Find the dimensions of a rectangle having an area of 10 square centimeters and whose width is 3 centimeters less that its length.
Let x = length, then width = x-3.
Area = Length * Width
10 = x(x-3) = x^2 - 3x
x^2 -3x -10 = 0
(x-5)(x+2) = 0
Take it from here.
Step 1: Let's assume the length of the rectangle as 'x' centimeters.
Step 2: According to the problem, the width of the rectangle is 3 centimeters less than its length. So, the width would be 'x-3' centimeters.
Step 3: The formula for the area of a rectangle is length multiplied by width. So, the area of the rectangle is (x * (x-3)).
Step 4: Given that the area of the rectangle is 10 square centimeters. So, we can set up the equation:
x * (x-3) = 10
Step 5: Now, let's solve the equation to find the value of x.
x^2 - 3x = 10
x^2 - 3x - 10 = 0
(x - 5)(x + 2) = 0
x = 5 or x = -2
Since the length can't be negative, we discard -2 as a valid solution.
Step 6: Therefore, the length of the rectangle is 5 centimeters.
Step 7: Now, substitute the value of x in the expression for the width: x - 3.
5 - 3 = 2
Step 8: Thus, the width of the rectangle is 2 centimeters.
Step 9: Therefore, the dimensions of the rectangle are length = 5 centimeters and width = 2 centimeters.
To find the dimensions of the rectangle, we can use the information given: the area and the relationship between width and length. Let's break down the steps to find the dimensions:
Step 1: Set up the equation for the area of a rectangle.
The formula for the area of a rectangle is A = length × width. We know that the area is 10 square centimeters, so we can write the equation as 10 = length × width.
Step 2: Express the width in terms of the length.
The problem states that the width is 3 centimeters less than the length. So we can write the width as length - 3.
Step 3: Substitute the expression for width into the equation for the area.
Using the expression for the width in terms of the length, we substitute it into the equation for the area:
10 = length × (length - 3).
Step 4: Solve the equation to find the length.
Now we have a quadratic equation. Simplify and solve for the length:
length² - 3length - 10 = 0.
Step 5: Factor or use the quadratic formula to solve for the length.
You can try factoring the quadratic equation or using the quadratic formula to find the values of length. After solving, you will find two possible solutions for the length.
Step 6: Find the corresponding width for each length value.
For each value of length found in the previous step, substitute it back into the expression for the width: width = length - 3.
Once you have both the length and width values, you have found the dimensions of the rectangle.