A kitchen table has length that is two feet less than twice its width. If a table cloth with an area less that 26 square feet covers the table top, what is the length of the table?
l=2w-2
A=lw=(2w-2)(w)=2w^2-2w<26
2w^2-2w<26
2w^2-2w-26<0
-3.14005<w<4.14005, substitute in the first equation to find the value of l
-8.2801<l<6.2801
To determine the length of the table, we'll need to set up equations based on the given information and solve for the length.
Let's assume the width of the table is represented by the variable 'w'.
According to the given information, the length of the table is two feet less than twice its width. So, the length can be represented as (2w - 2).
We also know that the area of the table top covered by the table cloth is less than 26 square feet.
The area of a rectangle is given by the formula: Area = Length × Width.
Substituting the values into the formula, we have:
(2w - 2) × w < 26.
Now, we can solve the inequality to find the possible values for 'w'.
2w^2 - 2w < 26.
Rearranging the inequality, we have:
2w^2 - 2w - 26 < 0.
To solve this quadratic inequality, we'll factor or use the quadratic formula.
Let's factor the quadratic expression:
2w^2 - 2w - 26 = 0.
(w - 3)(2w + 8) < 0.
Now, we have two factors (w - 3) and (2w + 8). To determine the sign of the inequality, we need to evaluate both factors individually.
Setting each factor separately less than zero:
(w - 3) < 0,
and
(2w + 8) > 0.
From the first factor: (w - 3) < 0, we find that w < 3.
From the second factor: (2w + 8) > 0, we find that w > -4.
So, the values that satisfy both conditions are -4 < w < 3, which means the width 'w' must be between -4 and 3.
However, since we are dealing with a physical object (a kitchen table), the width cannot be negative. Thus, we only consider the positive value for 'w', which is w = 1.
Now, substituting the value of 'w' into the equation for length: (2w - 2), we find:
Length = 2 × 1 - 2 = 2 - 2 = 0.
Therefore, the length of the table is 0 feet.
However, it's important to note that a table with a length of 0 feet is not practical or physically possible. So, in this case, it might be necessary to reevaluate the given information or assumptions.