Algebra 2 word problem.

Question

A large rectangular movie screen in an IMAX theatre has an area of 9744 square feet. Find the dimensions of the screen if it is 25feet longer than it is wide

Answer 1

Hi Kayla M.,

IMAX movie screen rectangle.

Area = 9744 ft² ; (given)

Width = w ; (unknown)

Length = w + 25 ft

Area = length*width

9744 = (w + 25)*w

9744 = w² + 25w, (w multiplied through)

0 = w² + 25w - 9744, (subtract 9744 from both sides)

0 = (w +112)*(w -87), (factor)

0 = w + 112 and 0 = w - 87, (set both to zero)

w = -112 and w = 87, (solve for w, choose the positive width)

Width (w) equals 87

Length = w + 25 = 112

IMAX screen is 87 ft by 112 ft.

I hope this helps, Joe.

Answer 2

Rather than just showing you how to do this specific problem, I want to provide you a Word Problem Solving Strategy that works with most word problems, not just this one, but using this specific problem as an example of how the overall strategy works and is applied. So, you'll get both how to do this problem and how to do other word problems.

Always start word problems by identifying what is being asked, then read (more than once if necessary) for understanding in context of what is being asked.

In this case, they want the dimensions of a rectangular screen.

Next, identify the relevant information and relationships that are given.

In this case,

the screen is a rectangle

its area is 9744 square feet

the length is 25 feet longer than the width.

It is best to write it In equation form, noting "is" translates as "=" and "longer than" translates as "+":

L = 25ft. + W, where L is length and W is width.

A=9744 sq.ft. , where A is Area.

Notice that I use meaningful algebraic symbols, "L" and "W" and "A" for length, width and area respectively. This helps me keep track of which symbols mean what rather than using x, y, or z and having to remember what they mean. If I insist on using x, y and z, then I should write down what I am using them for:

x=Length

y=Width

z=Area

But let's stick with L, W, and A.

Also notice that I am including the units of measure in my equations. That will help me notice if different pieces of information are given in different units (such as inches and feet) so that I do not forget to convert everything into the same units.

In this case all units relate to feet.

Next: Draw an illustration (or chart or table, etc ..) that helps visualize the problem, and label it.

In this case:

Draw a rectangle labeling the width as W and the length as L = 25ft. + W

Inside the rectangle, label A = 9744 sq.ft.

Again, I include the units in my labeling as I did in the equations, for the same reason.

Now we identify relevant relationships that we know apply from past learning or resources not provided in the problem.

In this case, we know that "Area = Length times Width", but write:

A=LW

I don't use the "X" symbol for multiplication as a habit, in the event I use it for an algebraic symbol, which I did not in this case.

Next, we always like to have one equation with one unknown to solve for that one unknown, then use that answer to find everything else we want to know using the given relationships and the relationships we know from past experience or learning.

In this case, we know

A= 9744sq.ft. and L= 25ft. + W

so, we make that substitution

A=LW which can be written A=(L)(W)

so

9744sq.ft. = ( 25ft. + W)(W)

Next we solve for W, the width, and then use L=25ft.+W to find L, the length.

We notice that we cannot do anything with the equation as is, so we change it's form by expanding the multiplication. Since we have everything in terms of feet, we can drop the units for simplicity.

9744. = 25W. + W^2, where "^2" signifies the "2" being an exponent of W.

When solving a polynomial (all exponents being whole numbers) and an exponent of the variable is other than "1", the norm is to re-form the equation with a variable expression set equal to "0", and to put the polynomial in standard form (variable terms from highest exponent to lowest).

W^2 + 25W - 9744 = 0

Now we solve for W, only accepting positive numbers as answers (because the width cannot be a negative number).

We can factor, if possible, setting each factor by itself equal to "0",

OR

We can use the quadratic formula, writing with "W" instead of "x".

aW^2+bW+c=0 and

W={[-b] +/- [SQRT(b^2-4ac)]}/(2a)

where, in our equation

W^2 + 25W - 9744 = 0

a=1, b=25 and c=-9744

In our case, there are two possible solutions to the equation,

W=87 or W= -112

But since width, W, can only be positive in our problem,

W=87 and L=25+W=112

The last step is to write the answer in the context of the original question:

"The length of the screen is 112ft. and the Width of the screen is 87ft."

That's All Folks.

David Mazur.