A bookcase is to be constructed as shown in the figure below. The height of the bookcase is 4 feet longer than the length of a shelf. If 20 feet of lumber is available for the entire unit (including the shelves, but NOT the back of the bookcase), find the length and height of the unit.
There is no "figure below". How many shelves are there?
To find the length and height of the bookcase, we can start by setting up equations based on the given information.
Let's assume the length of a shelf is x feet.
According to the problem, the height of the bookcase is 4 feet longer than the length of a shelf. So, the height of the bookcase is (x + 4) feet.
We also know that 20 feet of lumber is available for the entire unit, including the shelves. The bookcase consists of two shelves, so each shelf will use (x + x) = 2x feet of lumber. Additionally, we need to consider the height of the bookcase, which will require (x + 4) feet of lumber.
Therefore, our equation becomes: 2x + (x + 4) = 20.
Simplifying this equation, we get: 3x + 4 = 20.
Subtracting 4 from both sides of the equation, we have: 3x = 16.
Next, divide both sides of the equation by 3: x = 16/3 = 5.33 feet (rounded to two decimal places).
So, the length of a shelf is approximately 5.33 feet.
Now, to find the length and height of the unit, we substitute the length of a shelf (x) into the equations we set up earlier.
The length of the unit will be twice the length of a shelf: Length = 2x = 2 * 5.33 = 10.67 feet.
The height of the unit will be (x + 4) = 5.33 + 4 = 9.33 feet.
Therefore, the length of the unit is approximately 10.67 feet, and the height of the unit is approximately 9.33 feet.