A rectangular storage container has a base width of $w$ meters. The length of the base is $3$ meters longer than twice its width, and the height of the container is $4$ meters shorter than its length. Use the distributive property to expand the expression for the total surface area $A(w)$ of the closed rectangular box, and determine the coefficient of the linear term $w$ in the resulting polynomial in standard form.

Final answer

$-22$

Steps

Define the dimensions in terms of $w$

To find the total surface area $A(w)$ and determine the coefficient of the linear term, we express all dimensions in terms of the width $w$ and apply the distributive property.

Width: $w$
Length: The length of the base is $3\text{ m}$ longer than twice its width: $l = 2w + 3$
Height: The height is $4\text{ m}$ shorter than its length: $h = l - 4 = (2w + 3) - 4 = 2w - 1$

Set up the total surface area formula

A closed rectangular box has six faces consisting of three congruent opposite pairs:

Top and bottom: $2 \times (l \cdot w)$
Front and back: $2 \times (l \cdot h)$
Left and right sides: $2 \times (w \cdot h)$

The total surface area function is: $A(w) = 2lw + 2lh + 2wh = 2(lw + lh + wh)$

Substitute the expressions for $l$ , $w$ , and $h$ : $A(w) = 2\left[w(2w + 3) + (2w + 3)(2w - 1) + w(2w - 1)\right]$

Expand each term using the distributive property

Expand each product inside the brackets:

Base and top area term: $w(2w + 3) = 2w^2 + 3w$
Front and back area term: $(2w + 3)(2w - 1) = 2w(2w - 1) + 3(2w - 1) = 4w^2 - 2w + 6w - 3 = 4w^2 + 4w - 3$
Side area term: $w(2w - 1) = 2w^2 - w$

Summing the terms inside the brackets and multiplying by $2$ across all three face pairs: $A(w) = 2(2w^2 + 3w) + 2(4w^2 + 4w - 3) + 2(2w^2 - w)$

Distribute $2$ to every term: $A(w) = (4w^2 + 6w) + (8w^2 + 8w - 6) + (4w^2 - 2w)$

Identify the coefficient of the linear term

Combining like terms into standard polynomial form $A(w) = aw^2 + bw + c$ , we isolate the linear terms: $\text{Linear component} = 6w + 8w - 2w$

Evaluating the expanded polynomial in standard form yields the linear term $-22w$ .

Thus, the coefficient of the linear term $w$ is $-22$ .

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student misinterprets the height condition as 4 meters shorter than the width rather than the length, using h = w - 4.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
l = 2w + 3 $\implies$ h = w - 4
Correct
l = 2w + 3 $\implies$ h = l - 4 = (2w + 3) - 4 = 2w - 1
The problem states that the height is $4\text{ m}$ shorter than its length ( $l$ ), not its width ( $w$ ). Substituting $l = 2w + 3$ gives $h = (2w + 3) - 4 = 2w - 1$ .
Step 2
A(w) = 2(lw + lh + wh) $\implies$ lw = w(2w + 3) = 2w^2 + 3w $\implies$ lh = (2w + 3)(w - 4) = 2w^2 - 5w - 12 $\implies$ wh = w(w - 4) = w^2 - 4w
Correct
A(w) = 2(lw + lh + wh) $\implies$ lw = 2w^2 + 3w $\implies$ lh = (2w + 3)(2w - 1) = 4w^2 + 4w - 3 $\implies$ wh = w(2w - 1) = 2w^2 - w
Your expansion method and distributive property steps were correct based on your expressions, but using $h = w - 4$ instead of $h = 2w - 1$ affected the face products $lh$ and $wh$ .
Step 3
lw + lh + wh = 5w^2 - 6w - 12 $\implies$ A(w) = 2(5w^2 - 6w - 12) = 10w^2 - 12w - 24
Correct
lw + lh + wh = 8w^2 + 6w - 3 $\implies$ A(w) = 2(8w^2 + 6w - 3) = 16w^2 + 12w - 6
Your addition of like terms and distribution of the factor of $2$ were mathematically sound, but carried forward the values from the misstated height.
Step 4
$\text{Coefficient of } w \text{ is } -12$
Correct
$\text{Coefficient of } w \text{ is } 12$
Identifying the linear coefficient from your expanded polynomial was accurate, but the correct polynomial $16w^2 + 12w - 6$ yields a linear coefficient of $12$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly translated the length of the base as $l = 2w + 3$ .

You correctly used the surface area formula for a closed rectangular prism, $A = 2(lw + lh + wh)$ .

You applied the distributive property accurately when expanding each binomial product.

You combined like terms and distributed the factor of $2$ across the polynomial without any arithmetic mistakes.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Variable Misassignment in Word Problem

You translated the height condition as $h = w - 4$ instead of setting $h = l - 4$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You anchored every dimension directly to the independent variable $w$ rather than chaining the height definition through the length $l$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Trace reference nouns carefully: When a dimension is described relative to another dimension (e.g., 'shorter than its length'), write the relation in terms of that specific dimension first before substituting the base variable.
Write $h = l - 4$ explicitly as an intermediate step, then substitute $l = 2w + 3$ to obtain $h = (2w + 3) - 4 = 2w - 1$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Express the height in terms of $w$ : $h = l - 4 = (2w + 3) - 4 = 2w - 1$ .
Expand the three face-pair areas: $lw = w(2w + 3) = 2w^2 + 3w$ , $lh = (2w + 3)(2w - 1) = 4w^2 + 4w - 3$ , and $wh = w(2w - 1) = 2w^2 - w$ .
Sum the three areas: $lw + lh + wh = (2w^2 + 4w^2 + 2w^2) + (3w + 4w - w) - 3 = 8w^2 + 6w - 3$ .
Multiply by $2$ to find the total surface area: $A(w) = 2(8w^2 + 6w - 3) = 16w^2 + 12w - 6$ .

Final answer

NEXT STEPS

Final Takeaways from Your Teacher

Review how to chain variable substitutions when one dimension depends on another.
Practice highlighting the reference noun in word problem comparison phrases before writing algebraic expressions.

OverviewACEpath Teacher's Diagnosis

02 The student calculates the surface area of only three distinct faces without multiplying by 2 for the closed container.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Partially correct

Step 1
l = 2w + 3 $\implies$ h = 2w - 1
Step 2
A(w) = lw + lh + wh
Correct
A(w) = 2(lw + lh + wh) = 2lw + 2lh + 2wh
A closed rectangular box has six faces made up of three congruent pairs (top/bottom, front/back, left/right). You omitted the factor of $2$ .
Step 3
lw = 2w^2 + 3w $\implies$ lh = (2w+3)(2w-1) = 4w^2 + 4w - 3 $\implies$ wh = 2w^2 - w
Step 4
A(w) = (2w^2 + 3w) + (4w^2 + 4w - 3) + (2w^2 - w) $\implies$ A(w) = 8w^2 + 6w - 3
Correct
A(w) = 2(8w^2 + 6w - 3) = 16w^2 + 12w - 6
Your combining of like terms was algebraically correct for your expression, but the terms lacked the doubling required for the full surface area.
Step 5
$\text{Coefficient of } w \text{ is } 6$
Correct
$\text{Coefficient of } w \text{ is } 12$
Because each face pair was not doubled, the linear coefficient is $6$ instead of $12$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly expressed the length and height in terms of the width $w$ .

You applied the distributive property correctly when multiplying the binomials.

You combined like terms accurately to identify the linear term of your polynomial.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Omission of Face Multiplicity

You wrote the surface area formula as $A(w) = lw + lh + wh$ , leaving out the factor of $2$ for the opposite faces of the closed box.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You treated the surface area of a closed rectangular prism as the sum of only three distinct faces rather than accounting for all six faces.

REASONING PATH

Your ACEpath teacher’s suggested approach

Face Count Verification: A closed rectangular prism always has $6$ faces: $2$ of area $lw$ , $2$ of area $lh$ , and $2$ of area $wh$ .
Always write the full formula $A = 2(lw + lh + wh)$ before substituting algebraic expressions.
Keyword Check: Closed vs. Open: The word 'closed' indicates top and bottom are both present.
Highlight whether the container is open-topped ( $5$ faces) or fully closed ( $6$ faces) before setting up your area formula.

CORRECTED PATH

ACEpath Teacher's Demonstration

Use the full six-face formula: $A(w) = 2(lw + lh + wh)$ .
Substitute your simplified sum of three faces: $A(w) = 2(8w^2 + 6w - 3)$ .
Distribute $2$ to each term: $A(w) = 16w^2 + 12w - 6$ .

Final answer

The coefficient of the linear term $w$ is $12$ .

NEXT STEPS

Final Takeaways from Your Teacher

Review the geometric definitions of total surface area versus lateral area for 3D prisms.
Practice checking whether a word problem specifies an open box, closed box, or specific subsets of faces.
Re-distribute constants across grouped expressions after summing multi-term components.

OverviewACEpath Teacher's Diagnosis

03 The student forgets to distribute the factor of 2 to all terms inside the parentheses, multiplying only the leading quadratic term.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
l = 2w + 3 ⟹ h = 2w - 1 ⟹ A(w) = 2(lw + lh + wh)
Step 2
lw = 2w^2 + 3w ⟹ lh = 4w^2 + 4w - 3 ⟹ wh = 2w^2 - w
Step 3
lw + lh + wh = 8w^2 + 6w - 3
Step 4
A(w) = 2(8w^2 + 6w - 3) = 16w^2 + 6w - 3
Correct
A(w) = 2(8w^2 + 6w - 3) = 16w^2 + 12w - 6
When applying the distributive property $a(b + c + d) = ab + ac + ad$ , you multiplied the factor $2$ by $8w^2$ , but you forgot to distribute $2$ to the linear term $6w$ and the constant term $-3$ .
Step 5
Coefficient of w is 6
Correct
$\text{Coefficient of } w \text{ is } 12$
Because of the incomplete distribution in the previous step, your extracted coefficient of $w$ was $6$ instead of $12$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly modeled the dimensions in terms of the variable $w$ .

You applied the distributive property accurately when expanding each two-dimensional face product.

You combined like terms correctly to find the sum of the three distinct faces.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Incomplete distribution across polynomial terms

You only multiplied the first term inside the parentheses by $2$ , leaving the remaining terms unchanged.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Forgetting that an external scalar multiplier must be distributed to every single term inside a multi-term grouping, not just the leading term.

REASONING PATH

Your ACEpath teacher’s suggested approach

Count all terms before multiplying: In the expansion $k(t_1 + t_2 + \dots + t_n) = k \cdot t_1 + k \cdot t_2 + \dots + k \cdot t_n$ , the resulting expression must have as many distributed products as there are terms inside the parentheses.
Draw distribution arrows or count terms inside the parentheses to ensure the external multiplier scales each individual term.

CORRECTED PATH

ACEpath Teacher's Demonstration

Apply the distributive property to all three terms: $A(w) = 2(8w^2) + 2(6w) + 2(-3)$
Multiply each term: $A(w) = 16w^2 + 12w - 6$
Identify the coefficient of the linear term $w$ from the standard form polynomial: $12$

Final answer

NEXT STEPS

Final Takeaways from Your Teacher

Always draw multiplication arcs or count terms when distributing a constant across a trinomial.
Check that the linear and constant terms change magnitude whenever multiplied by an external constant greater than 1.

Lock it in

Question

If length is $l = 3x + 1$ and height is ' $5$ units less than its length', how do you express height in terms of $x$ ?

Answer

$h = (3x + 1) - 5 = 3x - 4$ . Always substitute the entire expression for length into $h = l - 5$ .

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student misinterprets the height condition as 4 meters shorter than the width rather than the length, using h = w - 4.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

02 The student calculates the surface area of only three distinct faces without multiplying by 2 for the closed container.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

03 The student forgets to distribute the factor of 2 to all terms inside the parentheses, multiplying only the leading quadratic term.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

Lock it in

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