The area of accent rug is 55 square feet. Find the dimensions of the rug if the length is one more than twice the width.

Answer 1

Let area = A, length = L, width = W. We are asked to find the dimensions, which are L and W.

Start with what we do know:

A = L*W

A = 55ft²

Therefore, L*W = 55ft²

We are told that length, L, is one more than twice the width, W. Therefore,

L = 1 + 2W

Now that we have written L in terms of W, we can substitute it into our equation for area and solve for W.

A = (1 + 2W)*W

Multiply each term in the parentheses by W.

A = 1*W + (2W)*W

A = W + 2W²

We know our value for area:

55ft² = W + 2W²

From here, we see that we have a quadratic equation for which we can solve for W, either by using the quadratic formula or by factoring.

To get our equation into quadratic form of Ax² + Bx + C = 0, we must subtract 55ft² from each side:

2W² + W - 55ft² = 0

Option 1: Solve for W by factoring.

Since our first term coefficient A = 2, it is easiest to multiply 2 by our constant, -55.

2 * -55 = -110. We will then find the two factors of -110 that, when added together, equal our middle term coefficient B = 1. Some factors of -110:

-110 and 1, -55 and 2, -22 and 5, -11 and 10, -10 and 11.

The only two factors that add together to make our middle coefficient B are -10 and 11 (because -10 + 11 = 1).

We can now rewrite our equation substituting -10 + 11 for B = 1.

2W² - 10W + 11W - 55 = 0

Next, we finish factoring by pulling out like terms:

(2W² - 10W) + (11W - 55) = 0

= 2W(W - 5) + 11(W - 5)

= (2W + 11) * (W - 5) = 0

Since either one or both terms must be zero in order for their product to be zero, we can set each term equal to 0 to solve for W.

2W + 11 = 0 W - 5 = 0

2W = -11 W = 5

W = -11/2 = -5.5 W = 5

Option 2: Solve for W with quadratic formula

x = [-B ± √(B² - 4AC)] ÷ 2A

= [-1 ± √(1² - 4(2)(-55))] ÷ 2(2)

= [-1 ± √(1 - -440)] ÷ 4

= (-1 ± 21) ÷ 4

= (-1 + 21) ÷ 4 = (-1 - 21) ÷ 4

= 5 = -5.5

We know that our width W cannot be a negative value, so our correct value for width is W = 5ft. (Note that we dropped units until now for the sake of more easily solving for W, but we cannot forget to bring units back in).

Now we can solve for length by substituting for W.

L = 1 + 2W

L = 1ft + 2*5ft

L = 1ft + 10ft

L = 11ft

We can check our work by testing A = L*W: A = 55ft² = 5ft * 11ft

Thus, our dimensions are L = 11ft and W = 5ft, so 11ft by 5 ft.

Answer 2

For the width let's use W

If the length is one more than twice the width then

L = 2W + 1

The area of the rug is (length) x (width) and is equal to 55

L x W = 55 Now let's replace the L with 2W+1

(2W+1) x (W) = 55 Now distribute (multiply) the W to the 2W+1

2W² + W = 55 This looks like it could be a quadratic equation. Subtract 55 from both sides

2W² + W - 55 = 0 Now we can factor the quadratic equation.

(2W + 11)(W - 5) = 0 Now we set both sets of parentheses equal to zero. You will have two equations

2W + 11 = 0 or W - 5 = 0 Now solve both equations

2W = -11

W = -11/2 or W = 5

There is no such thing as a negative width, so we reject -11/2

and our width is 5.

Width is 5

Length (from earlier) is 2W + 1 = 2(5) + 1 = 11

Answer 3

Given the equation, we can find that L (length) is equal to 2w(width)+1. The area is defined by length multiplied by width.

This would mean:

Area = L * w

By substitution, we can find that it is also equal to (2w+1) * w.

Distributing gives:

Area = 2w² + w

So 55 is equal to 2w² + w.

Rearranging the equation gives us that 2w² + w - 55 = 0

This can also be written as (2w +11)(w-5)

Since we can't have a negative distance, we find that w = 5ft.

Going back to the length equation, we find that length = 2*5+1, which is 11ft.

You can double check by multiplying the length and width to see if it is equal to 55 square feet.