The length of a rectangle is five less than twice the width. What values of the width will make the area less than 150 feet?

Question

Inequalities. Define variable, then write and solve a quadratic inequality.

Answer 1

Draw a rectangle. Label the short size as "w". At this point we don't actually know for sure if that is the short side or the long side. Just because the question says "w" is the width, do not assume that the width is shorter than the length. We will find out if it is when we solve the problem.

Now label the other side "2x-5" because the question tells is that the length of the rectangle is twice the width, "w" less 5. We have to assume that this means 5 feet because we are told that the area must be less than 150 square feet.

Now remember that the formula for the area of a rectangle is the product of the width "w" times the length "2w-5". Using this information we can write the following inequality to describe the problem in mathematical terms.

w*(2w-5) < 150

Before we solve this, just think about what it says for a minute. We want to know width and length in a particular relationship that produces an area less than 150 square feet. You should realize quickly that many numbers will produce that result; but we don't want or need to know all of them. We only need to know the largest one of them that does that. To find that number we need to solve an equation, not an inequality. Specifically we need to solve the equality

w*(2w-5) = 150

To do this multiply the left side and subtract 150 from each side to get

2w² -5w -150 = 150 - 150

2w² -5w -150 = 0

This is a "quadratic" equation because its highest power term is 2, a square. We can solve it in two ways: by factoring directly if we see that it is easily factorable; or by "completing the square" if it is difficult to factor easily. I'll do both so you will know how to solve this problem when the "coefficients," i.e. the numbers in the equation are"nice", i.e. easy to manipulate, and when they are not.

Staring at the equation for a few seconds or minutes, you might see that we need to find two numbers whose product is 75, half of 150, because we must multiply one of them by 2 when we multiply the factors of the equation. If you recognize that 15 and 10 will work, then you can write

2w² -5w -150 = (2w +15)*(w-10) = 0

Form here we need to find the values that make each factor zero because only those values satisfy the equation. They are

2w + 15 = 0

2w = -15

w = -7.5

and

w - 10 = 0

w = 10

While these are both solutions to our quadratic equation, only one of them makes sense in terms of the question because the question asks us to find dimensions of a rectangle and that requires positive numbers both "w" and 2w - 5. So if w = 10, then

2*w - 5 = 2*10-5 = 20 - 5 = 15

So the width for an area equal to 150 square feet is 10' and the length is 15'.

Returning to our inequality, we see, however, that we were asked to find a width that will make the total area LESS THAN 150 SQUARE FEET! But

10' * 15' = 150 square feet

No worries. All you need to do now is to point out in plain English that any width less than 10 feet satisfies the inequality. Don't try to find any one number because there are an infinite number of them. Just write

w < 10 feet

Now if this equation had not factored so easily we would have solved it by "completing the square" as follows. I am starting this solution with the equality.)

2w² - 5w - 150 = 0

2w² - 5w = 150

Divide both sides by 2 to get

w2 - (5/2)w = 75

Here's the tricky part. We need to find the number equal to half the coefficient of "w" which is now the fraction 5/2. That's easy.

(1/2)*(5/2) = (5/4)

Now we can rewrite the left side of the equation as

[w - (5/4)]² - (5/4)² = 75

I know this might look confusing and it is definitely harder than the first solution I already gave to this problem but it is important because, unlike the first solution that only works when the numbers are "nice", this one always works - even if there are no real solutions. All we've done here is to put the equation into a form that allows us to write out a solution quickly as follows. Subtract 75 from both sides to set the left side equal to zero again.

[w - (5/4)]² - (5/4)² - 75 = 0

Now combine the numbers into a single fraction.

[w - (5/4)]² - (25/16) - (16*75)/16 = [w - (5/4)]² - (1225/16)

Notice that we now have an equation of the form a² - b² where a = w - (5/4) and b = √(1225/16) (the square root of (1225/16). Now just remember that a² - b² = (a + b)*(a - b) and we are almost done. Substitute the value I've just defined for a (w - (5/4)) and b (√(1225/16)) into that formula and we get

[(w - (5/4)) + √(1225/16)]*[(w - (5/4)) - √(1225/16)] = 0

Now use your calculator to find the √1225 = 35 and you probably know √16 = 4 so now we just need to find the two values for "w" that set the factors equal to zero.

w - (5/4) + (35/4) = 0

w + (30/4) = 0

w = -(30/4) = -7.5

w - (5/4) - (35/4) = 0

w - (40/4) = 0

w = (40/4) = 10

Of course these are the same answers we found above. So we are done. For extra credit find the values for "w" that make the width of this rectangle longer than its length and also the smallest value that "w" can be. Explain why it cannot be smaller than that.

Answer 2

First, define your variable. To keep things simple, let w = the width of the rectangle, L = the length of the rectangle, and A = the area of the rectangle.

Based on these definitions, L (the length of the rectangle) can be expressed by L = 2w - 5 (five less than twice the width).

The formula for the area of a rectangle is A = L * w. Substitute in your new expression for L. You should get

A = (2w - 5) * w.

You want an area that is less than 50 ft. So, A < 150.

Plug in your expression for A and solve for the inequality.

(2w - 5) * w < 150

2w² - 5w < 150

2w² - 5w - 150 < 0

2w² - 5w - 150 = 0

(w - 10)(2w + 15) = 0 --> factor the expression

w - 10 = 0 2w + 15 = 0

w = 10 w = -15/2

Remember, the problem asks for the area of a rectangle. You can't have a negative width, so that eliminates answers involving w = -15/2.

Test values before and after w = 10

If w = 5

A = (2w - 5) * w = (2*5 - 5) * 5 = (10 - 5) * 5 = 5 * 5 = 25 < 150

If w = 20

A = (2w - 5) * w = (2*20 - 5) * 20 = (40 - 5) * 20 = 35 * 20 = 700 > 150

Thus, w < 10.

Alternatively, you can rationalize this by considering the relationship between the area and the width. The formula for the area of a rectangle is A = L * w. The area is directly proportional to the width, meaning that as the width increases, the area increases as well. Since you want the area to be less than 150, w cannot exceed 10.

The width is limited by the length, since the length cannot be less than zero. As a result, we get the inequality 2w - 5 > 0. When we solve this inequality, we get that w > 5/2.

Therefore, given the conditions in this problem, the 5/2 ft < w < 10 ft (don't forget units!) for the area of the rectangle to be less than 150 sq ft.