Question
A rectangle has an area of 40 square units. The length is 6 units greater than the width.
What are the dimensions of the rectangle?
8 by 5
10 by 4
11 by 9
13 by 7
What are the dimensions of the rectangle?
8 by 5
10 by 4
11 by 9
13 by 7
Answers
GPT-4o mini
Let the width of the rectangle be \( w \). According to the problem, the length \( l \) is 6 units greater than the width, so we can express the length as:
\[
l = w + 6
\]
The area of a rectangle is given by the formula:
\[
\text{Area} = l \times w
\]
We know the area is 40 square units, so we can set up the equation:
\[
(w + 6) \cdot w = 40
\]
Expanding this gives:
\[
w^2 + 6w = 40
\]
To rearrange this into standard quadratic form, we move all terms to one side:
\[
w^2 + 6w - 40 = 0
\]
Next, we can factor this quadratic equation. We need to find two numbers that multiply to \(-40\) and add up to \(6\). These numbers are \(10\) and \(-4\). Thus, we can factor the equation as follows:
\[
(w + 10)(w - 4) = 0
\]
Setting each factor to zero gives us:
\[
w + 10 = 0 \quad \Rightarrow \quad w = -10 \quad (\text{not a valid width})
\]
\[
w - 4 = 0 \quad \Rightarrow \quad w = 4
\]
Now that we have the width, we can find the length:
\[
l = w + 6 = 4 + 6 = 10
\]
Thus, the dimensions of the rectangle are:
\[
\text{Width} = 4 \text{ units}, \quad \text{Length} = 10 \text{ units}
\]
The correct dimensions of the rectangle are **10 by 4**.
\[
l = w + 6
\]
The area of a rectangle is given by the formula:
\[
\text{Area} = l \times w
\]
We know the area is 40 square units, so we can set up the equation:
\[
(w + 6) \cdot w = 40
\]
Expanding this gives:
\[
w^2 + 6w = 40
\]
To rearrange this into standard quadratic form, we move all terms to one side:
\[
w^2 + 6w - 40 = 0
\]
Next, we can factor this quadratic equation. We need to find two numbers that multiply to \(-40\) and add up to \(6\). These numbers are \(10\) and \(-4\). Thus, we can factor the equation as follows:
\[
(w + 10)(w - 4) = 0
\]
Setting each factor to zero gives us:
\[
w + 10 = 0 \quad \Rightarrow \quad w = -10 \quad (\text{not a valid width})
\]
\[
w - 4 = 0 \quad \Rightarrow \quad w = 4
\]
Now that we have the width, we can find the length:
\[
l = w + 6 = 4 + 6 = 10
\]
Thus, the dimensions of the rectangle are:
\[
\text{Width} = 4 \text{ units}, \quad \text{Length} = 10 \text{ units}
\]
The correct dimensions of the rectangle are **10 by 4**.