The length of a rectangular carpet is 4 feet greater than twice its width. If the area is 48 square feet, what is the carpet's length and width?

Question

Evelyn Arboleda · Accepted Answer

I found #12 and 4"feet"#

Have a look:

Violet Aldrich · Answer

undefined Let's denote the width of the rectangular carpet as $ w $ feet. According to the given information, the length of the carpet is 4 feet greater than twice its width, which means the length $ L $ can be represented as $ 2w + 4 $.

The formula for the area $ A $ of a rectangle is $ A = \text{length} \times \text{width} $. Given that the area is 48 square feet, we can write the equation:

$$ A = L \times w = 48 $$

Substituting the expressions for $ L $ and $ w $ into the equation:

$$ (2w + 4) \times w = 48 $$

Expanding and rearranging terms:

$$ 2w^2 + 4w - 48 = 0 $$

Now, we have a quadratic equation. To solve for $ w $, we can use the quadratic formula:

$$ w = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} $$

In this equation, $ a = 2 $, $ b = 4 $, and $ c = -48 $. Plugging these values into the formula:

$$ w = \frac{{-4 \pm \sqrt{{4^2 - 4 \times 2 \times (-48)}}}}{{2 \times 2}} $$

$$ w = \frac{{-4 \pm \sqrt{{16 + 384}}}}{{4}} $$

$$ w = \frac{{-4 \pm \sqrt{{400}}}}{{4}} $$

$$ w = \frac{{-4 \pm 20}}{{4}} $$

This gives us two possible solutions for $ w $:

$$ w_1 = \frac{{-4 + 20}}{{4}} = \frac{{16}}{{4}} = 4 $$

$$ w_2 = \frac{{-4 - 20}}{{4}} = \frac{{-24}}{{4}} = -6 $$

Since the width cannot be negative, we discard the negative solution. Therefore, the width of the carpet is 4 feet.

Now, we can find the length using the expression we derived earlier:

$$ L = 2w + 4 $$

$$ L = 2(4) + 4 $$

$$ L = 8 + 4 $$

$$ L = 12 $$

So, the length of the carpet is 12 feet and the width is 4 feet.

Violet Aldrich · Answer

undefined Let $ w $ be the width of the rectangular carpet.

Given that the length of the carpet is 4 feet greater than twice its width, the length $ l $ can be expressed as $ 2w + 4 $.

The formula for the area $ A $ of a rectangle is $ A = \text{length} \times \text{width} $.

Given that the area is 48 square feet, we have:

$$ A = l \times w = 48 $$

Substituting the expressions for $ l $ and $ w $, we get:

$$ (2w + 4) \times w = 48 $$

Expanding and rearranging terms:

$$ 2w^2 + 4w = 48 $$

$$ 2w^2 + 4w - 48 = 0 $$

Dividing both sides of the equation by 2:

$$ w^2 + 2w - 24 = 0 $$

This is a quadratic equation that can be factored as:

$$ (w + 6)(w - 4) = 0 $$

Setting each factor equal to zero and solving for $ w $, we find:

$$ w = -6 \quad \text{or} \quad w = 4 $$

Since the width cannot be negative, we take $ w = 4 $.

Substituting $ w = 4 $ into the expression for the length:

$$ l = 2w + 4 = 2(4) + 4 = 8 + 4 = 12 $$

Therefore, the width of the carpet is 4 feet and the length is 12 feet.