The sum of three consecutive integers is 51. What are the numbers? A rectangle's length is 4 more than 6 times the width. The perimeter is 64 inches. Find the dimensions.
Understand the Problem
The question is asking to find three consecutive integers that sum to 51 and the dimensions of a rectangle given specific conditions about its length and width, which relate to its perimeter of 64 inches.
Answer
The length is $28$ inches and the width is $4$ inches.
Answer for screen readers
The dimensions of the rectangle are:
- Length: 28 inches
- Width: 4 inches
Steps to Solve
-
Identify Variables for Rectangle Dimensions
Let ( x ) represent the width of the rectangle. Then, the length can be represented as ( 6x + 4 ). -
Set Up the Perimeter Equation
The formula for the perimeter ( P ) of a rectangle is given by:
$$ P = 2 \times (\text{length} + \text{width}) $$
Substituting the known values, we have:
$$ 2 \times ((6x + 4) + x) = 64 $$ -
Simplify and Solve the Equation
Distributing the 2:
$$ 2 \times (7x + 4) = 64 $$
This simplifies to:
$$ 14x + 8 = 64 $$
Now, subtract 8 from both sides:
$$ 14x = 56 $$ -
Divide to Find Width
Divide each side by 14 to solve for ( x ):
$$ x = \frac{56}{14} $$
Thus,
$$ x = 4 $$ -
Calculate Length
Now substitute ( x ) back to find the length:
$$ \text{length} = 6x + 4 = 6(4) + 4 = 24 + 4 = 28 $$
The dimensions of the rectangle are:
- Length: 28 inches
- Width: 4 inches
More Information
The perimeter of a rectangle is directly related to its length and width. Knowing the relationship between these dimensions can help in various mathematical problems involving geometry.
Tips
- Incorrectly applying the perimeter formula: Ensure the correct double multiplication of both dimensions.
- Misinterpreting relationships: Make sure to clearly understand how width and length relate before forming conditions.
AI-generated content may contain errors. Please verify critical information
