You have a rectangular space where you plan to create an obstacle course for an animal. The area of the rectangular space is represented by the expression 12x^2 - 15x. The width of the rectangular space is represented by the expression 3x
Part A: Write an expression to represent the length of the rectangular space. Then simplify your expression. Show all your work.
Part B: Prove that your answer in part A is correct by multiplying the length and the width of the rectangle. Show all your work.
Part A:
Expression: 12x^2 - 15x - 3x
Simplified Expression: 9x^2 - 15x
Part B:
Proof:
(9x^2 - 15x) * (3x)
= 27x^3 - 45x^2
= 12x^2 - 15x
Therefore, the expression 9x^2 - 15x is correct.
Part A:
To find the length of the rectangular space, we need to divide the area by the width.
Given:
Area of the rectangular space = 12x^2 - 15x
Width of the rectangular space = 3x
So, the expression for the length of the rectangular space would be:
Length = (Area)/(Width)
Length = (12x^2 - 15x)/(3x)
Now, let's simplify this expression:
Length = (12x^2)/(3x) - (15x)/(3x)
Length = 4x - 5
Therefore, the simplified expression for the length of the rectangular space is 4x - 5.
Part B:
To prove that the answer in Part A is correct, we need to multiply the length and the width of the rectangle and show that it is equal to the given area.
Length = 4x - 5 (from Part A)
Width = 3x (given)
Area = Length * Width
Area = (4x - 5) * (3x)
To multiply this expression correctly, we can use the distributive property:
Area = 4x * 3x - 5 * 3x
Area = 12x^2 - 15x
This is the same expression as the given area, 12x^2 - 15x. Therefore, our answer in Part A is correct.
Part A:
To find the length of the rectangular space, we divide the area by the width.
Length = Area / Width
Given:
Area = 12x^2 - 15x
Width = 3x
Substituting the given values into the formula:
Length = (12x^2 - 15x) / (3x)
Now, let's simplify the expression:
Length = (3x(4x - 5)) / (3x)
The common factor of 3x cancels out:
Length = 4x - 5
Therefore, the expression that represents the length of the rectangular space is 4x - 5.
Part B:
To prove that the expression 4x - 5 represents the length, we need to multiply it by the width and see if it gives us the area.
Length = 4x - 5
Width = 3x
Area = Length * Width
Area = (4x - 5) * (3x)
Using the distributive property:
Area = (3x)(4x) - (3x)(5)
Simplifying further:
Area = 12x^2 - 15x
This is the same expression we started with, which means our answer in Part A is correct.
Therefore, the expression 4x - 5 represents the length of the rectangular space.