2(m + 10)

What is the value of the variable $m$ ?

Show answer & explanation

Answer: 5

Answer

5

By correctly applying the distributive property on both sides of the equation, combining the resulting like terms, and applying algebraic operations to isolate the variable, the equation simplifies to

5m = 25

, which yields exactly

5

.

Step-by-Step Solution

1

Distribute the constants outside the parentheses into the terms inside.

10m - 5 - 3m = 2m + 20

This removes the parentheses and allows us to combine like terms.

2

Combine the like variable terms on the left side (

10m

and

-3m

).

7m - 5 = 2m + 20

Simplifying each side independently makes the equation easier to solve.

3

Subtract

2m

from both sides and add

5

to both sides.

5m = 25

This groups all variable terms on the left and all constant terms on the right.

4

Divide both sides by

5

.

m = 5

This isolates the variable

m

to give the final solution.

Key Concept

Solving one-variable linear equations requiring distribution and combining like terms

Estimated Time:1m 0s

Question 238Question →

Multiply the binomial $(3x - 4)$ by the trinomial $(5x^2 - 2x - 7)$ . After fully expanding the product and combining all like terms, what is the numerical coefficient of the $x$ term?

Show answer & explanation

Answer: -13

Answer

The coefficient of the x term is -13.

The coefficient of the

x

term is found by locating all the

x

terms in the expanded expression (

-21x

and

8x

) and adding their coefficients together (

-21 + 8 = -13

).

Step-by-Step Solution

1

Distribute the term

3x

to every term in the trinomial

(5x^2 - 2x - 7)

.

3x(5x^2) + 3x(-2x) + 3x(-7) = 15x^3 - 6x^2 - 21x

The distributive property requires multiplying each term of the first polynomial by each term of the second polynomial.

2

Distribute the term

-4

to every term in the trinomial

(5x^2 - 2x - 7)

.

-4(5x^2) - 4(-2x) - 4(-7) = -20x^2 + 8x + 28

This completes the distribution for the second term of the binomial, taking care to properly multiply the negative signs.

3

Write out the fully expanded polynomial before combining like terms.

15x^3 - 6x^2 - 21x - 20x^2 + 8x + 28

Listing all terms together makes it easier to identify which terms share the same variables and exponents.

4

Identify and combine the like terms that contain exactly the variable

x

to the first power.

-21x + 8x = -13x

The question specifically asks for the coefficient of the

x

term, so we combine

-21x

and

8x

.

Key Concept

Multiplying polynomials and combining like terms

Question 239Question →

A deck winch on a naval vessel applies a constant net horizontal force of $4,500 \text{ N}$ to drag a heavy supply pallet across a level deck. If the pallet has a mass of $1,500 \text{ kg}$ , what is the acceleration of the pallet?

Show answer & explanation

Answer: 3

Answer

The correct answer is 3.

By applying Newton's Second Law (

F = ma

) and rearranging for acceleration (

a = \frac{F}{m}

), we divide the net force of

4,500 \text{ N}

by the mass of

1,500 \text{ kg}

to find an acceleration of

3 \text{ m/s}^2

.

Step-by-Step Solution

1

Identify the given information

Force (

F

) =

4,500 \text{ N}

, Mass (

m

) =

1,500 \text{ kg}

Extracting the known variables is the first step in solving a physics calculation problem.

2

Select the appropriate formula

F = ma

Newton's Second Law relates force, mass, and acceleration.

3

Rearrange the formula to solve for acceleration

a = \frac{F}{m}

We need to isolate the unknown variable, acceleration (

a

).

4

Substitute the values and calculate

a = \frac{4500}{1500} = 3 \text{ m/s}^2

Plugging in the given numbers yields the final numerical answer.

Key Concept

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass (

F = ma

).

Question 240Question →

A high-speed rail train is traveling at an initial velocity of $216 \text{ km/h}$ . To safely navigate a sharp curve, the train must reduce its velocity to $108 \text{ km/h}$ . If the train applies its braking system to produce a constant deceleration of $1.5 \text{ m/s}^2$ , how many seconds will it take to reach the required speed?

Show answer & explanation

Answer: 20

Answer

It will take 20 seconds for the train to reach the required speed.

The correct answer is found by first converting both velocities to meters per second to match the acceleration's units.

216 \text{ km/h}

is equivalent to

60 \text{ m/s}

, and

108 \text{ km/h}

is equivalent to

30 \text{ m/s}

. The change in velocity is

-30 \text{ m/s}

. Using the formula

t = \Delta v / a

, dividing

-30 \text{ m/s}

by the deceleration of

-1.5 \text{ m/s}^2

results in exactly

20 \text{ seconds}

.

Step-by-Step Solution

1

Convert the initial velocity (

v_i

) from km/h to m/s.

216 \text{ km/h} \div 3.6 = 60 \text{ m/s}

Standardizing units to match the acceleration given in m/s^2 is required before calculation.

2

Convert the final velocity (

v_f

) from km/h to m/s.

108 \text{ km/h} \div 3.6 = 30 \text{ m/s}

All velocity variables must be in meters per second to be compatible with the acceleration.

3

Calculate the change in velocity (

\Delta v

).

\Delta v = 30 \text{ m/s} - 60 \text{ m/s} = -30 \text{ m/s}

Acceleration is the rate of change of velocity, so finding the total change is the necessary intermediate step.

4

Apply the acceleration formula

a = \frac{\Delta v}{t}

to solve for time (

t

).

t = \frac{-30 \text{ m/s}}{-1.5 \text{ m/s}^2} = 20 \text{ seconds}

Dividing the total change in velocity by the rate of deceleration yields the elapsed time.

Key Concept

Kinematics: Calculating time from constant acceleration and velocity change while managing unit conversions.

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept