a rectangle with an area of 45 square units has a length that is one less then half the width which expression represents the length of the rectangle?
What are your choices?
Let's break down the given information step by step:
Step 1: Let's represent the width of the rectangle as "w" (in units).
Step 2: According to the given information, the length of the rectangle is one less than half the width. This can be written as (1/2w - 1) units.
Step 3: Since we know the area of the rectangle is 45 square units, we can equate it to the product of the length and width of the rectangle.
Area = Length × Width
45 = (1/2w - 1) × w
Step 4: Simplifying the equation:
45 = (1/2w^2 - w)
Step 5: Multiplying both sides of the equation by 2 to eliminate the fraction:
90 = w^2 - 2w
Step 6: Rearranging the equation by subtracting 90 from both sides and setting it equal to zero:
w^2 - 2w - 90 = 0
Step 7: Factoring the quadratic equation:
(w - 10)(w + 9) = 0
Step 8: Setting each factor equal to zero and solving for "w":
w - 10 = 0 or w + 9 = 0
w = 10 or w = -9
Since the width of a rectangle cannot be negative, we discard w = -9.
Step 9: So, the width of the rectangle is 10 units.
Step 10: Finally, we can substitute the value of "w" into the expression we found in Step 2 to determine the length of the rectangle:
Length = (1/2w - 1)
Length = (1/2 × 10 - 1)
Length = (5 - 1)
Length = 4 units
Therefore, the expression representing the length of the rectangle is 4 units.
To find the expression that represents the length of the rectangle, we need to use the information given:
1. The area of the rectangle is 45 square units.
2. The length is one less than half the width.
Let's start by assigning variables. Let L represent the length and W represent the width of the rectangle.
From the second statement, we know that the length (L) is one less than half the width (W/2). So, we can write the equation as:
L = W/2 - 1
Now, we can use the formula for the area of a rectangle to create another equation:
Area = Length × Width
Substituting the values we know, we can rewrite this equation as:
45 = L × W
Now, we can substitute the expression for L we found earlier:
45 = (W/2 - 1) × W
Simplifying this equation will lead us to find the expression for the length of the rectangle.