Question

The length of one side of a rectangle is 16 km longer than four times the width of the other side. Find the dimensions of the rectangle, given that the area is 128 km2.

Answer 1

To find the dimensions of a rectangle with a given area and a relational expression for its sides, you set up a quadratic equation with the width as the variable, solve for the width, and then use that to find the length.

Step-by-step explanation:

The student is asked to find the dimensions of a rectangle where the length is 16 km longer than four times the width, and the area is 128 km2. To find the dimensions, let's denote the width as w kilometers. Then the length would be 4w + 16 km. Since we know the area of a rectangle is length times width, we set up the equation (4w + 16) × w = 128.

We can then solve for w using algebraic methods:

1. Multiply out the left side: 4w2 + 16w = 128.
2. Subtract 128 from both sides to set the quadratic equation to zero: 4w2 + 16w - 128 = 0.
3. Divide by 4 to simplify: w2 + 4w - 32 = 0.
4. Factor the quadratic equation or use the quadratic formula to find that w = 4 or w = -8. Since a width can't be negative, w = 4 km.
5. Substitute w back into the length equation to get the length: 4(4) + 16 = 32 km.

Therefore, the dimensions of the rectangle are 4 km by 32 km.