We want to find how long the hallway is if we used equal amounts of white and black paint. To do so, we will start by calculating the area of each rectangle. Let's recall the formula for the area of a rectangle.
A=bhConsider that the base of each rectangle will be equal to 6 feet. The height for the black rectangles will be equal to x+1, and the height for the white ones will be x.
Since we have 4 black rectangles, we can multiply its area by 4. In the case of white rectangles, we need to multiply its area by 5.
Area black rectangle= 4(6(x+1))
Area white rectangle= 5(6x)
Because the same amount of paint was used, the area painted in white is equal to the area painted in black. This means that we can equate the expressions of the areas.
4(6(x+1))= 5(6x)
Now, we can use inverse operations to isolate the variable on one side of the equation.
Finally, we will find the length of the hallway L by adding the base of each rectangle.
L= 4(x+1) + 5x
We will substitute x= 4 into the given expressions and calculate the length. Let's do it!
We want to know if the same hallway can be painted with the same pattern by using twice as much black paint as white paint. To do so, consider that the area of the black rectangles will be multiplied by 2. Then, we will have a new equality for the areas.
4(6(x+1))= 2(5(6x))We will solve this equation by using inverse operations to isolate the variable on one side of the equation.
Now that we have the new value of x, we will check if the length of the hallway is 40 feet. To do so, we will substitute x= 23 into the expression for the length that we obtained in Part A.
