What are the dimensions of the lid with an area of 392 square centimeters and a width to length ratio of 1:8?
Understand the Problem
The question is asking for the dimensions of a rectangular lid based on its area and width-to-length ratio. We have the area of 392 square centimeters and the ratio of width to length is 1:8. We need to calculate the width and length that fit these criteria.
Answer
$7 \text{ cm}$ by $56 \text{ cm}$
Answer for screen readers
The dimensions of the lid are $7 \text{ cm}$ by $56 \text{ cm}$.
Steps to Solve
- Define the variables
Let the width of the lid be $w$ and the length be $l$. According to the ratio given, we can express the length in terms of the width: $$ l = 8w $$
- Set up the area equation
We know the area of the rectangle is given by the formula: $$ \text{Area} = w \times l $$
Substituting the expression for $l$, we get: $$ 392 = w \times (8w) $$
- Rearrange the equation
This simplifies to: $$ 392 = 8w^2 $$
Dividing both sides by 8 gives: $$ w^2 = \frac{392}{8} $$ $$ w^2 = 49 $$
- Solve for width
Taking the square root of both sides: $$ w = \sqrt{49} $$ $$ w = 7 \text{ cm} $$
- Calculate the length
Now that we have the width, we can find the length using the equation for $l$: $$ l = 8w = 8 \times 7 = 56 \text{ cm} $$
The dimensions of the lid are $7 \text{ cm}$ by $56 \text{ cm}$.
More Information
The problem involves using the properties of ratios and the area of rectangles to find the dimensions based on a given area and ratio. The width to length ratio of 1:8 means that for every 1 cm of width, there are 8 cm of length.
Tips
- Mistaking the ratio (1:8) as width being larger than length, when it is the other way around.
- Forgetting to square root the area when rearranging the equation.
