Algebra II

Question

A door in 7 feet high and 30 inches wide, with trim of w inches wide on three sides of the door frame. Write the total area (in square inches) of the door and trim in standard form.

Answer 1

Start with the area formula for a rectangle, A=BH, where B represents the base (or width) and H represents the height.

 

In this particular example, we will use b (distinct from B) for the door's width, and h (distinct from H) for it's height. 

 

But the rectangle we're interested in includes more than just the dimensions of the door.  We must make the assumption that if the door only has three sides with trim, they are the left, right, and top.  

 

On either side of the door we have a piece of trim of width w.  This adds two w's to the base rectangle that is the door itself.  This gives us the equation:

B = b + w + w

B = b + 2w

 

On the top of the door only we have a piece of trim that also has the width w.  This gives us the equation:

H = h + w

 

Starting with the area formula above, we substitute the above expressions in for B and H.

 

A = BH

A = (b + 2w)(h + w)

 

(Parentheses are necessary because we are replacing multiplied variables with expressions that contain addition or subtraction)

 

At this point you may wish to substitute in the specific values stated in the question.  Remember that the door's width (b) is 30 inches and it's height (h)  is 7 feet.  You might then get this result:

A = (30 + 2w)(7 + w)

 

But this is incorrect.  Why?  

 

The answer involves units of measurement of which we have two: inches and feet.  We must convert one to the other before we proceed.  since the question asks that the answer be in square inches, we must convert the 7 ft to inches by multiplying by 12.  

 

7ft = 7*12 in = 84 in.

 

Now our area formula becomes:

 

A = (30 + 2w)(84 + w)

 

Then to convert to "standard form" we must multiply the above out.  For the sake of completeness I do this below in detail.

 

A = (30 + 2w)(84 + w)

A = (30 + 2w)84 + (30 + 2w)w    (The distributive property.  (30 + 2w) is distributed to both terms of (84 + w).)

A = 2520 + 168w + 30w + 2w^2 (The distributive property.)

A = 2520 + 198w + 2w^2           (Simplification or "combining like terms")

A = 2w^2 + 198w + 2520

 

Therefore the area of the door is (2w^2 + 198w + 2520) in^2

Answer 2

Draw a diagram and label.

 

The height of the door and the trim is 7.25 feet, or 87 inches

The width of the door and the trim is 30.5 inches

 

The area of the door and frame is 87 · 30.5 in²