All practice questions

Question 1921Question →

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 1922Question →

If $x = -3$ , $y = 4$ , and $z = -2$ , what is the value of the expression $\frac{2x^2y}{z} + 5x$ ?

-87

-51

21

51

Show answer & explanation

Answer: -51

Answer

The evaluated result of the algebraic expression is -51.

The correct value is determined by carefully substituting the negative variables, properly applying the exponent to get a positive

9

, calculating the fraction

\frac{72}{-2}

to get

-36

, and adding the

-15

from the second term to reach

-51

.

Step-by-Step Solution

1

Substitute the given numerical values for each variable in the expression.

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

Variable substitution replaces letters with their specific numerical values to prepare for arithmetic calculation.

2

Evaluate the exponent according to the order of operations (PEMDAS).

(-3)^2 = 9

Exponents must be calculated before multiplication or division.

3

Perform the multiplication in the numerator of the fraction and in the second term.

Numerator:

2(9)(4) = 72

. Second term:

5(-3) = -15

. The expression is now

\frac{72}{-2} + (-15)

.

Multiplication is the next step in the order of operations.

4

Perform the division, then combine the remaining terms.

\frac{72}{-2} = -36

. Finally,

-36 + (-15) = -51

.

Division occurs before addition, completing the evaluation.

Key Concept

Evaluating Algebraic Expressions

Estimated Time:1m 0s

Question 1923Question →

Which of the following represents the fully expanded and simplified form of the polynomial expression $(2x - 3)(x^2 - 4x + 5)$ ?

2x^3 - 11x^2 + 22x - 15

2x^3 - 5x^2 - 2x - 15

2x^3 - 15

2x^3 + 11x - 15

Show answer & explanation

Answer: $2x^3 - 11x^2 + 22x - 15$

Answer

The simplified polynomial is

2x^3 - 11x^2 + 22x - 15

.

The correct answer is found by distributing each term in the binomial to each term in the trinomial and combining like terms. Distributing

2x

yields

2x^3 - 8x^2 + 10x

, and distributing

-3

yields

-3x^2 + 12x - 15

. Combining

-8x^2

with

-3x^2

gives

-11x^2

, and combining

10x

with

12x

gives

22x

, resulting in the final simplified expression

2x^3 - 11x^2 + 22x - 15

.

Step-by-Step Solution

1

Distribute the first term of the binomial,

2x

, to each term in the trinomial

(x^2 - 4x + 5)

.

2x(x^2) + 2x(-4x) + 2x(5) = 2x^3 - 8x^2 + 10x

To multiply polynomials, every term in the first polynomial must be multiplied by every term in the second.

2

Distribute the second term of the binomial,

-3

, to each term in the trinomial.

-3(x^2) - 3(-4x) - 3(5) = -3x^2 + 12x - 15

It is critical to carry the negative sign when distributing to ensure the signs of the resulting terms are correct.

3

Add the results from both distribution steps and group the like terms together.

2x^3 + (-8x^2 - 3x^2) + (10x + 12x) - 15

Grouping terms with the same variable and exponent makes it easier to simplify.

4

Combine the coefficients of the grouped like terms.

2x^3 - 11x^2 + 22x - 15

Combining like terms provides the expression in its fully simplified, standard polynomial form.

Key Concept

Multiplying Polynomials

Question 1924Question →

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 1925Question →

Expand the given product of a binomial and a trinomial, combine like terms, and determine the values of the missing coefficients.

Fill in the blanks below

When the expression

(3x - 5)(2x^2 + 4x - 7)

is fully expanded and simplified, the resulting polynomial in standard form is

6x^3 +

x^2 +

x + 35

.

Show answer & explanation

Answer

The coefficient for the x^2 term is 2, and the coefficient for the x term is -41.

By distributing each term of the binomial

(3x - 5)

to every term of the trinomial

(2x^2 + 4x - 7)

, we get six terms. Grouping and combining the like terms yields an

x^2

coefficient of

12 - 10 = 2

and an

x

coefficient of

-21 - 20 = -41

.

Step-by-Step Solution

1

Distribute the first term of the binomial,

3x

, to each term in the trinomial.

3x(2x^2) + 3x(4x) + 3x(-7) = 6x^3 + 12x^2 - 21x

To apply the distributive property for the first part of the polynomial multiplication.

2

Distribute the second term of the binomial,

-5

, to each term in the trinomial.

-5(2x^2) - 5(4x) - 5(-7) = -10x^2 - 20x + 35

To apply the distributive property for the second part, being careful to carry the negative sign.

3

Combine the results and group the like terms.

6x^3 + (12x^2 - 10x^2) + (-21x - 20x) + 35

To arrange the expression so that coefficients of terms with the same exponent can be simplified.

4

Simplify the grouped terms.

6x^3 + 2x^2 - 41x + 35

By adding the coefficients:

12 - 10 = 2

and

-21 - 20 = -41

.

Key Concept

Distributive Property with Polynomials

Alternative Method

You can use the 'box method' (a grid) by creating a 2x3 table with

3x

and

-5

on the rows, and

2x^2

,

4x

, and

-7

on the columns. Multiply the row and column headers to fill each cell, then add all the cells together along the diagonals to combine like terms.

Estimated Time:1m 30s

Question 1926Question →

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 1927Question →

Which of the following represents the fully simplified form of the expression $(3x - 4)^2 - x(2x + 5)$ ?

7x^2 - 19x + 16

7x^2 - 5x + 16

7x^2 - 29x - 16

7x^2 - 29x + 16

Show answer & explanation

Answer: $7x^2 - 29x + 16$

Answer

The correct simplified expression is

7x^2 - 29x + 16

.

Expanding the squared binomial yields

9x^2 - 24x + 16

. Distributing the

-x

yields

-2x^2 - 5x

. Combining the like terms gives

(9x^2 - 2x^2) + (-24x - 5x) + 16

, which simplifies perfectly to

7x^2 - 29x + 16

.

Step-by-Step Solution

1

Expand the squared binomial

(3x - 4)^2

.

(3x)^2 - 2(3x)(4) + 4^2 = 9x^2 - 24x + 16

Using the perfect square formula

(a - b)^2 = a^2 - 2ab + b^2

is required to properly multiply the binomial by itself.

2

Distribute the term

-x

across the binomial

(2x + 5)

.

-x(2x) - x(5) = -2x^2 - 5x

The distributive property must be applied, ensuring the negative sign is carried to both terms inside the parentheses.

3

Combine the results from the previous steps and combine like terms.

(9x^2 - 24x + 16) + (-2x^2 - 5x) = (9x^2 - 2x^2) + (-24x - 5x) + 16 = 7x^2 - 29x + 16

Like terms (those with the same variable raised to the same power) must be grouped and added together to find the fully simplified polynomial form.

Key Concept

Expanding perfect square binomials and distributing monomial terms correctly.

Question 1928Question →

A logistics specialist is organizing four replacement fuel hoses by length. The lengths are $\frac{17}{4}$ meters, $4 \frac{3}{8}$ meters, $\frac{14}{3}$ meters, and $4 \frac{5}{12}$ meters. Arrange these hose lengths in ascending order from shortest to longest.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct ascending order is

\frac{17}{4}

meters,

4 \frac{3}{8}

meters,

4 \frac{5}{12}

meters, and

\frac{14}{3}

meters.

By converting all values to mixed numbers with a common denominator of 24, the fractional parts become

\frac{6}{24}

,

\frac{9}{24}

,

\frac{10}{24}

, and

\frac{16}{24}

. Ordering these from smallest to largest provides the correct sequence.

Step-by-Step Solution

1

Convert the improper fractions to mixed numbers for easier comparison.

\frac{17}{4} = 4 \frac{1}{4}

and

\frac{14}{3} = 4 \frac{2}{3}

.

Comparing values is simpler when whole numbers and fractional parts are separated.

2

Identify the fractional parts to be compared.

Since all lengths share a whole number of 4, we only need to compare the fractions:

\frac{1}{4}

,

\frac{3}{8}

,

\frac{5}{12}

, and

\frac{2}{3}

.

The whole numbers are identical, so the overall size depends entirely on the fractional parts.

3

Find a common denominator for the fractions 4, 8, 12, and 3.

The least common multiple (LCM) of the denominators 4, 8, 12, and 3 is 24.

A common denominator is required to directly and accurately compare the sizes of the numerators.

4

Convert each fraction to an equivalent fraction with a denominator of 24.

\frac{1}{4} = \frac{6}{24}

;

\frac{3}{8} = \frac{9}{24}

;

\frac{5}{12} = \frac{10}{24}

;

\frac{2}{3} = \frac{16}{24}

.

This standardizes the fractions so their relative sizes are visually clear.

5

Order the fractions from smallest to largest based on their numerators.

\frac{6}{24} < \frac{9}{24} < \frac{10}{24} < \frac{16}{24}

, which corresponds to

\frac{17}{4} < 4 \frac{3}{8} < 4 \frac{5}{12} < \frac{14}{3}

.

Ordering the standardized fractions reveals the correct ascending sequence for the original lengths.

Key Concept

Comparing and ordering mixed numbers and improper fractions by finding a common denominator.

Question 1929Question →

What is the exact numerical value of the variable $p$ that balances the given algebraic equation?

Fill in the blanks below

-2(3p - 5) + 4p = 3(p + 5) - 20

Value:

Show answer & explanation

Answer

3

The correct answer is found by first distributing the outer values into the parentheses, taking special care to distribute the negative sign accurately so that

-2(-5) = 10

. After expanding, the equation becomes

-6p + 10 + 4p = 3p + 15 - 20

. Combining the like terms on each side reduces the equation to

-2p + 10 = 3p - 5

. Adding

2p

to both sides and adding

5

to both sides isolates the terms, resulting in

15 = 5p

. Finally, dividing both sides by 5 yields a value of 3.

Step-by-Step Solution

1

Distribute the constants into the parentheses on both sides of the equation.

-6p + 10 + 4p = 3p + 15 - 20

Expanding the grouped terms is necessary to access and combine all the variable and constant terms.

2

Combine the like terms on each side of the equation separately.

-2p + 10 = 3p - 5

Simplifying each side independently reduces the complexity before moving variables across the equals sign.

3

Isolate the variable by moving all

p

terms to one side and all constants to the opposite side.

10 + 5 = 3p + 2p

, which simplifies to

15 = 5p

Adding

2p

and adding

5

to both sides gathers the unknown variable exclusively to the right and constants to the left.

4

Divide by the coefficient of the variable to solve for

p

.

p = 3

Dividing both sides by 5 leaves the variable

p

completely isolated, providing the final solution.

Key Concept

Applying the distributive property and balancing operations to solve multi-step linear equations.

Question 1930Question →

Which of the following represents the correct evaluation of the mathematical expression below?

-1.5 - 2.4 \times (6.2 - 3.45)

-8.1

-10.725

-8.34

5.1

Show answer & explanation

Answer: -8.1

Answer

The correct answer is -8.1.

The correct evaluation requires following the exact order of operations. First, correctly align the decimals to subtract inside the parentheses (

6.20 - 3.45 = 2.75

). Second, multiply that result by

2.4

to get

6.6

. Finally, calculate

-1.5 - 6.6

, which correctly yields

-8.1

.

Step-by-Step Solution

1

Evaluate the expression inside the parentheses.

6.20 - 3.45 = 2.75

According to the order of operations (PEMDAS), parentheses must be resolved first. A zero is added to

6.2

to properly align the decimals.

2

Perform the multiplication.

-2.4 \times 2.75 = -6.6

Multiplication takes precedence over addition and subtraction. The product of a negative and a positive number is negative.

3

Perform the final subtraction.

-1.5 - 6.6 = -8.1

Combine the remaining terms. Subtracting a positive number from a negative number moves further left on the number line.

Key Concept

Order of Operations with Decimals

Question 1931Question →

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 1932Question →

Solve the following linear equation for $k$ :

3 - 2(4k - 5) = -5(k + 1)

What is the exact value of $k$ ?

0

-\frac{2}{3}

6

\frac{8}{3}

Show answer & explanation

Answer: $6$

Answer

The correct value is

6

.

The equation is correctly solved by first using the distributive property on both sides to get

3 - 8k + 10 = -5k - 5

. Combining the constants on the left side yields

13 - 8k = -5k - 5

. Adding

8k

to both sides gives

13 = 3k - 5

. Finally, adding

5

to both sides results in

18 = 3k

, and dividing by

3

gives the solution

k = 6

.

Step-by-Step Solution

1

Distribute the coefficients into the parentheses on both sides of the equation.

3 - 8k + 10 = -5k - 5

To eliminate the parentheses and expose all terms for simplification.

2

Combine the constant terms on the left side of the equation.

13 - 8k = -5k - 5

To simplify the left side.

3

Add

8k

to both sides of the equation.

13 = 3k - 5

To gather all terms with the variable

k

on the right side.

4

Add

5

to both sides of the equation.

18 = 3k

To isolate the variable term.

5

Divide both sides by

3

.

k = 6

To solve for

k

.

Key Concept

Solving one-variable linear equations by applying the distributive property and combining like terms.

Question 1933Question →

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 1934Question →

Arrange the following mathematical expressions in ascending order, from the least value to the greatest value.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from least to greatest is:

1.2 \times 0.6

, then

0.35 + 0.38

, then

4.2 - 3.45

, and finally

0.48 \div 0.6

.

Each arithmetic expression must be evaluated independently applying the correct decimal rules for that specific operation. The results are

0.72

,

0.73

,

0.75

, and

0.80

. Sorting these decimal values chronologically by magnitude yields the correct ascending sequence.

Step-by-Step Solution

1

Evaluate the multiplication expression:

1.2 \times 0.6

0.72

Multiply the whole numbers (

12 \times 6 = 72

) and count a total of two decimal places from the factors.

2

Evaluate the addition expression:

0.35 + 0.38

0.73

Align the decimal points vertically and add the hundredths, then the tenths.

3

Evaluate the subtraction expression:

4.2 - 3.45

0.75

Rewrite

4.2

as

4.20

to align the decimal points properly, then subtract.

4

Evaluate the division expression:

0.48 \div 0.6

0.80

Multiply both numbers by

10

to make the divisor a whole number (

4.8 \div 6

), then divide.

5

Order the evaluated results from least to greatest.

0.72 < 0.73 < 0.75 < 0.80

Comparing the values digit by digit from the tenths place to the hundredths place establishes the correct magnitude sequence.

Key Concept

Evaluating all four basic operations with decimals and comparing their relative magnitudes.

Question 1935Question →

A carpenter uses a standard screwdriver to drive a screw into a piece of wood. The screwdriver handle, acting as the wheel, has a radius of $2 \text{ cm}$ . The metal shaft, acting as the axle, has a radius of $4 \text{ mm}$ . What is the ideal mechanical advantage (IMA) of the screwdriver?

5

0.5

2

0.2

Show answer & explanation

Answer: 5

Answer

The ideal mechanical advantage is 5.

The correct answer is 5. To find the ideal mechanical advantage, both radii must first be in the same unit. Converting the

2 \text{ cm}

handle radius yields

20 \text{ mm}

. Dividing the wheel radius (

20 \text{ mm}

) by the axle radius (

4 \text{ mm}

) gives a mechanical advantage of 5, meaning the screwdriver multiplies the user's turning force by 5 times.

Step-by-Step Solution

1

Convert the given measurements to the same unit (millimeters).

The handle (wheel) radius is

2 \text{ cm}

, which equals

20 \text{ mm}

.

Units must match before calculating any ratios to ensure accuracy.

2

Identify the values for the wheel and the axle.

Wheel radius =

20 \text{ mm}

; Axle radius =

4 \text{ mm}

.

In a screwdriver, the wider handle functions as the wheel, while the narrower shaft functions as the axle.

3

Calculate the Ideal Mechanical Advantage (IMA) by dividing the wheel radius by the axle radius.

IMA = 20 / 4 = 5

.

The formula for the IMA of a wheel and axle system driven by the wheel is the radius of the wheel divided by the radius of the axle.

Key Concept

Ideal Mechanical Advantage (IMA) of a Wheel and Axle

Question 1936Question →

Match each gear arrangement with its resulting mechanical effect on speed, torque, or rotational direction.

Click a left item, then click its matching right item

Show answer & explanation

Answer

A small driver to a large driven gear decreases speed and increases torque. A large driver to a small driven gear increases speed and decreases torque. Two directly meshed gears rotate in opposite directions. An idler gear between two gears allows the input and output to rotate in the same direction.

Each physical gear arrangement corresponds to a single, consistent mechanical law: size differences alter the speed/torque ratio inversely, while the number of meshing points dictates rotational direction.

Step-by-Step Solution

1

Analyze the size relationship for the first two scenarios.

A smaller driver turning a larger gear trades speed for torque (gear reduction). A larger driver turning a smaller gear trades torque for speed (overdrive).

Understanding the fundamental trade-off between speed and torque is required to match size differences with output changes.

2

Analyze the rotational direction for direct mesh versus an idler train.

Direct mesh always reverses direction. Adding a single idler gear reverses it again, making the output match the input direction.

Counting the number of gears in a train determines the final direction of rotation.

Key Concept

Basic Gear Ratios and Rotational Direction

Estimated Time:1m 0s

Question 1937Question →

When constructing a building framework in an earthquake-prone region, engineers typically choose structural steel over cast iron. Which of the following best explains why structural steel is the superior building material for this specific application?

Structural steel is highly ductile, allowing it to bend and absorb seismic shock without suddenly fracturing.

Structural steel is extremely brittle, which ensures the framework remains perfectly rigid and will not bend during an earthquake.

Structural steel functions exclusively under compression, whereas cast iron is designed to handle the high tension caused by seismic waves.

Structural steel has significantly more mass than cast iron, anchoring the building more firmly to the ground using its own heavy weight.

Show answer & explanation

Answer: Structural steel is highly ductile, allowing it to bend and absorb seismic shock without suddenly fracturing.

Answer

Structural steel is highly ductile, allowing it to bend and absorb seismic shock without suddenly fracturing.

Structural steel possesses high ductility, which is a material's ability to undergo significant plastic deformation before rupture. During an earthquake, a building framework experiences extreme and shifting lateral loads. A ductile material like structural steel will bend and yield, safely absorbing the seismic energy without experiencing a sudden, catastrophic fracture. Cast iron, by contrast, is brittle and would crack or shatter under the same dynamic, bending stress.

Step-by-Step Solution

1

Identify the primary forces exerted on a building during an earthquake.

Earthquakes introduce dynamic, unpredictable shocks and vibrations that cause structures to sway, creating rapidly alternating tension, compression, and shear forces.

Understanding the environmental stress is the first step in selecting the appropriate building material.

2

Compare the mechanical properties of structural steel and cast iron under dynamic stress.

Cast iron is hard but brittle, meaning it snaps or shatters under sudden, bending stress. Structural steel is ductile, meaning it can stretch and deform elastically and plastically before failing.

Matching material behavior to the required load type reveals which material will survive the event.

3

Determine which property prevents building collapse during seismic activity.

Ductility is the preferred property because the material absorbs the energy of the earthquake by flexing. A building framework that bends is significantly safer than one that snaps abruptly.

This confirms why structural steel is specifically chosen over brittle alternatives for seismic zones.

Key Concept

Ductility vs. Brittleness in Structural Materials

Estimated Time:45s

Question 1938Question →

A washing machine's spin cycle rotates wet clothes at high speeds inside a drum with small holes. This process effectively removes water from the fabric. Which of the following statements best explains the mechanical principles involved in this process?

The drum exerts an inward centripetal force on the clothes, while the water escapes through the holes by continuing in a straight line due to inertia.

The high-speed rotation generates an outward centripetal force that actively pulls the water toward the edge of the drum.

The circular motion creates an outward reaction force to balance the drum's spin, pushing the water away from the center.

The water has more mass than the fabric fibers, which causes it to experience a greater outward acceleration during the spin.

Show answer & explanation

Answer: The drum exerts an inward centripetal force on the clothes, while the water escapes through the holes by continuing in a straight line due to inertia.

Answer

The drum exerts an inward centripetal force on the clothes, while the water escapes through the holes by continuing in a straight line due to inertia.

For an object to move in a circle, there must be an inward force (centripetal force). In a washing machine, the solid walls of the drum provide this inward force on the clothes. When water droplets reach the holes, there is no wall to provide an inward force, so the water continues moving in a straight line (tangent to the drum) because of its inertia.

Step-by-Step Solution

1

Identify the actual forces acting on the objects inside the spinning drum.

The solid drum walls push inward on the clothes.

An inward net force (centripetal force) is required to make any object move in a circular path.

2

Determine what happens to the water when it encounters a hole in the drum.

The water is no longer pushed inward by the drum wall, so it moves in a straight line tangent to the circular path.

According to Newton's First Law (inertia), objects will continue moving in a straight line unless acted upon by a net force.

Key Concept

In circular motion, centripetal force always acts inward. Objects that lose this inward restraining force will travel in a straight tangential path due to inertia.

Question 1939Question →

Utility workers intentionally install overhead electrical wires with a slight sag between the poles. What is the primary mechanical reason the wires are left slack rather than being pulled perfectly tight?

To prevent the wires from contracting and snapping when the temperature drops in cold weather.

To provide room for the wires to contract and shorten as the outside temperatures increase.

To increase the surface area of the wire so it can cool down faster through convection.

To place the wooden utility poles entirely under compression rather than tension.

Show answer & explanation

Answer: To prevent the wires from contracting and snapping when the temperature drops in cold weather.

Answer

The wires are left slack to prevent them from contracting and snapping during cold weather.

The correct answer accurately applies the principle of thermal contraction. In cold winter weather, metal electrical wires shrink in length. If they were installed perfectly tight, this shrinkage would increase the tension beyond the wire's breaking point, causing it to snap. The sag provides the necessary extra length to safely absorb this contraction.

Step-by-Step Solution

1

Identify the physical principle affecting materials exposed to weather.

Metals undergo thermal expansion when heated and thermal contraction when cooled.

Electrical wires are made of metal and are exposed to seasonal temperature changes.

2

Analyze what happens to a tight wire when the temperature drops.

The cold causes the metal wire to contract (shorten). If it is already perfectly tight, shortening it will drastically increase the tension.

The fixed distance between the poles forces the contracting wire to bear extreme physical stress.

3

Determine the mechanical solution to this problem.

Leaving the wire with a sag (slack) provides extra length. When winter comes, the wire contracts and becomes tighter, but the provided slack prevents it from snapping.

Slack compensates for the natural thermal contraction of the metal.

Key Concept

Thermal Expansion and Contraction

Estimated Time:45s

Question 1940Question →

A heavy steel block must be moved along a level track. Rank the following methods or conditions based on the amount of frictional resistance they generate, from LEAST resistance to GREATEST resistance.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct order from least to greatest resistance is: floating on oil (fluid friction), moving on rollers (rolling friction), sliding continuously (kinetic friction), and overcoming the initial standstill (static friction).

Fluid friction separates the surfaces entirely, offering the least resistance. Rolling friction prevents surfaces from dragging, making it the second lowest. For the dry track, sliding (kinetic) friction is less than the static friction required to break the object's initial grip, making static friction the greatest resistance.

Step-by-Step Solution

1

Identify the primary type of friction acting in each scenario.

Oil film involves fluid friction; steel rollers involve rolling friction; continuous sliding involves kinetic friction; breaking free from a standstill involves static friction.

Categorizing the physical interaction is required to compare their relative magnitudes.

2

Determine the lowest sources of friction.

Fluid friction creates the least resistance, followed closely by rolling friction.

Fluids deform easily to allow movement and separate the surfaces entirely, while rollers prevent direct surface scraping.

3

Compare the resistance of a moving object versus a stationary object on the exact same dry surface.

Kinetic (sliding) friction is always less than the maximum static (standstill) friction.

Once the microscopic bonds between the resting surfaces are broken, it takes less force to keep the object moving than it did to start it.

4

Arrange the scenarios from least to greatest based on these physical principles.

The correct sequence is: Fluid, Rolling, Kinetic, Static.

This correctly orders the magnitudes of frictional resistance from lowest to highest.

Key Concept

Friction and Resistance

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